UBC Theses and Dissertations

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UBC Theses and Dissertations

Plane coordinates Parker, Sidney Thomas 1934

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'OH 'D3V 'OH iVD  U .B .CJLJB RA R Y BLAKE GOOEDIK&IES by S i d n e y Thomas P a r k e r  CAT. m.LElAlU^J&i&M ACC. NO!  A Thesis submitted f o r t h e Degree o f MASTER OF IRIS i n t h e Department  THE IMIYERSITX OF BRITISH COLUMBIA. October, 1934  l?$JdL  X»  TABLE OF CONTENTS CHAPTER I X«  Intr0duotion.  2 .  Fundamental D e f i n i t i o n s .  3 .  CoBrdinates o f Planes.  4.  P a r a l l e l Planes. D i r e c t i o n Cosines o f a L i n e .  6«  Angle Between Two D i r e c t e d L i n e s . P o l a r Coordinates of a Plane.  8«  R o t a t i o n o f Axes.  ?.  Standard Form of the E q u a t i o n of a P o i n t .  XO & E q u a t i o n s o f P o i n t s ( C o n t i n u e d ) . 11. 1 2  »  D i s t a n c e Between Two P o i n t s . D i v i s i o n o f a Segment i n a G i v e n R a t i o .  1 3 .  P l a n e Through Three P o i n t s .  14.  The E x p r e s s i o n ^ / ( x ~ x )*" + ' ( y ~ J Y *  1 5 «  D i s t a n c e Between P a r a l l e l " P l a n e s .  l 6 e  D i s t a n c e t o a P o i n t from a P l a n e .  If] e  A n g l e s Between L i n e and P l a n e j Plane and PXane *  18 ©  Two-Point E q u a t i o n s  19*  Equations of a Line (Continued).  20.  Two-Plane Form o f the E q u a t i o n s o f a L i n e  210  D i r e c t i o n Cosines o f a Line .  x  f  v  t  o f a Line»  II.  22.  Plane P a r a l l e l to a L i n e .  23.  P e n c i l of Planes.  24.  Three-Plane  25..  T r a n s l a t i o n o f Axes.  26 ,  The Degree o f a n E q u a t i o n i s Unchanged b y  0  j  Equation of a Point.  R o t a t i o n and T r a n s l a t i o n T r a n s f o r m a t i o n s ,  CHAPTER I I -  The g e n e r a l Second Degree E q u a t i o n  1*  E q u a t i o n o f the Tangent P o i n t .  2.  C o n d i t i o n t h a t a P o i n t L i e s on the S u r f a c e  3»  Locus o f M i d d l e P o i n t s of a System of P a r a l l e l Chords.  4.  The P r i n c i p a l P l a n e *  5 . ' The Roots o f ( 1 7 ) . 6.  E l i m i n a t i o n o f the y s , a z , z Terms.  7.  R e d u c t i o n when d  8.  R e d u c t i o n when d = 0.  9.  Center o f the C o n i c o i d o  10»  Polar Plane.  11.  Rectilinear  12*  Invariants©  ^ 0 .  Generators  XXX •  CHAPTER I I I C l a s s i f i c a t i o n of Surfaces 1.  Review o f P r e v i o u s f o r i .  2.  The Sphere.  3.  The E l l i p s o i d .  4.  The H y p e r b o l o i d o f One Sheet.  J?.  The H y p e r b o l o i d o f Two S h e e t s .  6.  The P a r a b o l o i d .  7*  I n v a r i a n t s f o r the V a r i o u s E q u a t i o n s .  CHAPTER I T Reduction o f the General Equation 1.  G e n e r a l Statement.  2.  R e d u c t i o n o f the P o i n t - C o n d i t i o n E q u a t i o n .  3.  To F i n d t h e E q u a t i o n o f t h e Center of a Conic o i d .  4.  The D i s c r i m i n a t i n g C u b i c .  5.  Discussion f o r  6.  D i s c u s s i o n f o r J © *= 0 .  7 •  Summary.  0 .  BIBLIOGRAPHY  PLAHE  OOOBDIHAIES  Chapter I 1.  Introduction: The p r i m a r y purpose o f t h i s t h e s i s i s t o d e v e l o p t h e  ordinary r e l a t i o n s o f s o l i d a n a l y t i c geometry "by the use o f plane-coordinates.  The significance o f v a r i o u s equations  of the C a r t e s i a n system w i t h r e f e r e n c e t o t h i s new system w i l l also be d i s c u s s e d . As f a r as possible, the treatment p a r a l l e l s the t r e a t ment of line-coBr&lnates, as contained i n the theses submitted, by ValgarcLsson of Manitoba and Heaslip and James o f B r i t i s h Columbia f o r the degree o f M a s t e r of A r t s . 2*  Fundamental D e f i n i t i o n s t We use the rectangular reference system, i . e . three  mutually perpendicular planes intersecting i n three mutually perpendicular straight l i n e s x'ox, Y/0Y, z'oz, which are c a l l e d the X, Y, Z axes,respectively.  The X a x i s i s formed  by the intersection of the Z X and XY planes; the Y axis by the intersection of the XY and YZ planes; and the Z axis by the intersection of the YZ and ZX planes.  The point 0 ,  common to a l l three planes, i s c a l l e d the o r i g i n .  The cus-  tomary conventions with regard to sign are observed.  For  example, the directions x' OX, Y'OY, Z'OZ are considered  p o s i t i v e , and the directions XOX  , YOY  , 20Z  are considered  negative* The ooBrdinates of a plane are defined to he the r e ciprocals of i t s intercepts on the coordinate axes. the plane ABO  Thus  i n figure ( l ) has coordinates (a, b, c ) ,  since  In Cartesian coSrdinates the point (a, b, e) i s such that its. directed perpendicular distances from the YZ, XY planes are a, b, c, r e s p e c t i v e l y . The ax * by * cz - 1 = 0  plane  ZX,  has intercepts i , - i . i on the coordinate axes* a D c 3»  CoOrdinates of P l a n e s ; Any plane whose intercepts on the coordinate axes  are a l l f i n i t e and different from zero i s seen to he represented uniquely by (a, b, c ) . The following i s a summary of some special (i)  cases:  Coordinate P l a n e s .  The XT plane i s denoted by  (a, b,<>0), where a and b are both f i n i t e . ( i i ) A plane through a coordinate axis and cutting the other axes obliquely. Such a plane through the X axis has t h e co8rdinates (a,oO  where a i s f i n i t e .  ( i i i ) The coordinates of a plane p a r a l l e l to that given i n ( i i ) are (o, b, c ) , where b and e a r e f i n i t e . (iv) The ooOrdinates of a plane p a r a l l e l to t h a t given i n ( i ) are (o, o, c ) , where o i s f i n i t e . (v)  The "plane at i n f i n i t y " has the coordinates  (o, o, o ) . (vi) A plane through the o r i g i n and oblique to a l l three axes has the coOrdinates (oO oo>oo)• t  It i s to be noted that the coordinates i n ( i i ) and (vi) do not represent one plane uniquely, and that the planes i n ( i ) and ( i i ) do not possess unique coordinates. 4.  P a r a l l e l Planes:Theorem:  The necessary and s u f f i c i e n t conditions  for the p a r a l l e l i s m (x , x  7  X  of two planes (x,, y, , z, ) and  » z ) are x  (2)  ±1 = I I  - l l  .  F i g . (2) The conditions are necessary.  For suppose that the  planes A, B, C, , i . e . (x , y , z, ), and A^B^Cj, ,: i . e . (  U , x  y^, z ) , are p a r a l l e l . Then they cut the coordinate (1) x  planes i n p a r a l l e l l i n e s , parallel.  that i s , AjB, and A B 2  t  are  Hence OAV OA/  (l)  (  „  0B . OB/ a  Wilson "Solid Geometry and Conic Sections™, p. 12.  In the same way  II  *  B  y' yT  Z  '  '  Therefore X/ x7  OS Z ; "zl *  The conditions are also s u f f i c i e n t . tions (2) hold.  Suppose r e l a -  Then A,B, i s p a r a l l e l to A ^ , and B, G,  i s p a r a l l e l to B^O,.• Hence plane A,B,C, i s p a r a l l e l to (1) plane A B C . t  l  3L  This theorem i s equivalent to the statement  that the  planes (a, b, c) and (ka, kb, ke) are p a r a l l e l * In the Cartesian system, two points whose coordinates s a t i s f y equations (2) are eollinear with the o r i g i n , and conversely.  I f two planes: A,ac *• B, y * 0, z - 1 « 0 A^x * B ^ * C^z - 1 = 0  are p a r a l l e l , then A, X~  -  B, B t  and conversely.  (1) Wilson, l o c . o i t . , p. 13.  = 2± U  x  '  f  j>»  D i r e c t i o n Gosines of a Mne:-'"  7  Let A be any directed l i n e i n space, and l e t  be  the l i n e through the o r i g i n with the same d i r e c t i o n as Let c< fi , y t  be the angles between the X, Y, Z axes, r e s -  pectively, and By d e f i n i t i o n these are the angles which with the axes.  makes  Ehey are c a l l e d the "direction angles" of  the l i n e st- , and t h e i r cosines are c a l l e d i t s " d i r e c t i o n cosines".  The d i r e c t i o n cosines w i l l be denoted by A A , ^ P/  respectively. z  (1)  As i n Snyder and Sisam "Analytic Geometry of Space".  (2)  See Snyder and Sisam, p. 3 .  I t i s e a s i l y proved that the r e l a t i o n  holds. 6*  CD  Angle between Two Directed Suppose that  d i r e c t i o n cosines  and  A,  Lines:' are two directed l i n e s with  , AJ, , and  A »y<  vs  i<4, respectively.  In s o l i d geometry the angle between two directed l i n e s i s defined to be the angle between the two s i m i l a r l y directed l i n e s through the o r i g i n .  ^' z  /  T  0  y  s F i g . (4)  (1)  Snyder and Sisam, p. 6.  (2)  As i n Snyder and Sisam*  8  In figure (4), ^ respectively.  'and  -A  are p a r a l l e l to  and  I f OP i s any segment taken along the  A.  positive  d i r e c t i o n of ^ ' , PQ i s perpendicular to £,', and PR i s perpendicular to the plane XOY SU are drawn to OQ,, 0S  The angle between  figure.  Bow cos e =  oa ^  ~ cos a =  —  A  Perpendiculars RT,  01, respectively,  S  diagram.  rtrtei  at R.  as shown i n the  and A. i s the angle  ou * m «  RS,  0  i n the  * TO, . Q P — »  therefore OU OS  UT SR  —  *  *  TO, PR g'oP  ;  and hence  O) 7.  cos © =  /M*.* ft'/*-*- +  •  Polar CoBrdinates of a Plane:Let the polar coordinates of a plane (x, y, z) be ©< % p  » Y)*  where ^  i s the length of the perpendicular  from the o r i g i n to the plane, and  << , ^  , Y  are the d i r e c -  t i o n angles of t h i s perpendicular. ABC  i s any plane (x, y, z) and OP i s the perpendicular  from the o r i g i n to the plane.  GP i s produced to meet AB at  Q, and 0 and Q, are joined. The plane Q.QC i s perpendicular to each of the planes XOY  and ABC.  Hence i t i s perpendicular to AB, t h e i r l i n e  of i n t e r s e c t i o n . to AB.  Therefore GQ. and OQ, are both perpendicular  Since the t r i a n g l e s GQB  and AOB are s i m i l a r , i t  Fig*  O)  follows that OB  AB  !  so that 1 A  OA.OB  rt  1  z *Y  V  x  l  l y*  In the t r i a n g l e Q,0C GO, ^ = 0G* * 00/ ;  therefore GO,  _ fL  HI  "1/  /x-»y + y"  1  z (x J  a  .  a  + y )" 1  Again, the t r i a n g l e s OPQ,, GOQ, are s i m i l a r ; therefore OP  _  0 0  from which, we obtain  „/  X  2.  V x  + y  *  Since OP i s perpendicular to the plane cos *  .  z  ABC  x  OP^ y x  + y  +  similarly  cos ^  7  a /x " + y 2  + z-  i  x  z  cos a- /x*  * y" * z x  Therefore  y "' x  COS °< a /  (4)  GOB0B A  * fi  Z  x  X  + z  x'  *  y  B  Vx  z  + y * + z"" 1  z  cos J = Vx " 2  (l)  + 7 y  + s  * yX  X  * Z  The perpendicular from the o r i g i n to a plane i s . always considered positive*  1 1 c  She inverse transformations are  oos  r  (<>)  7  cos ••/* cos  f. 8«  Rotation of Axes;-  12  F i g . (6) l e t the o r i g i n a l reference system he rotated about the o r i g i n to a new p o s i t i o n so that the new X a x i s has d i r e c t i o n cosines A, U,, t/  V  t  , the new T axis has d i r e c t i o n  12*  eosines A,. ,ya > ^ » and the new Z axis has d i r e c t i o n cosines v  ^3 »/S » ^ »  a  1  1  wi  "ttL respect to the old axes.  We  shall  denote the new axes by primed l e t t e r s . Suppose the X' axis cuts any plane (x, y, z) at as i n figure ( 6 ) . Denote the angle POA  x  by 0 .  tions (4), the d i r e c t i o n cosines of OP are cos o< ._ = •  X  . ...  * z"  •i/.x ^. * y  1  1  y  cos V x cos Y  « •.  • z  * y  , .. ,..,Z : .  v  x ^* y  1  From equation ( 3 ) we obtain obtain. *  A, x yx  +^.y + y  But, from figure (6), i t follows that OP  A  COS. U  st  •QJ"*/ SB  x -/x "*y ^z 1  2  i  By equating these two values f o r cos 0 , we get  similarly (6)  , y  and  = A  * / \ / * <2 , +  x  ,  By equa-  1 3 .  The inverse transformations are  17)  y  = yA« *' •*  +/  u  )  x'  }  Y/e can express r e s u l t s (6) and (7) i n tabulated form as follows; 2/' J 2  ( 8 )  These r e l a t i o n s are exactly the same as those obtained f o r 0artesian 9.  coOrdinates*  Standard fform of the Equation of a Point;The  standard equation of a point w i l l be that r e l a -  t i o n which involves the directed perpendioular the three coordinate planes to the point.  distances from  Let P be the  point whose directed perpendicular distances from the YZ, ZX, and XY planes are r , s, and t respectively. In figure l 7 )  s  OR « r , EQ, = s, QJ? = t .  axes so that the x' axis passea through P.  Rotate the  Then  x' coOrdinate of a l l planes which pass through P.  i s the Therefore  F i g . (7) But, from (6), we have  where Ax  «•  0  R  =  s  19)  4/1r  * s  * t  y r  * s  + t  Therefore rx * sy » t z r  + s  r  + s  +  and hence U P )  rx +• sy * t z - 1 =  0 ,  15.  We must now  show that a l l planes whose coordinates  s a t i s f y (10) pass through the given point. Let  (a, b, c) "be a plane whioh does not pass through  P, hut whose coordinates s a t i s f y (10)• (11)  Then  r a * sb .* t c - 1 * 0.  From s e c t i o n 4, the coordinates of a plane through P and p a r a l l e l to (a, b, c) are (lea, kb, ko).  Since these  coordi-  nates must s a t i s f y ClO), i t follows that (12)  k ( r a * sb * t c )  1 « 0.  The equations ( l l ) and (12) are both true only i f k = 1, i n which case the plane (ka, kb, kc) i s coincident with the plane (a, b, c ) .  Therefore the plane (a, b, c) must pass  through the point. 10.  Equations of Points (Continued):The standard equation of a point P i s given by (10)•  The d i r e c t i o n cosines of OP are given i n ( 9 ) . I f we denote the length of OP by f ,. equation (10) may be written (13)  Ax  +  yu.y  --L  a  0.  We s h a l l c a l l (13) the "directed" equation of the point. I f (^ , e<  t /  s  , y) are the polar coordinates of a  plane, whose intercept coordinates are (x, y, z ) , passing through the point rx * sy * t z - 1 » 0 then  16.  JL5 + y  v  .+  * z" f/x" * y"" +  .  §y  /x"- * y"~ * z "  tz V ? •*• y*" * z'  1  Therefore (14)  ^  = r oosc<  * a cos/?  We s h a l l e a l l (14) the "polar  Ir  + t cos ¥  equation of the point.  The equation of t h e o r i g i n i s ox * oy + os - 1 » 0, The equation of the "point at i n f i n i t y " i s Ax  •-t-yu-y -t- V-z-  The equation of a point on the X axis i s : TX  - 1 s o,  and the equation of a point i n t h e XT plane i s  rx t sy • 1  0,  B  In Cartesian coordinates the plane ox * oy * oz - 1 = G ClV i s known as the "plane at i n f i n i t y " . ' The plane  Ax + yu.  y  +1/2.  tt  0  passes through the o r i g i n and ^ ^u, 4/are the d i r e c t i o n 9  t  cosines of the normal to the plane. The plane rx i s p a r a l l e l to the YZ plane*  (l)  Snyder and Sisam, p. 34*  -1=0  17.  11*  Distance "between Two Let two points P  Points:and P  ;  r s * s y * t,z f  t  x  be denoted by the equations -1=0  and  Fig. Let the lengths of P, \ ,  OP, , OP^ be d»  pectively, and l e t angle P OP^be Q • ;  and hence, from ( 3 ) d"" * ( r  v #  (8)  9  f  res-  K  We have  and ( ? ) , i t follows that  * a,* * t,"*" )+(r ~ * s j " * t ^  )-2(r,  r  fc  + s s (  z  so that (15)  a =y ( r  t  - r ) % (a,. ^  S  /  ) % ( t ^ - t r.  +t,t ), a  1 8 .  12•  D i v i s i o n of a Segment i n a Given Ratio:Let the segment be defined by the two points given  i n section 11, and l e t the given r a t i o of d i v i s i o n be h : l u Suppose that P, the d i v i s i o n point, has the equation rx * sy + t z - 1 = 0«  Hence  On solving f o r r we obtain  1 9 .  (16)  _ 3cs, * hs. - • h + 1c let, + ht t =  Similarly  3  13.  Plane through. Three Points Let the equations of the three d i s t i n c t points P , ;  P, 2  ?  3  be r, x * s,y * t,z - 1 « r x * sj z  z  * t^z - 1  0 ,  =  0 ,  r ^ i • ajy.* t z f 1 • 0 , 3  respectively.  I f these equations are solved f o r x, y, z, we  obtain the coordinates of a plane passing "through the three points. F i n i t e solutions are possible provided that  r,  r  3  s.  t,  s,  t  s,  t,  %  I f .A ~ 0, then each element of any one row i s a l i n e a r combination of the corresponding elements of the other two rows. Suppose that r  3  «= k,r, +  t  3  = x,t, + k . t , .  Let us consider the point P whose equation i s •rx * ay + t z > 1 » 0, where  2 0 .  +  ~l  r =  3c r x  x  +  +  From (16) we see that the point P i s c o l l i n e a r with P, and P »  Therefore any plane through P, and P  r  P. •  • • •  ••  • ••  z  must pass through  •  From (9) we see that the vectors OP and 0P and the same straight l i n e . Pj are c o l l i n e a r . P  3  3  are one  Therefore the o r i g i n , P, and  Hence, any plane passing through P and  must pass through the o r i g i n , and one, at least, of x, y,  z must be i n f i n i t e . In Cartesian coordinates three planes determine a point except when one plane i s p a r a l l e l to the l i n e of i n t e r section of the other two. is 14.  The condition f o r this exception  /\ .** 0. The Expression /(x, - x, ) 1  x  + ( y - y, )*" + ( z - z t  t  (  Let © be the angle between the perpendiculars from the o r i g i n to two planes (x , y, , z, ) and ( x , y^ , z ) and f  a  2  l e t d be the distance between the feet of these perpendiculars.  Then d " «* • f>* + 1  where -f and t  A  - 2f, /  \  QOS  Q ,  are the lengths of the polar normals as  g i v e n i n (4) and c o s Q cos 0  ( 1 7 )  i s determined  by the r e l a t i o n  »• .•j/ x, + y, + 2"• / x j " + y j " + z^ v  v  that i s .  x, or ,x  i  -j  + y,  +  z, - x .  +  (xx  y- +  0*.  x," + y," -J- z, + x ^ + yj- •  +  + z z )  z,  + y  +zj  t  -2(x, x ^ + y, y^ + z, z j  v  ————————————  y,  +  + y y  _  —  + y," + z,- ) ( x V * y j - +  —  —  —  —  —  —  ^  )  1  w h i c h r e d u c e s to (IS) 13.  d. -  - x ) % (y  D i s t a n c e between P a r a l l e l  - y, ) + (z„ -z, f . X  x  Planes:-  The d i s t a n c e between the p a r a l l e l p l a n e s ( x , y, z ) and  (lex, Icy, k z ) i s e q u a l to t h e d i s t a n c e between the f e e t  of t h e i r p o l a r n o r m a l s .  From e q u a t i o n ( 1 8 ) we o b t a i n 3c - X  ( 1 9 )  x 16.  + y  + z  D i s t a n c e t o a P o i n t from a P l a n e : L e t the p o i n t be d e f i n e d by t h e e q u a t i o n rx + sy + t z -  1  =  0  and the p l a n e by the c o o r d i n a t e s ( x , y , z , ) . Through the (  (  p o i n t draw a p l a n e w i t h c o o r d i n a t e s , s a y , (toe, , Icy , k.z ) , ;  p a r a l l e l to t h e g i v e n p l a n e .  Then the d i s t a n c e t o t h e  (  22  p o i n t from the p l a n e i s e q u a l t o t h e d i s t a n c e between these two p l a n e s .  S i n c e t h e new p l a n e p a s s e s through t h e g i v e n  p o i n t , we have k ( r x , + sy, + t z , ) - l =  0 ;  that i s I  k -  r x , + sy, + t z On s u b s t i t u t i n g t h i s v a l u e f o r k i n (19) , we o b t a i n r x , + sy, + t z , - 1  d  ( 2 0 )  Theorem:  Two p o i n t s P, , P  r, x + s, y + t, z i \ x + s,y + t z t  -  1  1  whose e q u a t i o n s a r e  t  =  0 ,  =  0 ,  r e s p e c t i v e l y , a r e on the same s i d e o r on o p p o s i t e s i d e s o f the p l a n e (x, , y , z, ) , a c c o r d i n g as i t s c o o r d i n a t e s g i v e the f i r s t members o f t h e e q u a t i o n s o f t h e p o i n t s l i k e o r unlike signs.  F o r , l e t the p o i n t of i n t e r s e c t i o n o f the  l i n e P, P^ and t h e p l a n e be P whose e q u a t i o n i s rx + sy + t z -  1  =  0 ,  where r «= m, r,  * m^r^ ^  s = m.s, i m a , t  t = m,t, + m t t  4  x  }  and m  (  + m^ = 1 ( S e c t i o n 1 2 )  t  23.  Therefore (m, r, + m r )x + (m s + m a ) y + (m t + m t ) s  -  1  =  0 ?  that i s , m ( r x + s, y, + t, z - 1 ) + m^(r x, + s^y, + t (  u  I f r, x, + s, y, + t, z  (  r ^x, + s y + t z v  m,, m  t  (  l  v  have the same s i g n , and t h e p o i n t P  x  and P . z  /  -l)= 0 .  - 1 and r x , + s^y, + t ^ z , - 1 have u n l i k e  (  s i g n s , t h e n m, and m l i e s between P  i Z <  I f r, x,+ s, y, + t, z, - 1 and  - 1 have t h e same s i g n , t h e n t h e numbers  have o p p o s i t e s i g n s , hence the p o i n t P i s n o t between  P, and P . x  A p o i n t whose e q u a t i o n i s rx + sy + t z -  1  »  0  w i l l be c o n s i d e r e d to be on the p o s i t i v e or n e g a t i v e s i d e of the p l a n e (x,, y , z ) a c c o r d i n g as the e x p r e s s i o n t  r x , + sy, + t z ,  -  1  i s p o s i t i v e or negative r e s p e c t i v e l y . From (20) and the theorem J u s t proved we can s a y t h a t the d i s t a n c e to a p o i n t from a p l a n e i s p o s i t i v e o r n e g a t i v e a c c o r d i n g a s the p o i n t and t h e o r i g i n a r e on t h e same s i d e o r on o p p o s i t e s i d e s o f t h e p l a n e . 17.  Angles between L i n e and P l a n e ; P l a n e and P l a n e : The angle between a l i n e and a p l a n e i s the comple-  ment o f t h e angle between t h e l i n e and the p o l a r n o r m a l to the p l a n e .  I f A,yu, 1/  w h i c h makes an angle  a r e the d i r e c t i o n c o s i n e s o f a l i n e Q w i t h t h e p l a n e ( x , y, z ) , then from  24.  (3)  aM  (4) we  (21)  get s i n 6)=  /U  •yUy  +  V z ^  / x ^ + y" + z  x  The angle between two p l a n e s i s e q u a l 10 t h e angle between t h e i r p o l a r normals and i s g i v e n by ( 1 7 ) . 18.  Two-Point E q u a t i o n s of a L i n e : Two  d i s t i n c t p o i n t s w i l l determine  a straight  s i n c e the t o t a l i t y of p l a n e s , w h i c h pass t h r o u g h the points simultaneously, define a l i n e .  Hence the  line  two  simultaneous  equations  (22)  r,x +  3,7  *  r x + s y x  2  t, s - 1  + t z z  - 1  «  0 ,  =  0 ,  g i v e t h e e q u a t i o n s " of t h e l i n e . , We s h a l l r e f e r to (22) the "Two-Point" e q u a t i o n s 19.  Equations of Lines  as  of a l i n e „ (Continued);-  The most g e n e r a l e q u a t i o n s o f a l i n e are g i v e n by (1)  (22)»  The f o l l o w i n g i s a summary o f s p e c i a l e s s e s : (i)  A coordinate a x i s .  The X a x i s has  the  equations r x - 1 = 0; ox + oy + oz -  (1)  1  =  I t i s u n d e r s t o o d t h a t r , s, and f o l l o w i n g work.  0 .  t a r e not z e r o i n the  25»  (ii)  A l i n e p a r a l l e l to ( i ) and p a s s i n g t h r o u g h  the Y a x i s has the e q u a t i o n s x = sy-1  «=  0 ,  0 .  ( i i i ) A l i n e p a r a l l e l t o ( i ) and p u t t i n g the YZ plane has t h e e q u a t i o n s x =  0 ,  s y + t z - 1 «= 0 .  (iv)  A l i n e " t h r o u g h the o r i g i n and l y i n g i n a  coordinate plane.  Such a l i n e i n the XY p l a n e has t h e equa-  tions rx+sy-1 = 0 , ox+oy+oz-1 <= 0 .  (v)  A l i n e " through t h e o r i g i n o b l i q u e t o a l l  t h r e e axes has the e q u a t i o n s rx+sy+tz-1 = 0 , ox+oy+oz-1 = 0 .  ( v i ) A l i n e through the X and Y axes b u t not t h r o u g h the o r i g i n has the e q u a t i o n s rx-1  = 0  sy-1  =  8  0 .  ( v i i ) A l i n e t h r o u g h t h e X a x i s and p a r a l l e l t o t h e YZ p l a n e has the e q u a t i o n s rx-1  =  0 ,  r x + s y + t z - 1 «=• 0 .  26  20.  Two-Plane Form o f the •Equations o f a L i n e ; Let  and  ( x , y , z )»  f  I f the l i n e passes through t h e o r i g i n  then one o r more o f the c o o r d i n a t e s of e a c h p l a n e w i l l infinite.  z)  the l i n e be d e f i n e d by t h e p l a n e s ( x , , y,,  be  I f i t does not pass t h r o u g h the o r i g i n , a l l the  members o f a t l e a s t one s e t o f c o o r d i n a t e s w i l l be  finite.  Suppose the p o i n t s r x + s y + t z -  1  =  0,  r^x + s y + t z -  1  «  0 ,  (  (  (  x  l i e on the l i n e .  a  The p o i n t i n w h i c h t h e l i n e c u t s the XY  p l a n e c a n be f o u n d by e l i m i n a t i n g z from the two e q u a t i o n s , and  the p o i n t where I t c u t s the TZ p l a n e can be f o u n d by  e l i m i n a t i n g x.  L e t t h e s e two p o i n t s be denoted by the equa-  t i ons  = 0,  r^x + s y - 1 3  (23)  s^y + t ^ z - 1 «= 0V  respectively  Then r x + 3y r,x, r x 3  0,  =  - 1  s  3  + s y , - 1 - OV 3  %  * s y 3  - 1  x  I f these e q u a t i o n s i n r , s  0.  a r e t o be c o n s i s t e n t we must  have x  y  1  x,  y,  1  x  5V  x  1  27.  whence x - x, -  y - y,  c  2r  y, - y  v  v  I n the same way, from the second of" e q u a t i o n s  (23)  we o b t a i n y - y,  z - z,  =  Therefore (24)  x - x, x, - x Equations  (  &  y, - y .  x  x  z ~ z, z, -  (24) are c a l l e d the "Two-Plane  of a s t r a i g h t l i n e . of  ^ y - y  the denominators  x must be e q u a l t o x  fr  equations  O b v i o u s l y t h e s e have no meaning i f one i s zero. i  Suppose x, - x,. I s z e r o .  and i n s t e a d o f (24) we w r i t e x - x , (  7  Then  -  J,  y, - y  =  . .  z - z,  x  I n C a r t e s i a n c o o r d i n a t e s (24) g i v e the " t w o - p o i n t " equations o f a s t r a i g h t l i n e . 21.  D i r e c t i o n Cosines o f a l i n e : I f the l i n e - i s d e f i n e d by t h e two p o i n t s whose  e q u a t i o n s a r e ( 2 2 ) , t h e d i r e c t i o n c o s i n e s are f o u n d t o be  Suppose t h e l i n e i s d e f i n e d by ( 2 4 ) . f i r s t two f r a c t i o n s .  Equate t h e  Then  (x - x,) (y, - y j - ( y - y ) ( x , - x j . (  T h i s e q u a t i o n i s r e d u c i b l e t o t h e form y  ( 2 6 )  * " >_x + y  X  / "  x,y - x^y v  y - 1 - 0,  x y -x^y  ;  v  #  w h i c h i s t h e e q u a t i o n o f a p o i n t on t h e line»  I n t h e same  way t h e equa t i o n s ( 2  7  Z  )  ^ " y,  Z  ,  y * >  ~  y  z  ~  >  K  Z  ,  -  y,  z 2  ^  z  1  =  0 ,  ,  and g  ( 2 8 )  v " « z  x  X  X, Z -X Z ,  ' ~ ^i- E ~ X z, -x z  r e p r e s e n t p o i n t s on the l i n e .  1  ~ o;  We can t h e r e f o r e s e l e c t two  of these p o i n t s and f i n d the d i r e c t i o n c o s i n e s o f t h e l i n e j o i n i n g them by means o f ( 2 5 ) • I f t h e denominator x, y^ - x^y i.e , i f 0  B  '  y, y  8 r  f  has t h e value  zero,  £9  from s e c t i o n 3 we know t h a t x z cannot a l s o be z e r o .  - ,_z  7, 2 ^ - y z  K  #  v  (  x  f  I n t h i s case we c a n use t h e two p o i n t s  whose e q u a t i o n s are ( 2 7 ) and ( 2 8 ) . 22.  Plane P a r a l l e l to a l i n e : Theorem:  ( 2 9 )  The p l a n e  (k,x, + k^^,  k y ^ , k z, + k z J  k y,  v  v  i s p a r a l l e l t o the l i n e d e t e r m i n e d by t h e p l a n e s ( x and  / 9  y , z ) (  (  (x , y , z J . t  L  I f 6 I s t h e angle between t h e l i n e and p l a n e  from  9  e q u a t i o n ( 2 1 ) we o b t a i n si:  1"  y Let  y  -5-  + z  ( 2 7 ) and ( 2 8 ) be the e q u a t i o n s o f the l i n e .  Then, from  ( 2 5 ) we have 2 , " •rr  // z  - z,  v  y  Z  «*.  FT  , -v*  /z, - ,  »7  - ^  2  y, - y*  1/ (x, z - x s , j >, z - y z , / "fx, v  v  v  7,  t  z  (30)  /  11  5L. , / z  - z.  y, z ~ y z , t  z  v  " , s  ,v / z , ~ s  x  f /x, - u  X_ z x z - x j a , / (y, z^- y^z,/ Ax Ax z z - x fc  y.  x  v  (  ~  y„  y z^- y^z #  30  y,  *v.  /  ix, Z - x z l  x  - y.  x z - x z  y z - y z  ^  U,V  Z» l  The s u b s t i t u t i o n of (30)  y z/ x  y,s„-  i n the e x p r e s s i o n f o r s i n 0  y^, gives  us sin where -n  E,  Zi  —  x,z Q  v  =  . -J  P + Q, + R s.T .  (k,x, + I c ^ x J  x z  r  6 -  (x,y, - \ y  t  ,  S - y (3c, x, + k,x )" + (x, y, + \ y ^ ) + (x, z + I c j O V v  v  T  Ax  z - x ^ z / \y, \~ v  (  y^J  \2c,z^- x z k  ;  y  y z X. f  (  The numerator reduces to z e r o and hence s i n g> « 0, and the p l a n e i s p a r a l l e l to the l i n e . C o n v e r s e l y , i f the plane  ( x , y^ , z^) i s p a r a l l e l 3  to the l i n e of i n t e r s e c t i o n of ( x , y , z,) and f  (  i t s c o o r d i n a t e s must be o f the form ( 2 9 ) •  (x , y ,  We have  z  L  z ), t  3 1 .  and. t h e r e f o r e z - z,  z - z t  +  Xj  T-  y  3  a: z - x z 'x, - x ^ X z - X  /V  y, - y„ z 2-1  y,  Z  T - ~  I s ^  -  0 ,  y *  Tliis eauation reduces to x, ( 3 1 )  y•  i z.  2-  x  3  0e  y>  (^z^ - z ) c a n n o t a l w a y s be z e r o , s i n c e we do n o t have (  to r e s t r i c t the l i n e i n t h i s manner. the  (Therefore we must have  relation I aci-  y',  ,  y^  *i  y,  x  ( 3 2 )  s a t i s f i e d under a l l c o n d i t i o n s . z j must be a l i n e a r c o m b i n a t i o n  z  ,\ z.  s  =  0  3 I f ( 3 2 ) h o l d s , then x^, y „ 3  of the c o r r e s p o n d i n g  ele-  ments of the o t h e r two rows, and hence must be of the form  (29).. In C a r t e s i a n coordinates a p o i n t ( 2 9 ) i s co-planar w i t h t h e p o i n t s (x, , y, , z ), (  23.  y , zJ t  and t h e o r i g i n .  P e n c i l of Planes:Suppose the p l a n e  ( 2 9 ) passes through the l i n e o f  32*  i n t e r s e c t i o n o f t h e p l a n e s ( x , y, , z ) and ( x ^ y^, z ) • f  (  u  Then i t p a s s e s t h r o u g h a l l p o i n t s on the l i n e and i t s c o o r d i n a t e s must s a t i s f y t h e e q u a t i o n o f a n y p o i n t on t h e line«, L e t a p o i n t on the l i n e be d e f i n e d by t h e e q u a t i o n rx + s y + t z - 1 = 0. We 'must have • r x , + sy, + t s , - 1 = 0,  (33)  r x ^ * sy _+ t z - 1 •» 0, 1  t  r ( l c , x , * x ^ x ) + s(k,y,+ k j j + t(]c z + \ z j - 1 u  that i s (34)  (  ' -  '  lc, ( r x * sy, + t z , ) + k ( r x + sy^_+ t z j - 1 (  Equations  (33)  r  v  = 0.  and (34) h o l d s i m u l t a n e o u s l y o n l y i f  (35)  ^  This r e l a t i o n  = 0,  = 1.  i s t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t  a plane,. whose c o B r d i n a t e s a r e g i v e n by (29), w i l l pass t h r o u g h t h e l i n e o f i n t e r s e c t i o n o f t h e p l a n e s (x,, y , z, ) and  ( x ^ , y^, z j . In  ( 2 9 ) ,  i f we l e t x, =  —— h + x  s  h + 1c we have the system o f p l a n e s whose c o o r d i n a t e s a r e given, by x  3cx + hx «* _ ! h + x  ,  33.  (36)  * hy  Icy,  =  w  .  Ii * 3c  3cz . +  hz.  h + 3c w h i c h i s a p e n c i l o f p l a n e s , s i n c e r e l a t i o n (33)  still  holds. In Cartesian coordinates  a l l p o i n t s (56)  are  colli-  d i v i d e t h e segment j o i n i n g (x , y , z ) and  n e a r , and  (  (x^,  y^,  z^)  24.  Three-Plane E q u a t i o n of a P o i n t ; £et  (  t  i n t h e r a t i o h : 3c.  (x , y , z ), (x^, y^, z.J, and ( x , y , (  he the c o o r d i n a t e s  (  (  3  f  2j  )  o f t h r e e p l a n e s such t h a t no p l a n e i s  p a r a l l e l to t h e l i n e of i n t e r s e c t i o n o f t h e o t h e r two® The the  c o n d i t i o n s t h a t t h e s e t h r e e p l a n e s pass t h r o u g h  point\  rx + sy + t z - l = 0, are  rx, + sy, + tz, - 1 •» 0, rx^+  s y + tz,.- 1 «  0,  t  rx + sy, + tz,- 1 = 0. 3  The c o n d i t i o n t h a t r , s , t e x i s t so as t o s a t i s f y these f o u r simultaneous equations i s that  x (37)  y  z  1  z,  1 1 1  =  0.  3 4  T h i s i s the r e q u i r e d e q u a t i o n  s i n c e i t i s of  the  f i r s t degree i n x , y, z, and i s o b v i o u s l y s a t i s f i e d by  the  t  c o o r d i n a t e s of the t h r e e p l a n e s . If 2c,  z, ^  ^  x  x the p o i n t i s f i n i t e .  y,  y  i  z  3  ^  °i  •.. .  f]  a  I f w - 0 , ( 3 7 ) g i v e s an e q u a t i o n of the  form  r.i rx  + sy + t z =  0 ,  w h i c h has a l r e a d y been d e f i n e d as a p o i n t a t i n f i n i t y . CO  «= 0 , the elements o f any one  combination  o f the c o r r e s p o n d i n g  row  o f OJ must be a  If linear  elements o f the o t h e r  rows, and hence the p l a n e must be p a r a l l e l  two;  to the l i n e o f  |J hi •N  i n t e r s e c t i o n o f the o t h e r two* •, 25 •  ''  H  .  T r a n s l a t i o n of A x e s ; -  kj  Suppose the 0 r i g i n i s t r a n s l a t e d t o the p o i n t r x + sy + t z - 1 w i t h o u t any r o t a t i o n o f a x e s . by the p o l a r c o o r d i n a t e s (f>  }  f  }  w i t h r e s p e c t t o ihe o r i g i n a l and new Then  Oy  L e t any p l a n e be ot fi y  ; ,!  ) and ( ^  represented '  *, f'  )  systems, r e s p e c t i v e l y . ii;  3 5 «  From (20) we have - (rx  f  sy + tz - l ) x  + y  + z  Therefore ~ •» ( r x + s y + tz -  08) -  -  A  «= y. and hence  1 x  ,*  r x + s y + tz -  - -  (39)  y'  X 1  y"  r x + s y + tz - 1 z  a'  r x + sy + tz - 1 ons are X r x ' + sy' + tz't 1  y'  (40)  r x ' + sy' + tz' + 1 z' rx' + sy* + t z ' +  260  The Degree o f an Equation- i s Unchanged by T r a n s f ormat i o n s :-  (1)  1  ^  Tanner- and A l l e n " A n a l y t i c Geometry", p. 1 2 7 • Wentworth " A n a l y t i c Geometry", p. 109 »  3 6  L e t the degree o f the e q u a t i o n he n.  A g e n e r a l term  would be  (41)  i x W ,  where p, q, m a r e n o t n e g a t i v e and p •*- q + m  £  n.  I f we r o t a t e axes by e q u a t i o n s ( 7 ) ,  i n p l a c e o f (41) we  obtain A(/),x'*Ay'  / U ' ) ( / < . X ' * /-.Y+yO,  -0*(<*Wy'+ S  P  +  3 Z  'T  S i n c e each term i n each b r a c k e t i s o f the f i r s t degree, we cannot o b t a i n terms o f degree h i g h e r t h a n n. I f we t r a n s l a t e axes a c c o r d i n g t o e q u a t i o n s ( 4 0 ) , (41) becomes A x'  P  y'* z '  '  m  ( r x ' * sy* + t z ' ) P I f e v e r y term i n the new  +  <1  +  m  e q u a t i o n be m u l t i p l i e d by  (rx' + sy' + t z ' + l ) , n  the term (41) f i n a l l y becomes (42)  A x'  P  y'* z ' ( r x ' m  +  sy'  +  tz' + l )  n  ~  ( p  +  * *  Any term i n (42) cannot be o f degree h i g h e r than n.  m  ) Hence  the degree of an e q u a t i o n i s n o t r a i s e d by t r a n s l a t i o n or r o t a t i o n o f axes* Suppose the degree were l o w e r e d by a t r a n s f o r m a t i o n of coordinates.  Then, by a p p l y i n g the i n v e r s e t r a n s f o r m a t i o n ,  3 ? .  we should, be r a i s i n g the degree o f the e q u a t i o n . been p r o v e d i m p o s s i b l e . by r o t a t i o n and  T h i s has  T h e r e f o r e the degree i s unchanged  translation.  3 8 .  CHAPTER I I The G e n e r a l Second Degree E q u a t i o n The most g e n e r a l second degree e q u a t i o n i n x, y, z is (l)  ax%  h y + c z + 2 f y z + 2gzx + 2hxz + 2ux + 2vy w  u  •i- 2ws  + d!, « 0, "  where a t l e a s t one of a, b, c, f , g, h i s d i f f e r e n t from zero.  We  s h a l l show t h a t ( l ) always r e p r e s e n t s a c o n i c o i d  i n t h e p l a n a r system of c o o r d i n a t e s . 1.  E q u a t i o n o f the Tangent P o i n t The l i n e of i n t e r s e c t i o n o f the p l a n e s ( x , , y , z, )  and  ( x , y^, zj) t  i s g i v e n ( S e c t i o n 20, Chap. I ) by the equa-  tions .(2)  x -x,  =  y -  7 |  ^  y, - 5^  - x,_  - ,  z  2  Z/  - z  The c o o r d i n a t e s of any plane t h r o u g h x (3)  I f a plane  ^  =  v  (2) a r e  «. x, + p(x, - x ) , x  .y  =  y, + p(y, -  z  =  z, + p(z, - z J .  ( 3 ) touches  s a t i s f y equation ( l ) .  yj»  the s u r f a c e ( l ) , i t s c o B r d i n a t e s must S u b s t i t u t i n g ( 3 ) i n ( l ) we  obtain a  q u a d r a t i c e q u a t i o n i n p, w h i c h shows t h a t , i n g e n e r a l , through any l i n e two p l a n e s can be drawn t o t o u c h the s u r f a c e (1).  Suppose t h a t one o f t h e s e i s t h e p l a n e ( x , , y  z )  (i  I t f o l l o w s t h a t one r o o t o f the q u a d r a t i c i n p must be z e r o , and hence the c o n s t a n t term must be z e r o .  We t h e r e f o r e  have (4)  a x  % by,"* c z , " * 2fy,z,+ 2gs x, + 2hx y + 2ux + 2vy,  (  (  t  (  (  * 2w z , + d = 0« Suppose (3) d e t e r m i n e s one p l a n e o n l y .  I n t h i s cas  the p l a n e i s the t a n g e n t p l a n e (x , y , z ) , and both r o o t s t  of the quadratic are zero.  (  t  Both the c o n s t a n t  term and the  c o e f f i c i e n t o f p must be z e r o , so t h a t ax, ( x , ~ x j + by, (y, - y j + cz, (z,- z j + f{ y, (z,-  (3)  * z, (y, - y j j + h{ x ( y t  +  * g f z (x, - x j + x, (z, - z^ ) J (  - y^ ) + y, (* -  #  v(y, - y O  zj  )/  +  u(x,~ x j  + w(z, - z j = 0.  I t f o l l o w s from (2) t h a t ( x - x j : (y - y ) : (z, - z J f  #  u  - ( x - x ) : ( y - y, ) /  : ( z ~ z,)» and from ( 5 ) we g e t axx, + byy, + czz, + f (y, z + z, y) + g ( z , x + x, z ) + h(Xj y + y,x) + ux + v y + wz = ax,  11  + by, *" + c z  u (  + 2fy, z,+ 2gz x + 2hx,y, + ux, + vy, + w z . t  (  t  As a consequence o f ( 4 ) t h e r i g h t number o f ( 6 ) i s equal t o - (ux, + vy, * wz, + d) • Therefore  ( 6 ) reduces t o  40.  (7)  axx,+ byy, + c z z , * f (y,z + z, y ) + g ( z , x + x, z ) + h(x, y + y x ) + u(x. + x,) + v ( y + y, ) + w(z + z ) ;  ;  + d =• 0.' Formula (7)  i s the e q u a t i o n o f t h e p o i n t of tangency o f the  p l a n e (x, , y , z,) to t h e s u r f a c e  (l) .  (  (1) 2.  C o n d i t i o n t h a t a P o i n t L i e s on t h e S u r f a c e g L e t the e q u a t i o n o f the p o i n t on t h e s u r f a c e be  (8)  r x + s y + t z - 1 = 0.  Comparing e q u a t i o n s  (7) and (8) v;e have  ax + hy * g z + u  h x + by, + f z , + v  (  _______________________  = gx * f y  -5-  (  SX  cz, + w  t  _/ux * vy, + ws, + dj .  "  Put each f r a c t i o n e q u a l  to - A .  ax, + hy, + gz, * u +  Ar  1  Then = 0,  hx, + by, + f z , * v + A s  = 0,  gx, * f y , + cz,.+ w + A t  «= 0,  ux, + vy, + v/z, + d - A  = 0.  We a l s o have r x , + sy, + t z , - 1 E l i m i n a t i n g x, , y, , z , A (  = 0. from t h e above e q u a t i o n s ,  o b t a i n the r e q u i r e d c o n d i t i o n , namely  (1) C. S m i t h " S o l i d Geometry", p. 41.  ne  41  a  h  S  u  r  h  b  f  V  s  S  f  c  w  t  u  V  w  a  -1  r  s  t -1  0  w h i c h i s the same as (9)  Ar%  Bs*"+ C t %  2Fst + 2 G t r + 2Hrs + 2Ur  + 2Ts  + 2Wt  + D=0,  where A, B, C, e t c . , are- the c o - f a e t o r s of a, b, c, e t c . , r e s p e c t i v e l y , i n the d e t e r m i n a n t  The  a  h  g  u  h  b  f  v  g  f  e  w  u  v  w  d  r e l a t i o n ( 9 ) i s a c o n d i t i o n t h a t t h e p o i n t (8) l i e s  on  the s u r f a c e ( l ) Incidentally, C a r t e s i a n system.  ( 9 ) represents  a c o n i c o i d i n the  Hence, f o r a p o i n t to l i e on the  surface  ( 1 ) , i t must l i e on a conicoid-;' t h a t i s , ( l ) r e p r e s e n t s c o n i c o i d i n the p l a n a r system of  coBrdinates.  A proof that ( l ) represents i n s e c t i o n 3, where no r e f e r e n c e Cartesian  (1)  a  a c o n i c o i d w i l l be  given  i s made, as above, to  coordinates.  For a s i m i l a r d i s c u s s i o n see Snyder and Sisam, pp. I 3 0 ,  131.  42*  3»  Locus of M i d d l e P o i n t s o f a System o f P a r a l l e l Chords:o f t h e s u r f a c e be ( l ) , and l e t (8)  L e t the equation be the e q u a t i o n s a t i s f y (9).  o f any p o i n t on t h i s e o n i c o i d ; r , s, t must  Let Jt-x. + my + nz - 1 = 0  (10) be t h e e q u a t i o n a r e y\  , 4/»  o f a p o i n t on a l i n e whose d i r e c t i o n  cosines  The p o i n t (8) w i l l l i e on t h i s l i n e and  be d i s t a n t p from (10) i f  s - m <= p^6c , t - n = p-z/ , that i s  s  i f r =  + p A  s  S = m + P JUL. , (11)  v  t = m + p 4/ • I f we s u b s t i t u t e (11) i n (9) we o b t a i n a q u a d r a t i c e q u a t i o n i n p, w h i c h shows t h a t any g i v e n l i n e c u t s the s u r f a c e i n two p o i n t s .  I t follows that a l l s t r a i g h t  lines  i n a p l a n e c u t t h e s u r f a c e i n two p o i n t s , and t h e r e f o r e a l l plane s e c t i o n s of t h e s u r f a c e a r e c o n i c s e c t i o n s .  This i s  the d e f i n i t i o n o f a e o n i c o i d . We have (12.) p""(A / f + B ^ + C + Bm^_+  Cn  S\  + 2C-«/A 2H/1/-) + 2 p ( A i M +  F i i A + FmV + GlS + G-nA + H^A+  + v>_ + ' WT/) * ( A ^ * B m % v  * 2Vm + 2Wn + D) = 0,  HmA •+ IT A  Cn%"2Fmn * 2Gn/+ 2Him +  2  where A, B, C, ... , have the same v a l u e s as i n s e c t i o n 2. I f (10)  i s t h e e q u a t i o n of the m i d d l e p o i n t of the l i n e ,  the v a l u e s o f p o b t a i n e d from (12) but o p p o s i t e i n s i g n .  The c o n d i t i o n f o r t h i s i s t h a t the  c o e f f i c i e n t of p equals zero.  (13)  i(AA  must be e q u a l n u m e r i c a l l y  Hence  + Eyu + C V ) + m(H A + B^A. + F-*/)  * n(G A + Fyu. + 0 / )  * U A •«- Yyu- + f i / = 0,  T h e r e f o r e the p l a n e whose p o l a r c o B r d i n a t e s a r e g i v e n by p R*~ * s t  COS  oC  QL  =  R (14)  +  R  COS y_?  +  cos  S  If  R *~ + where A +  P  U  Q  A A  R  H A" + B  S  G A + F/U + c  passes t h r o u g h t h e p o i n t ( l O ) l ^ 1  + EyU. +  GV,  But (14) r e p r e s e n t s a  (1) C.f. e q u a t i o n ( 1 4 ) , Chapter I .  4 4 .  y\ , /b- , */  f i x e d p l a n e when  are f i x e d .  T h e r e f o r e the mid-  p o i n t s o f a l l p a r a l l e l chords whose d i r e c t i o n c o s i n e s are A * ^ *  ^  He  I n the p l a n e ( 1 4 ) .  A p l a n e w h i c h passes through the mid.-points o f a system o f p a r a l l e l chords o f a e o n i c o i d i s known as a diametral plane.  I f a d i a m e t r a l plane i s perpendicular to  the chords i t b i s e c t s , 4.  i t i s c a l l e d a p r i n c i p a l plane*  The P r i n c i p a l P l a n e ; I f t h e p l a n e (14) i s p e r p e n d i c u l a r to t h e chords  whose d i r e c t i o n c o s i n e s a r e A ,yu. , V c o s i n e s of i t s p o l a r normal must he AA  + H/c  * GV  -  -  \  H  , the d i r e c t i o n /\ ^u. , 4/*  A + B/"- + i V  G  A + F>- + c V ~  ~  P u t jT f o r the common v a l u e o f each of these  fractions;  then (A - J ) A (16)  Eliminating  + H/c +  - 0,  HA  +  GA  * F / c + ( C - J )</ - 0.  X  -  (B  » ^  w  e  S  [ A -/  0,  e t  H  H  B- J  G  F  G  F C-  w h i c h , when expanded, becomes t h e c u b i c ( 1 7 )  f -  Therefore  %  f!  - &  m  0 J  9  where  J  =  A  + B  + C,  + BC  + CA  2 F G H - A F ^ ~ EG*'-  CH .  and  o© - ABC  When. /  i s determined,  +  ^  = AB  - F-G~-  H*",  V  any two of the t h r e e r e l a t i o n s (16)  w i l l g i v e the c o r r e s p o n d i n g v a l u e s of \  ^  %  ,t/.  Since  one r o o t o f a c u b i c e q u a t i o n i s always r e a l , i t f o l l o w s t h a t t h e r e i s always a t l e a s t one p r i n c i p a l p l a n e . 5°  Roots o f ( I ? ) t - ^ ) 1  L e t £ , be any r o o t o f (17)  =  f, •  be complex.  0  * «S t h a t s a t i s f y  (not a l l z e r o ) be v a l u e s o f A when f  , <t/  and l e t A o , yu  I f ^ i s a complex number, A  0  0  (16)  ,yu„ ,  may  Let  A° ~ A, + i A*., «/* where i = <fT  and ^  Substitute £  w  , ,^ ,^  are r e a l .  and these v a l u e s o f 'A 6  A, ~ i y A , y u , - i A  add.  , ^„ ^  i n (16),  I* » A J^A by  ;  + i  fyA-o  m u l t i p l y the r e s u l t i n g v  , V, - i <.  »  for  equations  , r e s p e c t i v e l y , and  The r e s u l t i s  ( A % AC *^ + 2( /t//\, + V  v  ^ ) G + 2( /^y<*, + W  +* s)f /  l  - (A" * A )A v  )H.  The c o e f f i c i e n t o f / / i s r e a l and d i f f e r e n t from z e r o , and  (l)  Snyder and Sisam, p. 79.  the r i g h t member o f t h e e q u a t i o n is real. (17)  Since  i s also r e a l .  / i s any r o o t of (17), ;  Hence J  f  a l l the roots of  are r e a l . The c o n d i t i o n s  t h a t a l l t h e r o o t s of (17)  a r e zero  are  (18)  ABC + 2FGH - AF*'- BG^ - OH""  - 0,  AB . + " BC + CA - F*" - G "- H  «  1  A + B + C Square (18, 3 ) ,  = 0.  i.e«, the t h i r d e q u a t i o n  t r a c t t w i c e .('18, 2 ) , f r o m i t . V  o f (18),  and s u b -  The'result i s  A " + B~'+ C" + 2 P + 2G'"+ -  Q,  2H""=  0.  S i n e e A, B, 0, e t c . , a r e assumed to be r e a l , i t f o l l o w s that  '  (19)  A = B = C= I f (19)  F = G = H = 0.  i s t r u e , (9) reduces t o 2IJr + 2Ts + 2Wt + D = 0.  But. t h i s i s the, c o n d i t i o n t h a t the p l a n e whose p o l a r dinates a r e  coor-  47  passes through  t h e p o i n t whose e q u a t i o n i s r x + s y + t z - 1 = 0.  T h e r e f o r e the f i x e d p l a n e (20) w i l l pass through a l l the p o i n t s on t h e c o n i c o i d , and hence the e o n i c o i d reduces to a p l a n e . T h i s degenerate case i s o b t a i n e d by l e t t i n g a l l the r o o t s of the c u b i c be z e r o .  H e n c e f o r t h we s h a l l assume  t h a t a t l e a s t one r o o t o f the c u b i c i s d i f f e r e n t from z e r o . 6.  E l i m i n a t i o n of the y z , z x , z terms S i n c e a t l e a s t one o f t h e p r i n c i p a l p l a n e s i s n o t  a t i n f i n i t y , we can t r a n s l a t e and r o t a t e t h e system o f r e f e r e n c e so t h a t the new XY p l a n e i s a p r i n c i p a l p l a n e o f the s u r f a c e  0  L e t t h e e q u a t i o n of t h e e o n i c o i d r e f e r r e d t o the new axes be ( l ) .  Since the surface i s symmetrical w i t h  r e s p e c t t o the XY p l a n e , t h e two p a r t s i n t o w h i c h t h e XY p l a n e d i v i d e s the s u r f a c e must be e x a c t l y a l i k e . i s a tangent p l a n e ( x , , y  , z ) a t a p o i n t on one s i d e  (  (  of t h e XY p l a n e , t h e r e must be a c o r r e s p o n d i n g plane  tangent  ( x , y , - z ) a t a p o i n t on the other s i d e . (  stituting ax  I f there  v (  (  (  Sub-  each s e t o f c o o r d i n a t e s i n ( l ) , we o b t a i n  + by," + ez, " + 2fy, z  + 2vy  1  (  (  + 2wz  (  + 2g^ x (  (  * 2hx,y, + 2ux,  + d = 0  and a x " + b y " * e z " - 2 f y z, - 2 g z x + 2hx y (  + 2vy, - 2wz  4  + d = 0«  (  (  + 2ux ' (  48,  S i n c e these r e l a t i o n s a r e t r u e f o r a l l tangent p l a n e s , i t follows that f = g = w = 0. These r e s u l t s may follows.  be d e r i v e d i n a second way  as  C o n s i d e r the t h r e e p o i n t s r x + s,y + t, z - 1 = 0, (  (£1)  -1=0,  r _ x + s^y + t z u  r x  + s, y + t z - l = 0,  on t h e s u r f a c e and on one  s i d e of t h e XY p l a n e .  p o i n t s be c o n s i d e r e d as d i s t i n c t . t h a t t h e y approach c o i n c i d e n c e .  L a t e r we  Let  these  s h a l l require  On t h e o t h e r s i d e of the  XY p l a n e we must have the c o r r e s p o n d i n g p o i n t s = 0,  r,x + s y - t z - l  (22)  r _ x + s^y - t z - 1 •= 0, v  r, x + s y - t , z «" 1 = 0. The c o o r d i n a t e s (x, , J , S ) o f the plane (  X  the t h r e e p o i n t s (21)  a r e g i v e n ( S e c t i o n 13,  1  r r r  a, t, s  3  s  t 5  t  3  r, 1  t,  1  t^  1  \  r r  Chap. I ) by  s, t,  ;  3  through  5.  r r  s, 1 1  v S  I  3  A,  A,  t h r e e p o i n t s (22)  ( I Z.  »  1  s, - t ,  y*»  a r e g i v e n by  1 x ^ =  1  -t s  -J}  ,  -X  r, -t r  r  , ,  r  i  -t  1 -\ 1 -t —  A, i s  L  I  i —  Therefore Z i «* x,  (23)  f  y».  = y, i  Z  » -z,  L  4,  50  I n the case where  A  0, r e s u l t s s i m i l a r to (23)  can he o b t a i n e d b y u s i n g p o l a r c o B r d i n a t e s , L e t the p o i n t s (21) approach c o i n c i d e n c e ; then the p o i n t s (22) w i l l r e l a t i o n (23)  do l i k e w i s e .  A t a l l steps i n t h i s process  holds f o r the.coordinates of the planes  t h r o u g h the r e s p e c t i v e s e t s o f points,,  I n t h e l i m i t , I .e„  where tangency o c c u r s , the r e l a t i o n must s t i l l be t r u e . Therefore, f o r every tangent plane ( x , y , z ) at a p o i n t (  f  (  on one s i d e o f t h e XY p l a n e t h e r e must be a c o r r e s p o n d i n g t a n g e n t p l a n e ( x , y , - z ) a t a p o i n t on the o t h e r s i d e . (  (  (  I f f = g = w •= 0, e q u a t i o n ( l ) becomes (24) 7.  ax "* by "* cz^-t- 2hxy + 2ux + 2vy + d « 0. 2  1  R e d u c t i o n when d^-Q:I f we t r a n s l a t e the o r i g i n t o t h e p o i n t whose equa-  tion i s |x  I y  1-0,  (24) becomes (a - u ) x l  v  + (b - v ) y~ * cz*~+ 2/h - uv/xy + d = 0. 1  The term i n x y can be e l i m i n a t e d by r o t a t i n g the X, Y axes through a n a n g l e 0  d e t e r m i n e d by  51  a c c o r d i n g t o the r o t a t i o n formulae x - x'cos£  - y'sin 9 ,  y = x'sin6?  + y'cos _? ,  Z  = Z'.  D r o p p i n g p r i m e s , vje g e t an e q u a t i o n o f the form a x + b, y + c, z^+ d = 0. v  v  (  S i n c e d ^ 0, we c a n d i v i d e by -d and t h e r e s u l t i n g e q u a t i o n has the form (25)  a 2c + ^ y - * c z " = 1. v  1  6  s  Hence f o r d ^ O , under a l l c o n d i t i o n s we c a n reduce e q u a t i o n ( l ) to t h e form (2jj) • 8.  R e d u c t i o n when d - 0;~ The e q u a t i o n to be c o n s i d e r e d i s ax*~+ b y % c z " * 2hxy + 2ux + 2vy = 0. 1  ( i ) I f u = v = 0, by r o t a t i n g t h e X, Y axes  through  an angle 0 g i v e n by tan  2&  = 2h  a - b we e l i m i n a t e the x y term.  The r e s u l t i n g e q u a t i o n has the  form (26)  a x"+ b.yN- c . z ^ 0. D  ( i i ) I f v i s n o t z e r o we e l i m i n a t e the y term by r o t a t i n g the X, Y axes a c c o r d i n g t o the t r a n s f o r m a t i o n s  52.  vx'+ uy'  y  -J- V _ ; Z — 2 ,  and we o b t a i n an e q u a t i o n o f the f o r m a,x*+ b  e, z % 2 h x y + 2 u x = 0.  (  (  (  I f u,= 0 we have ease ( i ) . I f  0, by t r a n s -  l a t i n g the o r i g i n t o t h e p o i n t whose e q u a t i on i s ~ a_ 2u,  n, u,  - 1 «= 0,  we o b t a i n (27)  b, y % c z''* 2u,x = 0. 1  /  T h e r e f o r e e q u a t i o n ( I ) c a n be reduced forms ( 2 3 ) , ( 2 6 ) 5.  s  to one o f t h e  or ( 2 7 ) .  Center o f Gonicoid:C o n s i d e r e q u a t i o n ( 2 5 ) , namely ax*~+ b y + c z v  v  = 1.  The c e n t e r l i e s on t h e p l a n e midway between p a r a l l e l tangents to t h e s u r f a c e .  I f ( x , y, z ) i s t a n g e n t to t h e  s u r f a c e , (-x, -y, -2) i s a l s o t a n g e n t .  Therefore the  o r i g i n i s the c e n t e r o f t h i s t y p e of conicoid® C o n s i d e r e q u a t i o n ( 2 6 ) , namely a _ * by%- c z «= 0 v  v  e  As b e f o r e , the o r i g i n i s the c e n t e r . Suppose t h e c o n i c o i d reduces t o by*+ cz "* 2ux = 0 . 7  53»  L e t the p a r a l l e l p l a n e s ( x , y , z ) and (kx, , ky ;  f  ; }  }  c2  )  touch t h i s s u r f a c e ; t h a t i s by, + c z v  v (  + 2ux,  « 0, 2ukx,  bk^y^ * clc s v  =0.  I f xi = 0 t h i s e o n i c o i d " i s a degenerate o f ( 2 6 ) .  If u ^ 0  then k = 1, o r e l s e t h e p a r a l l e l tangent p l a n e s a r e a l l a t infinity.  Therefore  10.  Plane:-  Polar  t h e s u r f a c e has no f i n i t e  center*  We s h a l l show t h a t the p o i n t s o f c o n t a c t o f a l l tangent p l a n e s t h r o u g h a g i v e n p o i n t t o a e o n i c o i d l i e on a plane.  T h i s p l a n e i s c a l l e d the p o l a r plane of the p o i n t  w i t h r e s p e c t to the- e o n i c o i d ,  Conversely,  the point i s  c a l l e d the p o l a r p o i n t of. the p l a n e w i t h r e s p e c t t o the e o n i coid. ( i ) L e t the e q u a t i o n ax% The e q u a t i o n  of the c o n i c o i d be  by""+ cz' =• 1. L  of t h e t a n g e n t p o i n t o f t h e p l a n e ( x , , y , z ) f  i s g i v e n ( S e c t i o n 1, Chap. 2) by (28)  axx, + byy + c z z , (  -1=0.  Suppose the p l a n e ( x , y, , z,) passes t h r o u g h the p o i n t (  (29)  r x + s y + t z - 1 = 0;  then rx, * sy * tz, - 1 = 0 . The p o i n t ( 2 8 ) l i e s on the p l a n e f r , s_, _t )  since i t s  ;  n coordinates  s a t i s f y the equation.  Hence the p o i n t s o f t a n -  gency a l l l i e on the p l a n e (30) / r , s , t ), \a  b~  C /  w h i c h must t h e r e f o r e be t h e p o l a r p l a n e o f t h e p o i n t  (29)  w i t h r e s p e c t to t h e c o n i c o i d . ( i i ) L e t the e q u a t i o n  of t h e c o n i c o i d be  ax"+ by""+ cz ^ - 0. The e q u a t i o n  of the tangent p o i n t i s  (31)  axx, + byy, + czz,  The p o i n t (31)  - 0«  l i e s on t h e p l a n e ( 0 , o„ o ) , and t h e r e f o r e  a l l t a n g e n t p o i n t s are a t i n f i n i t y . w i l l be d i s c u s s e d  T h i s type o f c o n i c o i d  later*  ( i i i ) L e t t h e e q u a t i o n of the c o n i c o i d be —• ' x. ' by + cz -s- 2uz 0. The e q u a t i o n  of the tangent p o i n t i s  (32) If  byy, + czz, + u ( x + x ) = 0 . (  t h e p l a n e ( z , y , z,) a l s o passes through the p o i n t (  whose e q u a t i o n  i s (29)  (33)  t h e p o i n t (32)  9  l i e s on t h e p l a n e  / J L , us , u_t ) \  since i t s coordinates  r  br  cr J *  s a t i s f y (32).  Therefore  (33)  i s the  p o l a r plane o f the p o i n t (29) w i t h r e s p e c t t o t h i s c o n i c o i d . 11*  Rectilinear •  (34)  Generators;-  L e t the e q u a t i o n o f the s u r f a c e be ax + by-cz»=l,  55. w h i c h may be w r i t t e n i n the form (ax + c s ) ( a x - c z ) = ( l + b y ) ( l - b y ) , or  (35)  ax -i- c z 1 + by  1 - by  ^ =  ax - c s  _" t  !  s a  y*  Then ax + c z «= 7 | ( l + b y ) , (ax - c z ) 7j «• 1 - by .  (36)  F o r e v e r y value o f 1^ , these e q u a t i o n s E v e r y p o i n t l y i n g on t h e s u r f a c e (34)  define a l i n e .  must s a t i s f y t h e  r e l a t i o n ( S e c t i o n 2, Chap. 2 ) .  (37)  r* * _ t~ v „ I f the p o i n t whose e q u a t i o n - i s +  s  a  b  r x  + s y + t  (  l i e s on the l i n e (36),  (  z - l = 0,  i t f o l l o w s ( S e c t i o n 12, Chap, l )  that r.  (38)  = m —  + m an  s, = -m ,b + m, b t  ;  m R e l a t i o n (37) respectively.  = m, H.  }  - m c^ ^  + m, = 0.  h o l d s when we r e p l a c e r , s, t by r , s^ , t , Therefore  any p o i n t on t h e l i n e (36)  l i e s on t h e s u r f a c e (34), l i n e a r generators  of  Equat i o n (34)  and (36)  also  i s a system of r e c t i -  (34). may be w r i t t e n  56  (39)  ax * c z 1 - by  =  1 + by ax - c z  ^ =  9  s a y  *  Then ax + c z = _T ( 1 - b y ) , (40)  (ax - c z ) ^ - 1 + by, system of r e c t i l i n e a r - g e n e r a t o r s of ( 3 4 ) .  w h i c h i s a second  We can f i n d i n a s i m i l a r manner the equations o f t h e g e n e r a t i n g l i n e s o f the s u r f a c e (41)  b^y"" - c V  = 2ux.  The e q u a t i o n s o f the g e n e r a t o r s of one system a r e by - c s = 2 < r i , by + c z = JL. ; and o f the o t h e r  system by + c z - 2 r x , by ™ c z = _ .  12.  Invariants:L e t t h e e q u a t i o n o f the s u r f a c e be ax*"+ by"+ c z " « 1.  I f the a x e s ' a r e r o t a t e d t o new p o s i t i o n s a c c o r d i n g t o equat i o n s (8) o f Chapter I , the r e s u l t i n g e q u a t i o n i s o f t h e form a x"+ b y%- c z % 2f y z + 2g z x + 2h x y = 1, (  f  (  (  (  (  where a, = a j , + 1J y^v  b  = a /V„ + b v  (  +  + c c  %  51.  c, = a / )  + b /U  3  f", * a/1 /l + b/.y& x  3  + e  3  4  + c<^  g, = a/^A, + b / ^ , + c  3t  ,  Making use o f t h e r e l a t i o n  A, A.  4/, we  obtain I  a  D. £  h  h  a 0  x g, a be  l  o.  0 b' 0 0 0  «,  e  T h e r e f o r e 2) i s unchanged b y r o t a t i o n . I n t h e same way i t c a n be shown t h a t I  = a + b + c,  J  be . + ea + ab ~ i  g - h  are unchanged by r o t a t i o n . I t c a n r e a d i l y be shown t h a t these e x p r e s s i o n s a r e n o t i n v a r i a n t under t r a n s l a t i o n . The c o n d i t i o n t h a t a p o i n t r x + sy + t z - 1 = 0 l i e s on the g e n e r a l (42)  e o n i c o i d i s ( S e c t i o n 1, Chap. 2)  A r " + Bs'" + Ct  = 0,  where A, B, C, ...... a r e t h e c o - f a c t o r s o f a, b, e, , e i n the d e t e r m i n a n t  e * e  j?0.  a  B  u  h  b  f  V  S  X  c  w  XL V  w  d  L e t the axes be r o t a t e d to new p o s i t i o n s a c c o r d i n g to t h e f o r m u l a e (8) o f Chapter I , namely x = A, x' + y ~yu,  Ax,  -' + / - v y  y' +  , +  i z',  >fc» z',  z = i/, x' + /f/ y' + 4/ v  z'.  3  The p o i n t whose e q u a t i o n r e f e r r e d to the o l d system i s -  r x + s y + t z - 1 = 0,  becomes (r A  + s>/ + t  A/,  •)_' * ( r ^_.+  s/«v + t^«C ) y ' + ( r /13 + s / ^  + t V, )£' ~ 1 i n t h e new  system; t h a t i s  = 0,  '  r ' = r A1 + Sju, + t  ,  s' = r ^ + s ^ + t  ,  v  t ' = r > j + s^k, + t *4 . The i n v e r s e r e l a t i o n s a r e r e a d i l y found to be r = r M , + s' Ax.+ t'Aj  (43)  s  s = 2?/*, + s'/<v + t > , 3  t = r </, + s' 1  + tV  3  .  The degree o f e q u a t i o n (42) w i l l be u n a l t e r e d , as p r o v e d i n Chapter I , S e c t i o n 26,  by the s u b s t i t u t i o n s  (43).  I f , by a change of r e c t a n g u l a r axes through the same o r i g i n ,  5?.  Ar" " + B s " + a t " + 2 F s t + 2Gtr + 2Hrs 1  1  becomes changed i n t o A ' r " + E ' S " + C't ' + 2 F ' s t + 2G'tr * 2H'rs; 1  then, since (44)  s~+ t ^ I s u n a l t e r e d by t h i s change o f axes, *  Ar"+ B s n Ct^+ 2 F s t + 2 G t r * 2Hrs - J (r"+ s"+ t ~ ) w i l l be t r a n s f o r m e d  into  (45)  A' r ^ + B' s" + C t  v  + 2F'st + 2G-' t r + 2H'rs - J ( r % s % t " ) .  The e x p r e s s i o n s ( 4 4 ) and ( 4 5 ) w i l l  t h e r e f o r e be t h e  p r o d u c t o f l i n e a r f a c t o r s f o r the same v a l u e s o f J . The c o n d i t i o n t h a t ( 4 4 ) i s t h e p r o d u c t o f l i n e a r factors i s A - J  .H  -J  B  H G  G  •"'  F  F  C  -  that i s J  3  '« 0.  , • -  J*(A + B ~  + C) (ABC  + +  J(BC  2FGH  + CA  - AF^-  + AB  - F*"- G V  EG**- G H ) V  =  H")  0*  The c o n d i t i o n t h a t ( 4 5 ) i s the p r o d u c t o f l i n e a r f a c t o r s i s similarly f  3  -  J V +  B'  +  C' ) +  |(B'C'  +  C'A' + A'B' - F ' -  '-(A' B'C' ^ F ^ ' E' - A' F' - B' G'"-  C'/)  G'-  H' ) 1  0.  S i n c e t h e r o o t s o f t h e above c u b i c e q u a t i o n s i n J a r e the same, t h e c o e f f i c i e n t s must be e q u a l .  6o.  Hence  3  = A + B + C,  ^ = BC' + CA + AB - F"~ G'- H", 7  A  H  G  H  B  F  G  F  C  ,  a r e u n a l t e r e d by r o t a t i o n . T r a n s l a t i o n o f axes to t h e p o i n t whose e q u a t i o n i s «T x + /5 y + y _ - 1 = 0 c a n he a c c o m p l i s h e d ( S e c t i o n 2j?, Chap. I ) "by means o f t h e formulae o< x'+ /9 y'+  #z' + 1  x ' + ^ y' +  c< x' + /3y' +  '+ 1  /z'+  1  The p o i n t , whose e q u a t i o n r e f e r r e d t o the o l d axes i s r x + s y + t z - 1 = 0, has the.e qua t i on r x ' + sy' + t z ' - ( c< x' + /3 y '+ ^ z' + l ) = 0 r e f e r r e d t o t h e new axes; t h a t i s r ' = r ~ c< , s ' «= s - /3 » t' = t therefore  r i  6 1 .  = r + oC , (46)  s » s' +  ,  t = t ' + }' . The s u b s t i t u t i o n of (46) i n (42) doss n o t change any o f the c o e f f i c i e n t s o f t h e second degree terms*, , j^.  ,  ^j- » jj'  »  a r e u n a l t e r e d by t r a n s l a t i o n o f a x e s .  Thus  u n a l t e r e d by t r a n s l a t i o n or r o t a t i o n , and  a r e  are t h e r e f o r e  Therefore  invariants.  The p r o o f t h a t given f o r  ,  / \ i s i n v a r i a n t i s s i m i l a r to t h a t  The c o n d i t i o n  t h a t a p o i n t l i e s on a  eonicoid  is Ar Let  + Bs + Ct + 2Fs"C + 2Gtr + 2Hrs + 2Ur + 2Ts + 2Wt + D=0.  t h i s e q u a t i o n be t r a n s f o r m e d by a r o t a t i o n A'r%  B's^+  into  C' t^+ 2 F ' s t + 2G' t r + 2H'rs + 2U'r + 2V's • + 2W't  + "D'«  0.  ,This r o t a t i o n t r a n s f o r m s the e x p r e s s i o n Ar%  B s % Ct '* 7  2 F s t + 2Gtr + 2Hrs + 2IJr + 2Vs + 2Wt + D  (48)  - kCr^+'s"*  1)  into A' r % - B' s + C t + 2 F ' s t + 2G-' t r + 2H' r s + 2tf'r + 27's v  L  (49)  "  + 2W't.- + D - '3c(r"+ s~+ t*"+ 1 ) . A  The d i s c r i m i n a n t s A - 3c H H  o f (48) and (4?) a r e , r e s p e c t i v e l y G  TJ  B - 3c F  1  Or  F  C ~ 3c  W  JJ  1  W  D -  I and  62. A - k H' •  1  B'-k  G'  F'  IT'  Y'  The e x p r e s s i o n s l i n e a r expressions  H  C-'  -U'  p'  Y'  C' - k  W  7  \7'  B'-k  ( 4 8 ) and ( 4 ? ) a r e f a c t o r a b l e i n t o  f o r the same v a l u e s  o f k.  The c o n d i t i o n  t h a t each i s f a c t o r a b l e i s t h a t i t s d i s c r i m i n a n t equals zero. Hence, s i n c e t h e c o e f f i c i e n t o f k * i n e a c h case i s u n i t y , the c o n s t a n t terms o f t h e s e d i s c r i m i n a n t s must be e q u a l ; is  _\ =  A •» Hence, 1  I n order  that  i s i n v a r i a n t under r o t a t i o n .  to p r o v e t h a t  /_»  i s i n v a r i a n t under t r a n s -  l a t i o n , l e t t h e axes be t r a n s l a t e d t o the p o i n t whose equation i s  The c o n d i t i o n t h a t t h e p o i n t l i e s on t h e c o n i c o i d becomes Ar"+ B s % C t + 2 F s t + 2 G t r + 2Hrs + 2(A<* w  (50)  + K/i  + G Y + u)r  + 2(H*L + B/3 + F r + V ) s + 2 ( G K + F ^3 + C *  + W)t  + D'- 0, where D* i s t h e l e f t member of ( 4 7 ) when r , s, t a r e r e p l a c e d "by  c< , ^3 , y  o  The d i s c r i m i n a n t o f (j?0) i s  A  H  G  A<^ +H/6+G y +TJ  H  B  F  H *<- +B  G  F  C  G^+F/S+C flf+W  A*< +H^» +G  r +u,  H  ,< +B^a +F r +v,  G  ^ +F/3 +C ar +w,  ~  Y +IT  i>'  Multiply t h i r d by  the f i r s t Y  »  an(3  -  column by  , the second by(3  , the  s u b t r a c t t h e i r sum from t h e l a s t column»  I n the r e s u l t i n g d e t e r m i n a n t , m u l t i p l y the f i r s t row by o( t h e second by  , t h e t h i r d by  sum from t h e l a s t row. Hence t r a n s l a t i o n and  Y  s an3. s u b t r a c t t h e i r  The r e s u l t i n g d e t e r m i n a n t i s /\ «  , so t h a t rotation.  i s i n v a r i a n t under both  64  CHAPTER H I C l a s s i f i c a t i o n o f Surfaces 1.  Review o f P r e v i o u s Work:I n C h a p t e r I I we have seen t h a t the c o n d i t i o n  that  a p o i n t whose e q u a t i o n i s (1)  r x + sy + t z - 1 = 0  l i e s on t h e s u r f a c e whose e q u a t i o n i s g i v e n by (2)  ax''+ by'-!- cz" + 2 f y z + 2gzx + 2hxy + 2ux + 2vy + 2wz +d=0 v  1  Is (3)  i r + Bs"" + Ct" + 2 F s t + 2 G t r + 2Hrs + 2Ur + 2?s + 2Wt +D«0 r  where A, B, C, .... , a r e t h e c o - f a c t o r s o f a, b , c, .... , i n t h e determinant  5" =  a  h  g  u  h  b  f  v  g  f  c  w  u  v  w  d  For b r e v i t y we s h a l l r e f e r t o (3) as the • • p o i n t - c o n d i t i o n equation.  We have a l s o seen t h a t  3 =  (A + B + C ) ,  j = (AB + BC + CA -F"-G""- H"),  A  H  G  H  B  F  G  F  C  and  A  H  G- U  H  B  F  ¥  G  F  C  W  TJ  Y  W  DV  are i n v a r i a n t under t r a n s l a t i o n and r o t a t i o n .  88  2.  The Sphere;The sphere i s d e f i n e d to he the l o c u s o f a p o i n t  w h i c h moves so as t o remain a t a c o n s t a n t d i s t a n c e from a fixed point.  This d i s t a n c e i s known as the r a d i u s and the  f i x e d p o i n t i s t h e c e n t e r o f the s p h e r e .  L e t the e q u a t i o n  o f the c e n t e r he c^x + {2> y +  z - 1 = 0,  and l e t the r a d i u s he R; then we have / ( r -d.  (s ~ 7 * r + ( t - * T  - R,  or ( r - . * 0 % ( s - y 3 ) % ( t - y ) " - R.\ Therefore  t h e g e n e r a l p o i n t - c o n d i t i o n e q u a t i o n of a sphere  A r % A s % A t % £Ur + 2Ys + 2Wt * D = 0, where A i s d i f f e r e n t r x + sy + t z  k  from z e r o .  C o n v e r s e l y , any p o i n t  1, where r , s, t s a t i s f y the c o n d i t i o n equa-  t i o n , l i e s on a s p h e r e . The p o i n t - c o n d i t i o n e q u a t i o n of a sphere whose c e n t e r i s the o r i g i n , i s seen t o he T  * S + t = R e  The sphere may a l s o b e " d e f i n e d as t h e envelope o f p l a n e s w h i c h move so as always to remain a t a c o n s t a n t d i s tance f r o m a f i x e d p o i n t .  Thus  y x N - yN- Z"-  1 .  oc x + /? y + y z-1  R  1  that i s (4)  R \ x % y % 0 =  ( ^ x + ^ y  + y z - 1)".  The e q u a t i o n o f a s p h e r e , c e n t e r a t the o r i g i n , i s seen to he R"^"*  y~ + z"") « 1,  or (5)  a x + a y % a s + d = 0. v  v  ( I f a and d have the same s i g n the sphere i s i m a g i n a r y . ) The p o i n t - c o n d i t i o n e q u a t i o n o f the sphere (5) i s A r % - BS^-J- Ct'-*- 2 F s t + 2Gtr + 2Hrs + 2Ur + 2Vs 1  > where A = a^~d, B  2Wt + S = 0, '  a^d, C = a 4  D = a , and  v  3  (  F = G = H-  TT-V«=W=0.  Therefore 3  - 3a d, l  o^) = a ' d , 3  ^ 3.  = a d » f  J  The E l l i p s o i d ; C o n s i d e r t h e s u r f a c e whose e q u a t i o n i s  (6)  a x + by + c z  = 1.  The p o i n t - c o n d i t i o n e q u a t i o n o f t h i s s u r f a c e i s found t o he (7)  b c r + c a s + a b t  For a, b, c a r e a l l d i f f e r e n t  = a b c .  from z e r o , and a, b, e i n  6. 7  descending  o r d e r o f magnitude, we  a  have  _ a" _  a  1  and  Hence a p o i n t on t h e s u r f a c e can n o t be a t a d i s t a n c e from the o r i g i n g r e a t e r t h a n a n o r l e s s than c.  The s u r f a c e i s  t h e r e f o r e l i m i t e d i n e v e r y d i r e c t i o n ; and,  s i n c e a l l plane  s e c t i o n s of a c o n i c o i d are c o n i e s , i t f o l l o w s t h a t a l l p l a n e s e c t i o n s of (6) a r e e l l i p s e s .  This i s the u s u a l d e f i n i t i o n  of an ellipsoid© The s u r f a c e i s c l e a r l y s y m m e t r i c a l w i t h r e s p e c t to the t h r e e c o o r d i n a t e p l a n e s , the t h r e e c o o r d i n a t e axes, the o r i g i n .  and  The p o i n t s i n w h i c h i t c u t s t h e axes are found  by l e t t i n g s = t = 0, e q u a t i o n (7)•  t = r - = 0,  r = s = 0, r e s p e c t i v e l y , i n  These p o i n t s are d e t e r m i n e d by the r e l a t i o n s r  =  ±  s  =  t  =  a,  ±ax  -1=0,  _r b,  ± by  -1=0,  ±  ±cz  c,  - 1  B  0,  respectively. Consider  the system o f tangent p l a n e s through  the  point (8) on the Z a x i s .  mz - 1 = 0, The c o o r d i n a t e s of a l l p l a n e s through  this  68.  p o i n t and t o u c h i n g the s u r f a c e a r e ( x , J, a x + b y + e_  1 ) where m  = i .  -  The p o l a r p l a n e o f the p o i n t (8)  i s ( 0 , 0, m_\ . T r a n s l a t e cc  1  the o r i g i n t o the p o i n t  2l  z  - i = or  m the new  XY plane w i l l be the p o l a r of the p o i n t (8).  e q u a t i o n of (8)  The  becomes ______-„ z - 1 = 0; m  t h a t i s , the c o o r d i n a t e s of a l l p l a n e s through (8) (V v  m » ^* T ~y» m - e v  will  L e t t h e s e p l a n e s t o u c h the s u r f a c e whose  be new  equation i s a^x" * b^y"*"* c" z*~ = QL. z + l ] ; 1  so t h a t (9)  a x + b y^=  - v  m ~ e  «=  .—  ^ „ m "  Therefore we have an e l l i p s e , ^(1)' F o r m > c the e l l i p s e i s r e a l , and f o r m < c i t i s i m a g i n a r y . axes remains c o n s t a n t , namely a : b. i s equal to  (1)  a /1 - —  The r a t i o of the s e m i The major s e m i - a x i s  , w h i c h i s seen t o be z e r o f o r m = c  Valgardsson "Line Coordinates"•  and  equal t o a f o r m i n f i n i t e l y l a r g e *  n i t e l y l a r g e t h e p o l a r plane  As m becomes i n d e f i -  ( 0 , 0, m_ ) approaches c  coinci-  dence w i t h t h e XY p l a n e . I n t h e same way we c o u l d show t h a t the s e c t i o n o f the s u r f a c e made by t h e YZ plane  i s an e l l i p s e o f semi axes  b a n d c and t h a t t h e s e c t i o n made by t h e ZX plane i s am e l l i p s e o f semi-axes c, a . of t h e e l l i p s o i d .  We c a l l a, b, c t h e "semi-axes"  I f a ~ b, t h e s e c t i o n s p a r a l l e l  to the  XY p l a n e a r e c i r c l e s and the s u r f a c e i s a s u r f a c e of r e v o l u tion.  I f a = b = c we have a s p h e r e . For the e l l i p s o i d Jr  '  = - ( a b + b c + c a ;  \  = ( a " b V ) (a"+ b^+ c*")  &  - - a'^e^.  I f c = 0, ( 6 ) becomes a x + b j - 1, and  the p o i n t - c o n d i t i o n e q u a t i o n  ( ? ) becomes  a ^ b ^ t " = 0. 1  I f a, b a r e d i f f e r e n t f r o m z e r o , then t = 0.  Hence f o r  c = 0, t h e s u r f a c e must l i e w h o l l y i n t h e XY plane* case  S  ~ - a"" b \ 1  }P  A  °-  -  o, - o.  In this  70.  I f b = c = 0, (6)  becomes a X  le  Hence the s u r f a c e has degenerated i n t o the two p o i n t s ax t1 Let (6)  a «= a A  , c « e, A  .  Equation  t h e n becomes  (10) Let  , b = h, A  (  = 0.  a^x^ A  fixed.  b^ y  w  +  °  i n c r e a s e i n d e f i n i t e l y but l e t a, , b , c, remain (  I n t h e l i m i t vie have a x  * b, y  (  -s- e, z  ^ 0.  Hence t h i s e q u a t i o n i s the l i m i t i n g case o f a n e l l i p s o i d as the semi -axes a, b, c become i n f i n i t e l y l a r g e .  I t i s t o be  n o t i c e d t h a t t r a n s l a t i o n does n o t a f f e c t the l a t t e r equation.  Th.e o n l y p l a n e w h i c h i s tangent t o t h e s u r f a c e i s the t  p l a n e (0, 0,  0).  The p o i n t - c o n d i t i o n e q u a t i o n of (10) I s b e ' A-  1  r  _  c a, a b J L s / _J L t A" K  „ « a b e, . v  I n the l i m i t , when A becomes i n f i n i t e l y l a r g e , t h i s t i o n becomes Or'" + 0s*~ •* 0t*~ = a/ b",  ,  whioh can be s a t i s f i e d o n l y by p o i n t s a t i n f i n i t y . I n t h i s case  - ^  a  ~ /S  0.  equa-  71  The H y p e r b o l o i d o f One  Sheet:-  C o n s i d e r the s u r f a c e whose e q u a t i o n i s  (11)  a x + b y - c z  = 1.  The p o i n t - c o n d i t i o n e q u a t i o n o f t h i s s u r f a c e i s found to be (12)  D c r + c a s - a b f  = a b e.  L e t a, b, e be a l l d i f f e r e n t ' from z e r o .  The s u r -  f a c e i s c l e a r l y s y m m e t r i c a l w i t h r e s p e c t t o the c o o r d i n a t e planes„ c o o r d i n a t e a x e s , and the o r i g i n .  By the same method  as employed i n S e c t i o n 3, we c a n show t h a t t h e p l a n e s e c t i o n s of the s u r f a c e p a r a l l e l to the XY plane are e l l i p s e s whose axes have minimum v a l u e s i n t h e XY plane s e c t i o n , and i n c r e a s e i n d e f i n i t e l y as the s e c t i o n i s moved f u r t h e r away from the XY p l a n e .  Thus a x + b y  = m + c  i s the e q u a t i o n of the e l l i p s e when the p l a n e passes  through  the p o i n t —- ,2, 23 ** 1 m  —  0&  The semi-axes are i n the r a t i o a : b and t h e semi-major a x i s has the v a l u e  a / m"+  c  f  which becomes i n f i n i t e l y  m l a r g e as m approaches z e r o . I n t h e same way we f i n d the YZ p l a n e a r e h y p e r b o l a s .  t h a t s e c t i o n s p a r a l l e l to  In p a r t i c u l a r ,  the s e c t i o n made by t h e p l a n e / —  i f we c o n s i d e r  , 0, 0 ) , we o b t a i n the  72.  equation b y  - e z  = m  This c u r v e  a*"  i s w e l l - d e f i n e d e x c e p t f o r m = a, and t h i s i s seen  to be the case where the p l a n e  i s a t a d i s t a n c e from t h e YZ  plane e q u a l to t h e s e m i - a x i s a of the e l l i p s e w h i c h i s formed by the i n t e r s e c t i o n  o f the s u r f a c e by t h e XT p l a n e .  We c a n d i s c u s s t h i s case e a s i e r w i t h r e f e r e n c e t o the p o i n t condi t i o n e q u a t i o n w h i c h i s c a s - a b t  = a b c  - b c r .  When r = a we have  b  c  £ t  _ b_ c  tha t i s +  The s y s t e m o f p o i n t s whose e q u a t i o n s a r e ty + tz - 1 » 0  ( 3) 1  c and (14)  -A c  ty + tz - 1 = 0  can be shown t o d e f i n e two l i n e s . of t h e l i n e j o i n i n g  (13)  F o r the d i r e c t i o n  cosines  to t h e o r i g i n ( S e c t i o n 21, Chap. I )  are cos oC -  0,  73»  cos  =  b  ,  <j/ b + e " v  cos y  «  1  o'"  ,  j/b + which, a r e c o n s t a n t . d e f i n e s a line„  I n t h e same way we can show t h a t (14)  T h e r e f o r e when ra = a, we have a p a i r o f  s t r a i g h t l i n e s through the o r i g i n . For t h i s surface  J)  --= b^c*" + c" a" - a^b ,  ^  =  Z_ = Suppose c = 0.  a b " e ( c - a" - b " ) v  v  v  9  a b c »  This case has a l r e a d y been d i s c u s s e d  under t h e e l l i p s o i d . I f b = 0 and a and e a r e d i f f e r e n t from z e r o , t h e e q u a t i o n becomes (14)  a* ! " - c^z'' = 1 -  1  (which i s V a l g a r d s s o n t s h y p e r b o l a  i n l i n e coordina t e s ) .  Itoen  b = 0 $  '«= a^c ,  I f a •= b = 0, t h e s u r f a c e i s i m a g i n a r y .  If  b = c = 0 we have t h e case ax  — x,  w h i c h r e p r e s e n t s a p a i r o f p o i n t s , as we have a l r e a d y seen.  i f we l e t a = a, A  , b = b , A  s  c = c , A  , then i t  follows that as  + b, y  - c, E  «  —  The s e c t i o n of t h i s s u r f a c e made by a p l a n e p a r a l l e l t o the XT p l a n e has t h e e q u a t i o n a, x  * b y  = — A  ,  where k depends o n l y on the p o s i t i o n of the c u t t i n g p l a n e . T h i s i s an e l l i p s e whose semi-axes a r e A a, and k both o f w h i c h become i n f i n i t e as  A  A b, , k  becomes i n f i n i t e  c  I n the same way we c a n show t h a t t h e major axes o f the h y p e r b o l i c s e c t i o n s p a r a l l e l , t o t h e o t h e r c o o r d i n a t e p l a n e s become i n f i n i t e as A  becomes i n f i n i t e .  In t h e l i m i t we have a^x^ For  this l a s t  1  equation  _ J J5.  * b/y" ~ e^z " = Oo  »j  «*£ = A = 0.  The H y p e r b o l o i d o f Two S h e e t s : Consider the equation  (15)  a x " - b^y" "- c^z " = 1. v  1  1  The p o i n t - c o n d i t i o n e q u a t i o n f o r (lj?) i s  (16)  b"" <Tr " 1  - o"a s^ - a" b" t " = a" b  0 »  75  This s u r f a c e i s s y m m e t r i c a l w i t h r e s p e c t t o t h e c o o r d i n a t e p l a n e s , c o o r d i n a t e axes, a n d t h e o r i g i n . we can f i n d  As b e f o r  the s e c t i o n s made by p l a n e s p a r a l l e l t o t h e  c o B r d i n a t e planes..  The s e c t i o n s p a r a l l e l t o t h e XY and ZX  p l a n e s a r e f o u n d t o be h y p e r b o l a s ,  and t h e s e c t i o n s by p l a n e s  para 11 e l to t h e YZ' p l a n e a r e e l l i p s e s .  Suppose t h e p l a n e  p a r a l l e l t o the YZ p l a n e p a s s e s through  the point  r x - 1 = 0. I t i s r e a d i l y seen t h a t t h e e l l i p s e s  are imaginary  unless  r ^ >^ a*". I f r = a , the e l l i p s e s For  this  degenerate i n t o p o i n t s on trie X a x i s .  surface J)-  ^  £  =  =  (G & ~ U  +  L  S^\TG'  a"  -  1  t^cT)  (a"- b""- c")  = - a'b'e* = - a'bV .  When b o r c i s z e r o , cases a r e o b t a i n e d w h i c h have been d i s c u s s e d a l r e a d y .  L e t us c o n s i d e r the case when t h e  semi-axes become i n f i n i t e ;  suppose the e q u a t i o n i s  ax  - b y  - c z =  —  A Then t h e r e i s no p a r t o f t h e s u r f a c e between t h e p l a n e s parallel  t o t h e YZ p l a n e and p a s s i n g through the p o i n t s ± a A x - 1 = 0.  If  A approaches i n f i n i t y the d i s t a n c e between these p o i n t s  becomes i n f i n i t e .  I n t h e l i m i t we have the h y p e r b o l e i d o f  76,  two s h e e t s a t i n f i n i t y . 6.  We have J  = ^ ==<^-/\== 0.  She P a r a b o l o i d ; C o n s i d e r the s u r f a c e d e f i n e d by t h e e q u a t i o n b^ y*"  (17)  i-  c's " + 2ux - 0 . 1  The p o i n t - c o n d i t i o n e q u a t i o n of ( 1 7 ) i s (18)  G^VL'S"  *  b ^ u ^ t + 2b"c"ur ~ 0.  I f b, c, u a r e a l l d i f f e r e n t from a e r o , we may w r i t e , i n s t e a d of (18),  . £ .  b"  0.  +  _ c"~  u  The s u r f a c e ( 1 7 ) I s s y m m e t r i c a l w i t h r e s p e c t t o t h e XY a n d ZX p l a n e s and t h e X a x i s .  mx - 1 » 0  (19)  is  The p o l a r of the p o i n t  , 0 , 0j«  ( S e c t i o n 1 0 , Chap. I I ) t h e p l a n e /  Translate  the o r i g i n t o t h e p o i n t - mx - 1 « 0o Then the p o l a r p l a n e w i l l be t h e new XY plane.,  Equations  ( 1 9 ) and ( 1 7 ) , r e f e r r e d t o the new a x e s , a r e r e s p e c t i v e l y 2mx - 1 = 0 , b ^ y " * ^ z " - 2umx*~ + 2ux = 0 . 1  l e t a l l t h e tangent p l a n e s pass t h r o u g h t h e p o i n t ( l ? ) ; t h a t i s x = 1_ . 2m  T h e r e f o r e we have ^ b y~ + c"2 ^  B  -  u 2m •  Hence p l a n e s e c t i o n s p a r a l l e l t o t h e YZ p l a n e a r e e l l i p s e s  77-  of  semi-axes  J-  2m and c  2m «  tt  This e l l i p s e  XL  degenerat es  to a p o i n t when m = 0; t h a t i s , the YZ plane touches the s u r f a c e a t the o r i g i n .  The e l l i p s e i n c r e a s e s i n s i z e as the  c u t t i n g p l a n e i s moved f u r t h e r from t h e o r i g i n .  I t i s t o he  n o t e d t h a t m and u must be o p p o s i t e i n s i g n f o r r e a l  ellipses.  I f u I s p o s i t i v e t h e s u r f a c e l i e s w h o l l y on the p o s i t i v e s i d e of the YZ plane*, C o n s i d e r any p l a n e p a r a l l e l to the JZ p l a n e , ( 0 , m, 0), say.  T r a n s l a t e the o r i g i n t o t h e p o i n t •L y - 1 « 0 ; m  t h a t i s the new XZ p l a n e i s t h i s p l a n e .  E q u a t i o n ( l 8 ) becomes  (Chapter I I )  For any p o i n t i n t h e new XZ p l a n e S = 0 .  T h e r e f o r e the  p o i n t - c o n d i t i o n e q u a t i o n o f t h e p l a n e s e c t i o n by the new JZ p l a n e becomes (20)  t  2r  c  XL  +  1 m"b  I t c a n eas i l y be shown the l i n e - c o n d i t i o n e q u a t i o n f o r a p a r a b o l a has the same form as ( 2 0 ) .  (l)  T h e r e f o r e the  T h i s c a n be dose by a method s i m i l a r to t h a t employed i n Chapter I I , S e c t i o n 2 . See Snyder and Sisam, p. 9 1 .  78  s e c t i o n by t h i s p l a n e i s a parabola,,  I n the same way we c a n  show t h a t s e c t i o n s p a r a l l e l t o t h e XY plane y i e l d We c a l l the s u r f a c e whose e q u a t i o n i s (17)  parabolas• an e l l i p  t i c p a r a b o l o i d , because the s e c t i o n s p a r a l l e l to one c o o r d i n a t e plane a r e e l l i p s e s and t h e s e c t i o n s p a r a l l e l t o t h e o t h e r two c o B r d i n a t e p l a n e s a r e p a r a b o l a s . I n the same way we c a n i n v e s t i g a t e the s u r f a c e whose e q u a t i o n i s (£1)  b y  - c i ' ' + 2ux = 0.  v  v  S e c t i o n s p a r a l l e l t o t h e YZ plane y i e l d h y p e r b o l a s and s e c t i o n s p a r a l l e l t o the o t h e r two c o o r d i n a t e p l a n e s parabolas,.  T h e r e f o r e (21)  r e p r e s e n t s an h y p e r b o l i c p a r a -  boloid® For ( I ? )  J  j  - ~ u (b%v  -  &• »  O,  b'oV; 0,  /\ <= b' G u . 6  C  F o r (21)  J  « - u"(b"-  o ),  ^ = - b^u*, ^  -  yield  0,  A - - b* c u " . 4  I f u = 0, we have by  + c z  =0  by  - c s  =0.  or  u  The f i r s t i s a s p e c i a l case o f the i n f i n i t e e l l i p s o i d ,  and  t h e second r e p r e s e n t s a p a i r o f i n f i n i t e l y d i s t a n t p o i n t s F o r t h e s e two cases  - ^  -  =  =  0  o  When e = 0, we have b^y " + 2ux  = 0.  This i s a p a r a b o l a i n t h e XY p l a n e I n - b u , jj< = ?  = A  u  •= 0.  t h i s ease 3- =  The p o i n t - c o n d i t i o n e q u a t i o n  reduces t o  t h a t i s , t = 0, and the p o i n t s a l l l i e i n the XY p l a n e .  (3) 7.  I n v a r i a n t s f o r the V a r i o u s E q u a t i o n s ; -  Equation = 1  -  b y - c z = l  +  a x"+ b > ^ v  ax  +  A  c"z  v  _  +  ?  ?  a x - b y ~ c z < = l  -  -  ?  ?  b^y^-  c^z'-s- 2ux  « 0  -  0  +  _  b^y"-  c"z""+ 2ux  = 0  +  0  l  ?  (1)  V a l g a r d s s o n " l i n e G o B r d i n a t e s " , Ch. I I I .  (2)  V a l g a r d s s o n , Ch. I I , S e c t . 4.  (3)  I t i s u n d e r s t o o d t h a t a l l c o e f f i c i e n t s a p p e a r i n g i n the f o l l o w i n g t a b l e a r e d i f f e r e n t from z e r o .  8o,  Equation ax  v  A  + by" ' + ess " = 0 1  ax  v  3  3-  1  0  0  0  0  + by" = 0  0  0  0  0 •*  a x + b V  = 1  0  .0  0  a x - b y  <= 1  0  0  0  b"y%- 2ux  = 0  0  0  0  •f  8, X  ~ X  0  0  0  0  T  v  S i n c e these a r e a l l t h e p o s s i b l e e q u a t i o n s , we c a n say t h a t when. £ ^ 0,  £ 0 we have an e l l i p s o i d o r an hyper-  b o l o i d . o f one o r tw o s h e e t s .  If A ^ 0, «® = 0 we must have  e i t h e r an e l l i p t i c o r h y p e r b o l i c p a r a b o l o i d .  If ^  =  =. jj - 0, and J? =f^ 0 we have a plane c u r v e , w h i c h c a n be an e l l i p s e , p a r a b o l a , or hyperbola.  I fA  -  = ^  *= _9 ^ 0,  the e q u a t i o n r e p r e s e n t s two p o i n t s , o r e l s e may be s a t i s f i e d o n l y by p o i n t s a t i n f i n i t y . The o r i g i n a l e q u a t i o n r e p r e s e n t s two p o i n t s when i t has two l i n e a r  f a c t o r s i n x , y, z, f o r w h i c h a n e c e s s a r y  c o n d i t i o n i s t h a t the d i s c r i m i n a n t & v a n i s h .  81  CHAPTER I T R e d u c t i o n of t h e G e n e r a l 1.  Equation  G e n e r a l Statement I n t h i s c h a p t e r we s h a l l c o n s i d e r t h e r e d u c t i o n o f  the g e n e r a l e q u a t i o n when represents 2.  A  G, t h a t i s , when t h e e q u a t i o n  an e l l i p s o i d , h y p e r b o l o i d ,  or paraboloid*  Reduction of the Point-Condition  Equation;-  Let the equation Ar"+ (1)  BS^-J- Ct"+ 2 F s t + 2Gtr + 2Hrs * 2Ur + 2Ys "  '  + 2Wt *" D = 0  be t h e p o i n t - c o n d i t l o n equat i o n of t h e s u r f a c e ax"* by"*'+ cz '-!- 2 f y z + 2gzx + 2hxy + 2ux + 2vy 1  (2)  "  "  * 2wz +"d « 0.  We have seen ( S e c t i o n 4, Chap. I I ) t h a t there i s a t l e a s t one p r i n c i p a l p l a n e .  Take t h i s p l a n e f o r the XY p l a n e  i n a new system o f c o o r d i n a t e s .  The degree of ( l ) w i l l be  u n a l t e r e d by t h e t r a n s f o r m a t i o n . By s u p p o s i t i o n the XY p l a n e b i s e c t s a l l chords p a r a l l e l t o the Z a x i s j therefore i f r x (  + s  (  y * t  (  z - l  = 0  be any p o i n t on t h e s u r f a c e , the p o i n t r x * s y - t, z - 1 = 0 (  w i l l a l s o be on the s u r f a c e .  From t h i s we see t h a t i n t h e  transformed equation F = G- = W = 0. The reduced e q u a t i o n t h e r e f o r e i s A r % Bs" + Ct"+ 2Hrs + 2Ur + 27s + D = 0. l\Tow r o t a t e t h e X, Y axes through a n a n g l e O g i v e n by t h e relation tan  2&  =  2H  .  A-vB a c c o r d i n g t o t h e t r a n s f o r m a t i o n s (4j>) o f Chapter I I , namely r = r'oos & + s ' s i n Q , s = - r' s i n 0 + s' cos 6 , t « t'. ; Dropping p r i m e s , we get a n e q u a t i o n of the form O)  A r % B s + C t + 2Ur + 2Ys + D = 0» w  v  ( i ) L e t A, B, C be a i l zero.  f i n i t e and d i f f e r e n t  We c a n t h e n w r i t e e q u a t i o n (j>) i n the form A  *  (r  V  TJ  ) \  A /  B/s + T ) \ V. B"7  Ct"= U~4 ¥ " ~ A . B"  DHD'.  Hence, by changing the o r i g i n t o t h e p o i n t |x  +  J y - 1 - 0  by means o f f o r m u l a e ( 4 6 ) o f Chapter I I , we o b t a i n Ar"" + B s " + C t " » 3)'. If D  1  be n o t z e r o we have  from  83.  w h i c h we can w r i t e i n t h e form (4) a  b  c  r" a  s^ b  r "  s"*  or (5)  „  t c  or (6)  1  ac c o r d i n g as D , D' , D' A 1" U ;  t  -  1,  E  a r e a l l p o s i t i v e , two p o s i t i v e  one n e g a t i v e , o r one p o s i t i v e  and two n e g a t i v e ,  respectively*  ( i f a l l t h r e e are n e g a t i v e the s u r f a c e i s c l e a r l y I f 3' be a e r o , we  (7)  imaginary.)  have  A r " + Bs' -s- C t "  =0.  u  ( i i ) L e t A, any one  o f the c o e f f i c i e n t s , he z e r o .  W r i t e the e q u a t i o n i n the form 2Ur  + B^s  + Y  y  Ct  V  + D - Y  v  = 0.  I f U be not z e r o , by c h a n g i n g the o r i g i n t o the p o i n t Qx + TJ y -* 1 = 0, whe re Q, = 1 / D - T , , 2tr we  (8)  and  can reduce the e q u a t i o n to B s " + C t " + 2Ur  = 0,  84.  I f U = 0, we have the form (?)  Bs" + C t  v  + D' = 0,  o r , i f I)'= 0, the form (10)  B s " + C t " = 0. ( i i i ) L e t B, C, two ©f t h e t h r e e c o e f f i c i e n t s , be  zero.  We then have A ^ r + ] J j \ ZJa * D' - U A  - 0,  I f we t r a n s l a t e the o r i g i n t o the p o i n t  the e q u a t i o n reduces to the form (11)  r.*" = 23ts. I f , however, ¥ = 0, t h e e q u a t i o n i s e q u i v a l e n t to  (12) 3o  r*" = 3c' © go F i n d the E q u a t i o n s of the Center o f a Conicoid.:I f the o r i g i n i s the c e n t e r of the s u r f a c e , i t i s  the middle p o i n t o f a l l chords p a s s i n g t h r o u g h i t ; i f r,2c-t-s y*t z-l 1  |  = 0  be any p o i n t on the s u r f a c e , the p o i n t - r x - s y - t, z - 1 (  will  = 0  (  a l s o be on the surface» Hence we have Ar," + Bs," * C t ^ + 2Fs, t, * 2Gt, r + 2Hr s, + 2Ur + 2Vs, (  ~ '  "  "  •'• 2wt, +  D  «  (  o;  85  and Ar," + Bs^ + Ct, + 2Fs, t, + 2Gt, r, + 2Hr, s, - 2Ur, v  - 27a, -• 2Wt + D « 0; (  the r e f o r e Vs + Wt, «= 0.  TJr  ;  S i n c e t h i s r e l a t i o n h o l d s f o r a l l p o i n t s on t h e s u r f a c e , we must have U, V, W a l l z e r o .  Hence, when t h e  o r i g i n i s the c e n t e r o f a c o n i c o i d , t h e c o e f f i c i e n t s  of r ,  s, t a r e a l l z e r o . Let o r x + ^ y + y z - l « = 0 be t h e e q u a t i o n o f the c e n t e r o f the s u r f a c e ; then i f we take.the center f o r o r i g i n , the c o e f f i c i e n t s the t r a n s f o r m e d  e q u a t i o n w i l l a l l be z e r o .  o f r , s, t i n The transformed  e q u a t i o n w i l l be ( S e c t i o n 46, Chap. I I ) A ( r V «< )  V +  B(s + ^  f + C ( t + / ) + 2 F ( s + ^ ) ( t i «f ) x  + 2 G ( t + y ) ( r + a<) + 2 H ( r + ^ " ) ( s + ^ + 2tJ(r + <<) * 2 7 ( s + /3 ) + 2W(t +  ) + D « 0.  Hence"the e q u a t i o n s g i v i n g the c e n t e r a r e  (13)  Ae(.  * H^  + G T  H<<  + B /3  + F 2T + 7 = 0,  G *c + F y 3  + IT « 0,  + C r  + W •» 0.  The r e f ore (13a)  )  H  G  TJ  A• G  U  A  H  TJ  A  H  G  B  F  ¥  H  F  T  H  B  t.  H  B  F  F  C  G  C  ¥  G  F  W  G  F  C  Qf  The p o i n t - c o n d i t i o n e q u a t i o n o f the e o n i c o i d when the c e n t e r i s a t the o r i g i n i s (14)  Ar"+ B s + Ct"+ 2 F s t + 2 G t r + 2Hrs + D ' = 0, v  where D' i s o b t a i n e d from (3) by p u t t i n g r = <*" , s -/3  ,  t = tY • M u l t i p l y e q u a t i o n s (13)  i n o r d e r by •< , ^  ,  %  and s u b t r a c t the sum from D j t h e n we have. (13)  V From (13)  = TJe< + Y/3 + W * + D*  and (15) we have A  H  G TJ  H  B  F  Y  G  F  G  W  TJ:-  Y  w D-D'!  I » 0;  therefore (16)  D' A  H  G-  ,A  H  G  IT  H  B  F  S  B  •0  7  G  F ,c  G  F" 0  W  U" Y  w  D  w h i c h may be w r i t t e n (17)  v'c£>  «  A .  I t i s seen t h a t t h e e q u a t i o n o f the c e n t e r i s g i v e n by  (18) where  x  +  V  y +  s -  £>  - 0,  e t c . , a r e the c o - f a c t o r s o f IT, Y, e t c , I n Z\  87.  4.  The D i s c r i m i n a t i n g C u b i c ; We have seen ( S e c t i o n 2) t h a t by a p r o p e r c h o i c e o f  r e c t a n g u l a r axes Ar*+ B s ^ * G t % - 2 F s t + 2 G t r + 2Hrs can always be reduced t o t h e form c< r + ^ s""+ y t "; v  -1  and t h i s r e d u c t i o n can be e f f e c t e d w i t h o u t changing t h e o r i g i n , f o r t h e terms o f second degree a r e n o t a l t e r e d by t r a n s f o r m i n g t o any p a r a l l e l a x e s . How r + s^+ t *Is u n a l t e r e d by a change o f r e c t a n g u v  lar  axes t h r o u g h t h e same o r i g i n .  Hence, when t h e axes a r e  so changed t h a t A r + Bs^+ C t % 2 F s t + 2Gtr ->- 2Hrs v  becomes r^+ (h s^+ V t \ (19)  Ar^+ Bs^+ GtN- 2 F s t +' 2Gtr + 2Hrs - J ( r " + s" + t" )  w i l l become (20)  ^ r + ^3 s % v  f t * " - J'(r '+ a*"* t x  .  B o t h t h e s e e x p r e s s i o n s w i l l t h e r e f o r e be t h e p r o duct o f l i n e a r f a c t o r s f o r t h e same v a l u e s o f J  •  The  c o n d i t i o n t h a t (19)  i s the product o f l i n e a r f a c t o r s i s  (21)  |A - J  H  II  B -  G  F  G  f  F  1=0,  C -jT  But (20) i s the p r o d u c t o f l i n e a r f a c t o r s when J i s  88.  e q u a l to oC , ^  , or  roots of (21).  The e q u a t i o n when expanded i s  f*  ,  Hence oc , p  - f (A + 3 + C) +  a r e the t h r e e  s  f (AE + BC + CA - F*~- G-*"~ H*)  - (ABC + 2FGH - A3?*"- SG^-CH") = 0, or  a s ) y -&r*w -  -  T h i s e q u a t i o n i s c a l l e d the " d i s c r i m i n a t i n g c u b i c " . 5"  D i s c u s s i o n f o r JQ £ From e q u a t i o n  0:-  (18) we see t h a t t h e r e i s a d e f i n i t e  center a t a f i n i t e distance, unless one  «= 0.  o f ^ , <V , Q/0 I s d i f f e r e n t from zero  i s a d e f i n i t e center a t an i n f i n i t e If  If  0 and  (i.e. A  0) t h e r e  distance.  be n o t z e r o , change to p a r a l l e l axes  through  t h e c e n t e r , and the e q u a t i o n becomes Ar"+  B s ^ * C t % 2 F s t + 2Gtr + 2Hrs + 35 ' « 0  where D' i s found'as i n S e c t i o n 2.' How, k e e p i n g  S  the o r i g i n  f i x e d , change the axes i n s u c h a manner t h a t t h e e q u a t i o n i s reduced to t h e form  oC r % fl> s % Then, by S e c t i o n 3, c*T , /3 , Y  Vt  + 3)' = 0.  w i l l be the t h r e e r o o t s o f  the d i s c r i m i n a t i n g c u b i c . Since  D'-  /\ , the l a s t e q u a t i o n may be w r i t t e n  JiU  r % &/!> s"* ^ / t "  i n the form  I f the t h r e e q u a n t i t i e s  "P*  + Z\  =0.  , 'QJJL. ,  °^J^  are  8?.  a l l n e g a t i v e , the s u r f a c e i s an e l l i p s o i d ;  i f two o f them  are n e g a t i v e , t h e s u r f a c e i s an h y p e r b o l o i d o f one s h e e t ; i f one i s n e g a t i v e , the s u r f a c e i s an h y p e r b o l o i d of two s h e e t s ; and i f t h e y a r e a l l p o s i t i v e , t h e s u r f a c e i s an imaginary  ellipsoid.  We have shown i n Chapter I I t h a t the g e n e r a l  equation  c a n be reduced t o one o f the t h r e e forms (23)  az + by* + oz"  -1  (24)  ax"+ by"-*- c s ~ 0,  (23)  b y % cz*" + 2ux*= 0*  v  = 0,  w  We see from S e c t i o n 7"of Chapter I I I t h a t always r e q u i r e s A b.  0  £  -fc 0, w h i c h i s t r u e o n l y f o r (23)«.  D i s c u s s i o n of the Case j9 = 0 : When  must be z e r o .  = 0, one r o o t o f the d i s c r i m i n a t i n g c u b i c From S e c t i o n 4, Chapter I I , we see t h a t one  p r i n c i p a l p l a n e must be the p l a n e  (0, 0, 0)«  0, vie  If  must have tw0 f i n i t e p r i n c i p a l p l a n e s , and t h e r e f o r e the c e n t e r i s a t i n f i n i t y and must l i e on the l i n e of i n t e r s e c t i o n of the two f i n i t e p r i n c i p a l planes© If $  - 0 and Z\  center i s a t i n f i n i t y .  0, e q u a t i o n (18) shows t h a t t h e  S i n c e one r o o t o f the d i s c r i m i n a t i n g  c u b i c i s z e r o , the e q u a t i o n can e a s i l y be s o l v e d ; l e t t h e r o o t s be 0, oC > ^3 .  F i n d the d i r e c t i o n c o s i n e s of t h e  p r i n c i p a l a x i s by means o f e q u a t i o n s  (16),  Chapter I I , and  take t h e Z a x i s p a r a l l e l to Hie p r i n c i p a l a x i s *  The  90,  e q u a t i o n w i l l then become v  or,  by  s  +  (t> t* + 2 U ' r + 2V's + ZW't + JD = 0 ,  change* o f o r i g i n , «=<• s + v  t % - 2U 'r = 0 .  Hence we have the s u r f a c e , w h i c h , expressed  I n plane  coordinates, i s ay + bs^* 2ux = v  since A 7.  ^  O,^ ^ 1  0.  Summary;L e t us i n v e s t i g a t e the g e n e r a l e q u a t i o n  of a e o n i c o i d .  I f A ^ 0 and d 7^ 0 , i t f o l l o w s t h a t JQ £ 0 and w e have an e l l i p s o i d , or hyperboloid.  If &  i s p o s i t i v e we have the  h y p e r b o l o i d o f one s h e e t .  If &  i s n e g a t i v e we d i s c o v e r t h e  nature  o f the s u r f a c e by s o l v i n g t h e d i s c r i m i n a t i n g c u b i c -  t h r e e r o o t s w i t h t h e same s i g n denote an e l l i p s o i d and r o o t s w h i c h d i f f e r i n s i g n denote an h y p e r b o l o i d o f two sheets,, I f Z\ £ 0 but d = 0 , i t f o l l o w s thatj@ = 0 „ ^ g i v e s us an e l l i p t i c A  This  or h y p e r b o l i c p a r a b o l o i d a c c o r d i n g as  i s n e g a t i v e or p o s i t i v e , r e s p e c t i v e l y . The p l a n e c u r v e s a r e found to be those s u r f a c e s f o r  w h i c h a l l the i n v a r i a n t s except 3 plane  curve i s a p a r a b o l a .  vanish.  I f d = 0 the  I f d / 0 the plane curve i s an  (1)  Snyder and Sisam, p. I 5 0 .  (2)  S e c t i o n 7, Chapter I I I .  91  e l l i p s e o r h y p e r b o l a a c c o r d i n g as J) i s n e g a t i v e o r p o s i t i v e , respectively* A p a i r o f points i s given Mien p r o v i d e d t h a t the e q u a t i o n i s f a c t o r a b l e  = ^  ~  = £± ~ 0  e  Otherwise the e q u a t i o n r e p r e s e n t s an i n f i n i t e c o i d or an i n f i n i t e  conic *  coni-  BIBLIOGRAPHY 1.  Lambert " A n a l y t i c  Geometry , 13  The M a c m i l l a n Co., 1904. 2.  S m i t h " S o l i d Geometry", M a c m i l l a n and Co., 188?o  3c  Snyder and Sisam " A n a l y t i c  Geometry o f Space",  Henry H o l t and Co.,  1  4.  1914.  Tanner and A l l e n " B r i e f Course i n A n a l y t i c Geometry",' A m e r i c a n Book Co.,  5.  valgardsson "Line Coordinates", M.A.  6.  1911.  Thesis a t U n i v e r s i t y o f Manitoba.  W i l s o n " S o l i d Geometry and Conic S e c t i o n s " , M a c m i l l a n and Co., 1898.  

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