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Plane coordinates Parker, Sidney Thomas 1934

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'OH 'D3V 'OH iVD BLAKE GOOEDIK&IES by Sidney Thomas Parker U . B . C J L J B R A R Y CAT. m.LElAlU^J&i&M ACC. NO! l?$JdL A Thesis submitted f o r the Degree of MASTER OF IRIS i n the Department THE IMIYERSITX OF BRITISH COLUMBIA. October, 1934 X » TABLE OF CONTENTS CHAPTER I X « Intr0duotion. 2 . Fundamental D e f i n i t i o n s . 3 . CoBrdinates of Planes. 4. P a r a l l e l P lanes. D i r e c t i o n Cosines of 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 of Axes. ?. Standard Form of the Equation of a P o i n t . XO & Equations of P o i n t s (Continued). 11. Distance Between Two P o i n t s . 1 2 » D i v i s i o n of a Segment i n a Given R a t i o . 1 3 . Plane Through Three P o i n t s . 14. The Expression^/ ( x x ~ x f )*" + ' ( y v ~ Jt Y * 1 5 « Distance Between P a r a l l e l " P l a n e s . l 6 e Distance to a P o i n t from a Plane. If] e Angles Between Line and Planej Plane and PXane * 18 © Two-Point Equations of a Line» 19* Equations of a Line (Continued). 20. Two-Plane Form of the Equations of a Line 210 D i r e c t i o n Cosines o f a Line . I I . 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. j 24. Three-Plane Equation of a P o i n t . 25.. T r a n s l a t i o n of Axes. 260, The Degree of an Equation i s Unchanged by Ro t a t i o n and T r a n s l a t i o n Transformations, CHAPTER I I - The general Second Degree Equation 1* Equation o f the Tangent P o i n t . 2. C o n d i t i o n that a P o i n t L i e s on the S u r f a c e 3 » Locus of Middle 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 Plane* 5 . ' The Roots o f ( 1 7 ) . 6. E l i m i n a t i o n of the ys, az, z Terms. 7 . Reduction when d ^ 0 . 8. Reduction when d = 0. 9. Center of the Conicoido 10» P o l a r Plane. 11. R e c t i l i n e a r Generators 12* Invariants© XXX • CHAPTER I I I C l a s s i f i c a t i o n of Surfaces 1. Review of Previous f o r i . 2. The Sphere. 3 . The E l l i p s o i d . 4. The Hyperboloid of One Sheet. J?. The Hyperboloid of Two Sheets. 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 Various Equations. CHAPTER IT Reduction of the General Equation 1. General Statement. 2. Reduction of the P o i n t - C o n d i t i o n Equation. 3 . To Find the Equation o f the Center of a Conic o i d . 4. The D i s c r i m i n a t i n g Cubic. 5 . D i s c u s s i o n f o r 0 . 6 . Discussion f o r J © *= 0 . 7 • Summary. BIBLIOGRAPHY PLAHE OOOBDIHAIES Chapter I 1. Introduction: - The primary purpose of t h i s t h e s i s i s to develop the ordinary r e l a t i o n s of s o l i d a n a l y t i c geometry "by the use of plane-coordinates. The significance of var i o u s equations of the C a r t e s i a n system w i t h reference to t h i s new system w i l l also be d i s c u s s e d . As far as possible, the treatment parallels the treat- ment of line-coBr&lnates, as contained i n the theses sub- mitted, by ValgarcLsson of Manitoba and Heaslip and James of B r i t i s h Columbia for the degree of Master of Arts. 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 in three mutually perpendicular straight lines x'ox, Y/0Y, z'oz, which are called the X, Y, Z axes,respectively. The X a x i s i s formed by the intersection of the ZX 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 called the origin. The cus- tomary conventions with regard to sign are observed. For example, the directions x' OX, Y'OY, Z'OZ are considered positive, and the directions XOX , YOY , 20Z are considered negative* The ooBrdinates of a plane are defined to he the re- ciprocals of i t s intercepts on the coordinate axes. Thus the plane ABO i n figure ( l ) has coordinates (a, b, c), since In Cartesian coSrdinates the point (a, b, e) is such that its. directed perpendicular distances from the YZ, ZX, XY planes are a, b, c, respectively. The plane ax * by * cz - 1 = 0 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 repre- sented uniquely by (a, b, c). The following is a summary of some special cases: (i) 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 the co8rdinates (a,oO where a i s f i n i t e . ( i i i ) The coordinates of a plane parallel to that given i n ( i i ) are (o, b, c), where b and e are f i n i t e . (iv) The ooOrdinates of a plane parallel 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 origin and oblique to a l l three axes has the coOrdinates (oO too>oo)• It is to be noted that the coordinates in ( 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. Pa r a l l e l Planes:- Theorem: The necessary and sufficient conditions for the parallelism of two planes (x,, y, , z, ) and (x x, 7X » z x) are (2) ±1 = I I - l l . Fig. (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 x ) , are p a r a l l e l . Then they cut the coordinate (1) planes in parallel lines, that i s , AjB, and A 2B t are pa r a l l e l . Hence OAV „ 0Ba . OA/ OB/ (l) Wilson "Solid Geometry and Conic Sections™, p. 12. In the same way II * Z' ' Therefore X / B y ' OS Z ; x7 yT "zl * The conditions are also sufficient. Suppose rela- tions (2) hold. Then A,B, is parall e l to A ^ , and B, G, is parallel to B̂ O,.• Hence plane A,B,C, i s parallel to (1) plane A tB lC 3 L. This theorem i s equivalent to the statement that the planes (a, b, c) and (ka, kb, ke) are parallel* In the Cartesian system, two points whose coordinates satisfy equations (2) are eollinear with the origin, and conversely. If two planes: A,ac *• B, y * 0, z - 1 « 0f A^x * B ^ * C^z - 1 = 0 are parallel, then A, - B, = 2± X~ B t U x ' and conversely. (1) Wilson, loc.oit., p. 13. j>» Direction Gosines of a Mne:-'"7 Let A be any directed line in space, and l e t be the line through the origin with the same direction as Let c< tfi , y be the angles between the X, Y, Z axes, res- pectively, and By definition these are the angles which makes with the axes. Ehey are called the "direction angles" of the line st- , and their cosines are called i t s "direction cosines". The direction cosines w i l l be denoted by A P /A, ^ respectively. z (1) As in Snyder and Sisam "Analytic Geometry of Space". ( 2 ) See Snyder and Sisam, p. 3 . It i s easily proved that the relation holds. CD 6* Angle between Two Directed Lines:' Suppose that and are two directed lines with direction cosines A, , AJ, , and A »y<v s i<4, respectively. In solid geometry the angle between two directed lines is defined to be the angle between the two similarly directed lines through the origin. ^ ' s z 0 / T y Fig. (4) (1) Snyder and Sisam, p. 6. (2) As in Snyder and Sisam* 8 In figure (4), ^ 'and -A are parallel to and A. respectively. If OP i s any segment taken along the positive direction of ^ ' , PQ i s perpendicular to £,', and PR i s perpendicular to the plane XOY at R. Perpendiculars RT, RS, SU are drawn to OQ,, 0SS 01, respectively, as shown i n the diagram. The angle between and A. i s the angle 0 i n the figure. Bow therefore rtrtei A oa ou * m * TO, . cos e = ^ « Q P — » ~ OU OS UT SR TO, PR cos a = — * — * g'oP ; and hence O) cos © = /M*.* ft'/*-*- + • 7. 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 origin to the plane, and << , ̂  , Y are the direc- tion angles of this perpendicular. ABC i s any plane (x, y, z) and OP is the perpendicular from the origin 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 is perpendicular to AB, their line of intersection. Therefore GQ. and OQ, are both perpendicular to AB. Since the triangles GQB and AOB are similar, i t Fig* O) follows that OB ! AB so that 1 1 Art OA.OB z *Y l V x l y* In the triangle Q,0C GO, ̂  = 0G* * 00/ ; therefore GO, _ fL H I / x - » y a . "1/ + y" 1 z J ( x a + y 1 ) " Again, the triangles OPQ,, GOQ, are similar; therefore OP _ 0 0 from which, we obtain „/ 2. X V x + y * Since OP i s perpendicular to the plane ABC OP x z cos * . ^ y x + y + similarly cos ̂  a 7 /x 2" + - i - x cos a- - y + z z / x * * " x y * z Therefore COS °< a y + s "' x (4) *0Bfi 7 GOB A B y / Z X x' * x + y + z Vx z + y * + z"1" cos J = z Vx 2" * y X X * Z ( l ) The perpendicular from the origin to a plane i s . always considered positive* 1 1 c She inverse transformations are (<>) 7 oos r cos ••/* cos f. 8 « Rotation of Axes;- 12 Fig. (6) l e t the original reference system he rotated about the origin to a new position so that the new X axis has direction cosines A, t/U,, Vt , the new T axis has direction 12* eosines A,. ,yav > ̂  » and the new Z axis has direction cosines 3̂ »/S » ^ » a 1 1 wi"ttL respect to the old axes. We shall denote the new axes by primed letters. Suppose the X' axis cuts any plane (x, y, z) at , as i n figure (6). Denote the angle POAx by 0 . By equa- tions (4), the direction cosines of OP are cos o< = cos cos Y From equation ( 3 ) we obtain. ._ • X . .. •i/.x1^. * y y * z1" V x * y « •. , .. ,..,Z : . • z v x ̂  * y 1 in * A, x +^.y y x + y But, from figure (6), i t follows that A OP x COS. U st •QJ"*/ SB - / x 1 " * y 2 ^ z i By equating these two values for cos 0 , we get similarly (6) , y = A x * + / \ / * <2 , and 1 3 . The inverse transformations are 17) y = yA« *' •* + /u) x'} Y/e can express results (6) and (7) i n tabulated form as follows; ( 8 ) 2/' J 2 These relations are exactly the same as those ob- tained for 0artesian coOrdinates* 9. Standard fform of the Equation of a Point;- The standard equation of a point w i l l be that rela- tion which involves the directed perpendioular distances from the three coordinate planes to the point. 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. Rotate the axes so that the x' axis passea through P. Then i s the x' coOrdinate of a l l planes which pass through P. Therefore Fig. (7) But, from (6), we have where Ax «• 0 R = 19) Therefore s 4/1 r * s * t y r * s + t and hence U P ) rx * sy » tz r + s r + s + rx +• sy * tz - 1 = 0 , 15. We must now show that a l l planes whose coordinates satisfy (10) pass through the given point. Let (a, b, c) "be a plane whioh does not pass through P, hut whose coordinates satisfy (10)• Then (11) ra * sb .* tc - 1 * 0. From section 4, the coordinates of a plane through P and parallel to (a, b, c) are (lea, kb, ko). Since these coordi- nates must satisfy ClO), i t follows that (12) k(ra * sb * tc) 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 direction cosines of OP are given in (9). If we denote the length of OP by f ,. equation (10) may be written (13) A x + yu.y --L a 0. We shall c a l l (13) the "directed" equation of the point. If (^ , e< t / s , y) are the polar coordinates of a plane, whose intercept coordinates are (x, y, z), passing through the point rx * sy * tz - 1 » 0 then 16. JL5 . + §y . tz + y v * z" f/x" * y"" + /x"- * y"~ * z1" V ? •*• y*" * z' Therefore (14) ^ = r oosc< * a cos/? + t cos ¥ We shall e a l l (14) the "polar I r equation of the point. The equation of the origin 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 is : TX - 1 s o, and the equation of a point i n the XT plane i s rx t sy • 1 B 0, 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 origin and ̂  9^u,t 4/are the direction cosines of the normal to the plane. The plane rx - 1 = 0 is parallel to the YZ plane* ( l ) Snyder and Sisam, p. 34* 17. 11* Distance "between Two Points:- and Let two points P; and P x be denoted by the equations r f s * st y * t,z - 1 = 0 Fig. (8) Let the lengths of P, \ , OP, , OP^ be d» 9 fK res- pectively, and let angle P; OP^be Q • We have and hence, from ( 3 ) and (?), i t follows that d"" * ( r # v * a,* * t,"*" )+(r ~ * sj" * t ^ ) - 2 ( r , r f c + s ( s z +t,t a), so that (15) a =y ( r t - r ) % (a,. ̂  S / ) % ( t ^ - t r. 1 8 . 12• Division of a Segment in a Given Ratio:- Let the segment be defined by the two points given in section 11, and let the given ratio of division be h : lu Suppose that P, the division point, has the equation rx * sy + tz - 1 = 0« Hence On solving for r we obtain 1 9 . ( 1 6 ) Similarly t = _ 3cs, * hs. - • h + 1c let, + ht 3 1 3 . Plane through. Three Points Let the equations of the three distinct points P; , P 2, ? 3 be r, x * s,y * t,z - 1 « 0 , r z x * szj * t^z - 1 = 0 , r ^ i • ajy.* t 3 z f 1 • 0 , respectively. If these equations are solved for x, y, z, we obtain the coordinates of a plane passing "through the three points. Finite solutions are possible provided that r, s. t, s, t % r3 s, t, If .A ~ 0, then each element of any one row i s a linear com- bination 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 is •rx * ay + tz > 1 » 0, where 2 0 . r = ~ l + 3c xr x + + From (16) we see that the point P is collinear with P, and P r» Therefore any plane through P, and P z must pass through P. • • • • • • • • • From (9) we see that the vectors OP and 0P3 are one and the same straight l i n e . Therefore the origin, P, and Pj are collinear. Hence, any plane passing through P and P3 must pass through the origin, 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 is parallel to the line of inter- section of the other two. The condition for this exception is /\ .** 0. 14. The Expression 1/(x, - x, ) x + (y t - y, )*" + ( z t - z( Let © be the angle between the perpendiculars from the origin to two planes (xf , y, , z, ) and (x a, ŷ  , z 2) and let d be the distance between the feet of these perpendi- culars. Then d1" «* • f>* + - 2f, / \ QOS Q , where -ft and A are the lengths of the polar normals as given i n (4) and cos Q i s determined by the r e l a t i o n ( 1 7 ) cos 0 »• .•j/ x, v + y,v + 2"• / x j " + yj" + z^ that i s . -j i ( x x + y y + z z ) x , + y, + z, -x. + y- + 0*. + y, + z, + y t +zj or ,x x," + y," -J- z,v + x^ + yj- • -2(x, x^ + y, y^ + z, z j — — — — — — — — — — — — _ — — — — — — — ^ + y," + z,1- ) ( x V * yj- + ) which reduces to (IS) d. - - x ) % (y x - y, ) X + (z„ -z, f . 1 3 . Distance between P a r a l l e l P l anes:- The distance between the p a r a l l e l planes (x, y, z) and (lex, Icy, kz) i s equal to the dis t a n c e between the f e e t of t h e i r p o l a r normals. From equation ( 1 8 ) we ob t a i n ( 1 9 ) 3c - X x + y + z 1 6 . Distance to a P o i n t from a Plane:- Let the poin t be defined by the equation r x + sy + t z - 1 = 0 and the plane by the coordinates (x ( , y( , z , ) . Through the p o i n t draw a plane w i t h coordinates, say, (toe, , Icy; , k.z( ), p a r a l l e l to the given plane. Then the distance to the 22 p o i n t from the plane i s equal t o the distance between these two planes. Since the new plane passes through the gi v e n p o i n t , we have k ( r x , + sy, + t z , ) - l = 0 ; that i s k - I r x , + sy, + t z On s u b s t i t u t i n g t h i s value f o r k i n (19) , we o b t a i n ( 2 0 ) d rx, + sy, + t z , - 1 Theorem: Two p o i n t s P, , P t whose equations are r, x + s, y + t, z - 1 = 0 , i \ x + s,y + t t z - 1 = 0 , r e s p e c t i v e l y , are on the same side or on opposite sides of the plane (x, , y , z, ), acc o r d i n g as i t s coordinates g i v e the f i r s t members of the equations of the p o i n t s l i k e or u n l i k e s i g n s . For, l e t the p o i n t of i n t e r s e c t i o n of the l i n e P, P^ and the plane be P whose equation i s r x + sy + t z - 1 = 0 , where r «= m, r, * m^r^ ^ s = m.s, i m ta 4 , t = m,t, + m t t 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^(ru x, + s^y, + t i Z < - l ) = 0 . I f r, x, + s, y, + t, z( - 1 and r v x , + s^y, + t ^ z , - 1 have u n l i k e s i g n s , then m, and mx have the same s i g n , and the p o i n t P l i e s between P ( and P z . I f r, x,+ s, y, + t, z, - 1 and r ̂ x, + s vy ( + t l z / - 1 have the same s i g n , then the numbers m,, mt have opposite s i g n s , hence the p o i n t P i s not between P, and P x . A p o i n t whose equation i s r x + sy + t z - 1 » 0 w i l l be considered to be on the p o s i t i v e or negative side of the plane (x,, y , zt ) according as the expression 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 Just proved we can say that the distance to a p o i n t from a plane i s p o s i t i v e or negative according as the p o i n t and the o r i g i n are on the same side or on opposite s i d e s of the plane. 1 7 . Angles between Line and Plane; Plane and Plane:- The angle between a l i n e and a plane i s the comple- ment of the angle between the l i n e and the p o l a r normal to the plane. I f A,yu, 1/ are the d i r e c t i o n cosines of a l i n e which makes an angle Q w i t h the plane (x, y, z ) , then from 24. (3) a M (4) we get (21) s i n 6)= / U • y U y + V z ^ / x ^ + y " + z x The angle between two planes i s equal 10 the 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 Equations 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 s t r a i g h t l i n e s i nce the t o t a l i t y of planes, which pass through the two p o i n t s s i m u l t a n e o u s l y , d e f i n e a l i n e . Hence the simultaneous equations (22) r,x + 3,7 * t, s - 1 « 0 , r x x + s 2 y + t z z - 1 = 0 , give the equations" of the l i n e . , We s h a l l r e f e r to (22) as the "Two-Point" equations of a l i n e „ 1 9 . Equations of L i n e s (Continued);- The most general equations of 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 esses: ( i ) A coordinate a x i s . The X a x i s has the equations r x - 1 = 0; ox + oy + oz - 1 = 0 . (1) I t i s understood t h a t r , s, and t are not zero i n the f o l l o w i n g work. 25» ( i i ) A l i n e p a r a l l e l to ( i ) and passing through the Y a x i s has the equations x = 0 , s y - 1 «= 0 . ( i i i ) A l i n e p a r a l l e l to ( i ) and p u t t i n g the YZ plane has the equations x = 0 , s y + t z - 1 «= 0 . ( i v ) 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 plane has the equa- t i o n s r x + s y - 1 = 0 , ox+oy+oz-1 <= 0 . (v) A l i n e " through the o r i g i n oblique to a l l three axes has the equations rx+sy+tz - 1 = 0 , ox+oy+oz-1 = 0 . ( v i ) A l i n e through the X and Y axes but not through the o r i g i n has the equations r x - 1 = 0 8 s y - 1 = 0 . ( v i i ) A l i n e through the X a x i s and p a r a l l e l to the YZ plane has the equations r x - 1 = 0 , rx+sy+tz - 1 «=• 0 . 26 20. Two-Plane Form of the •Equations of a L i n e ; - Let the l i n e be defined by the planes (x,, y,, zf) and (x , y , z )» I f the l i n e passes through the o r i g i n then one or more of the coordinates of each plane w i l l be i n f i n i t e . I f i t does not pass through the o r i g i n , a l l the members of at l e a s t one set of coordinates w i l l be f i n i t e . Suppose the p o i n t s r ( x + s ( y + t ( z - 1 = 0, r^x + s x y + t a z - 1 « 0 , l i e on the l i n e . The p o i n t i n which the l i n e cuts the XY plane can be found by e l i m i n a t i n g z from the two equations, and the p o i n t where I t cuts the TZ plane can be found by e l i m i n a t i n g x. Let these two p o i n t s be denoted by the equa- t i ons (23) Then r^x + s 3 y - 1 = 0, s^y + t^z - 1 «= 0V r e s p e c t i v e l y r 3 x + 3sy - 1 = 0, r,x, + s 3y, - 1 - OV r 3 x % * s 3 yx - 1 0. I f these equations i n r , s are to be c o n s i s t e n t we must have x y 1 x, y, 1 x x 5V 1 27. whence x - x, c y - y, - 2 r v y, - y v In the same way, from the second of" equations (23) we o b t a i n y - y, = z - z, Therefore (24) x - x, ^ y - y ( & z ~ z, x, - x x y, - y x. z, - Equations (24) are c a l l e d the "Two-Plane f r equations of a s t r a i g h t l i n e . Obviously these have no meaning i f one of the denominators i s zero. Suppose x, - x,. Is zero. Then x must be equal t o x i and i n s t e a d of (24) we w r i t e x - x ( , . . 7 - J , = z - z, y, - y x I n C a r t e s i a n coordinates (24) give the "two-point" equations of a s t r a i g h t l i n e . 21. D i r e c t i o n Cosines of a l i n e : - I f the lin e - i s define d by the two p o i n t s whose equations are (22), the d i r e c t i o n cosines are found t o be Suppose the l i n e i s defined by (24). Equate the f i r s t two f r a c t i o n s . Then (x - x,) (y, - y j - (y - y ( ) (x, - x j . This equation i s r e d u c i b l e to the form ( 2 6 )  y * " y>_x + X/ " y - 1 - 0, x , y v - x^y ;- x y v -x^y # which i s the equation of a p o i n t on the line» In the same way the equa t i o n s ( 2 7 )  Z ^ " z > y * y> ~ z - 1 = 0 , y, Z , ~ K Z , - y, 2 ^ z , and ( 2 8 )  g v " z« x X ' ~ ^i- E ~ 1 ~ o; X, Z -X Z , X z, -x z represent p o i n t s on the l i n e . We can therefore 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 cosines of the l i n e j o i n i n g them by means o f ( 2 5 ) • I f the denominator x, y^ - x^y f has the value zero, i . e 0 , i f B y, ' y r 8 £ 9 from s e c t i o n 3 we know that x # z v - K , _ z ( 7, 2^ - y x z f cannot also be zero. In t h i s case we can use the two p o i n t s whose equations are ( 2 7 ) and ( 2 8 ) . 2 2 . Plane P a r a l l e l to a l i n e : - Theorem: The plane ( 2 9 ) (k,x, + k^^, k y, k v y ^ , k z, + k v z J i s p a r a l l e l t o the l i n e determined by the planes ( x / 9 y ( , z ( ) and ( x t , y L , z J . I f 6 I s the angle between the l i n e and p l a n e 9 from equation ( 2 1 ) we o b t a i n s i : y 1" -5- y + z Let ( 2 7 ) and ( 2 8 ) be the equations of the l i n e . Then, from ( 2 5 ) we have 2 , " Z , •rr FT «*. -v* »7 // z v - z, y / z , - 2 , - ^ y, - y* 11- 1/ (x, z v - x v s , j >, z v - y tz,/ "fx, 7, 5L. z, / z / - z. (30) y, z t ~ y x z , z v " s , ,v /z, ~ s v f /x, - x u y. ~ y„ Ax z - X x z f c- x j a , / (y, z^- y^z,/ x_z ( y # z^- y^z 30 *v. y, - y. / x z - x z y z - y z ix, Z l- x x z ^ Z l» y x z / U , V y,s„- y ^ , The s u b s t i t u t i o n of (30) i n the expression f o r s i n 0 gives us P + Q, + R s i n 6 - s.T . where -n E, — Zi x , z r - x v z (k,x, + Ic^xJ Q . - J = (x,y, - \ y t , S - y (3c, x, + k,x v)" + (x, y, + \ y ^ ) v + (x, z + IcjOV T A x z v - x^z ( / \y, \~ y^J \2c,z^- x k z ; y ( y z X. f The numerator reduces to zero and hence s i n g> « 0, and the plane i s p a r a l l e l to the l i n e . Conversely, i f the plane ( x 3 , ŷ  , z^) i s p a r a l l e l to the l i n e of i n t e r s e c t i o n of (x f , y ( , z,) and ( x z , y L , z t ) , i t s coordinates must be o f the form (29)• We have 3 1 . and. t h e r e f o r e z - z, a: z - x z z - z t T-Xj + y 3 'x, - x ^ y, - y„ X z - X z / V 2-1 y, Z T - ~ y * I s ^ - 0 , T l i i s eauation reduces to ( 3 1 ) x, y • i 2- x 3 y> z. 0 e (^z^ - z () cannot always be zero, since we do not have to r e s t r i c t the l i n e i n t h i s manner. (Therefore we must have the r e l a t i o n I aci- y', z,\ ( 3 2 ) x , y^ z. * i y, s3 = 0 s a t i s f i e d under a l l c o n d i t i o n s . I f ( 3 2 ) h o l d s , then x^, y 3 „ z j must be a l i n e a r combination of the corresponding e l e - ments of the other 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 the p o i n t s (x, , y, , z( ), y t, z J and the o r i g i n . 23. P e n c i l of P l a n e s : - Suppose the plane ( 2 9 ) passes through the l i n e of 32* i n t e r s e c t i o n of the planes (x f, y, , z ( ) and ( x ^ y^, z u ) • Then i t passes through 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 - nates must s a t i s f y the equation of any poin t on the line«, Let a poin t on the l i n e be defined by the equation rx + sy + t z - 1 = 0. We 'must have • r x , + sy, + t s , - 1 = 0, (33) rx^* sy1_+ t z t - 1 •» 0, r ( l c , x , * x^x u) + s(k,y,+ k j j + t(]c z ( + \ z j - 1 = 0, t h a t i s ' - ' (34) lc, ( r x ( * sy, + t z , ) + k r(rx v+ sy^_+ t z j - 1 = 0. Equations (33) and (34) h o l d simultaneously only i f (35) ^ = 1. This r e l a t i o n i s the necessary 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 coBrdinates are gi v e n by (29), w i l l pass through the l i n e of i n t e r s e c t i o n of the planes (x,, y , z, ) and ( x ^ , y^, z j . In ( 2 9 ) , i f we l e t x, = —— s h + x h + 1c we have the system of planes whose coordinates are given, by 3cx + hx x «* _ ! , h + x 33. (36) = Icy, * h y w . Ii * 3c 3cz . + h z . h + 3c which i s a p e n c i l o f planes, s i n c e r e l a t i o n (33) s t i l l h o lds. In C a r t e s i a n coordinates a l l p o i n t s (56) are c o l l i - near, and d i v i d e the segment j o i n i n g (x( , y ( , zt ) and (x^, y^, z^) i n the r a t i o h : 3c. 24. Three-Plane Equation of a P o i n t ; - £et (x( , y( , z( ), (x^, y^, z.J, and ( x 3 , y f, 2 j) he the coordinates of three planes such t h a t no plane i s p a r a l l e l to the l i n e of i n t e r s e c t i o n of the other two® The c o n d i t i o n s that these three planes pass through the p o i n t \ rx + sy + t z - l = 0, are rx, + sy, + tz, - 1 •» 0, rx^+ sy t + tz,.- 1 « 0, rx 3 + sy, + tz,- 1 = 0. 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 y z 1 (37) z , 1 1 1 = 0. 3 4 This i s the r e q u i r e d equati o n t since i t i s of the f i r s t degree i n x, y, z, and i s obviously s a t i s f i e d by the coordinates of the three planes. I f ^ °i •.. . f] the p o i n t i s f i n i t e . I f w - 0 , ( 3 7 ) gives an equation of the form r.i r x + sy + t z = 0 , which 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 . I f CO «= 0 , the elements of any one row o f OJ must be a l i n e a r combination of the corresponding elements of the other two; rows, and hence the plane must be p a r a l l e l to the l i n e of |J i n t e r s e c t i o n of the other two* hi •N •, '' H 25 • T r a n s l a t i o n of Axes;- . kj Suppose the 0 r i g i n i s t r a n s l a t e d t o the point ; ,! r x + sy + t z - 1 Oy without any r o t a t i o n of axes. Let any plane be represented by the p o l a r coordinates (f>} otf fi} y ) and ( ̂  ' *, f' ) w i t h respect t o ihe o r i g i n a l and new systems, r e s p e c t i v e l y . Then ii; 2c, y, z, x ^ ^ x i y 3 z a 3 5 « From (20) we have - (rx sy + tz - l ) f x + y + z Therefore 0 8 ) and hence (39) 1 x y' a' (40) ~ •» ( r x + sy + tz - - - A «= y. ,* X r x + sy + tz - 1 - - y" r x + sy + tz - 1 z r x + sy + tz - 1 ons are X rx' + sy' + tz' t 1 y' r x ' + sy' + tz' + 1 z' rx' + sy* + tz' + 1 260 The Degree of an Equation- i s Unchanged by Transf orma- t i o n s :- ^ ( 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 Let the degree of the equation he n. A general term would be (41) i x W , where p, q, m are not negative and p •*- q + m £ n. I f we r o t a t e axes by equations ( 7 ) , i n place of (41) we o b t a i n A(/),x'*Ay' + / U') P(/<.X'* /-.Y+yO, -0*(<*Wy'+ S 3 Z ' T Since each term i n each bracket i s o f the f i r s t degree, we cannot o b t a i n terms of degree higher than n. I f we t r a n s l a t e axes a c c o r d i n g to equations (40), (41) becomes A x ' P y'* z' m ' ( r x ' * sy* + t z ' ) P + <1 + m I f every term i n the new equation 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 ' m ( r x ' + sy' + t z ' + l ) n ~ ( p + * * m ) Any term i n (42) cannot be o f degree higher than n. Hence the degree of an equation i s not 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 of axes* Suppose the degree were lowered by a transformation of c o o r d i n a t e s . Then, by applying the i n v e r s e transformation, 3 ? . we should, be r a i s i n g the degree o f the equation. This has been proved impossible. Therefore the degree i s unchanged by r o t a t i o n and t r a n s l a t i o n . 3 8 . CHAPTER I I The General Second Degree Equation The most gen e r a l second degree equation i n x, y, z i s ( l ) a x % hy w+ cz u+ 2fyz + 2gzx + 2hxz + 2ux + 2vy •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 represents a c o n i c o i d i n the p l a n a r system of coordinates. 1. Equation of 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 of the planes (x,, y , z, ) and ( x t , y^, zj) i s g i v e n ( S e c t i o n 20, Chap. I) by the equa- t i o n s .(2) x - x , = y - 7 | ^ z - 2 , = ^ - x,_ y, - 5̂  Z / - z v The coordinates of any plane through (2) are x «. x, + p(x, - x x ) , (3) . y = y, + p(y, - yj» z = z, + p(z, - z J . I f a plane ( 3 ) touches the surface ( l ) , i t s coBrdinates must s a t i s f y equation ( l ) . S u b s t i t u t i n g ( 3 ) i n ( l ) we o b t a i n a quadratic equation i n p, which shows t h a t , i n general, through any l i n e two planes can be drawn to touch the surface ( 1 ) . Suppose that one of these i s the plane (x,, y(i z ) I t f o l l o w s that one r o o t o f the quadratic i n p must be zero, and hence the constant term must be zero. We ther e f o r e have (4) a x ( % by,"* c z , " * 2fy,z,+ 2gs( x, + 2hxty( + 2ux(+ 2vy, * 2w z , + d = 0« Suppose (3) determines one plane only. In t h i s cas the plane i s the tangent plane (xt , y ( , zt ), and both roots of the quadratic are zero. Both the constant term and the c o e f f i c i e n t of p must be zero, so that ax, ( x , ~ x j + by, (y, - y j + cz, (z,- z j + f{ y, (z,- zj (3) * z, (y, - y j j * g f z( (x, - x j + x, (z, - z^ ) J + h{ x t ( y # - y^ ) + y, (* - ) / + u(x,~ x j + v(y, - yO + w(z, - z j = 0. I t f o l l o w s from (2) t h a t ( x f - x j : (y # - y u ) : (z, - z J - (x - x / ) : (y - y, ) : (z ~ z,)» and from ( 5 ) we get axx, + byy, + czz, + f (y, z + z, y) + g(z,x + x, z) + h(Xj y + y,x) + ux + vy + wz = ax, 1 1 + by, *" + cz ( u + 2fy, z,+ 2gz tx ( + 2hx,y, + ux, + vy, + wz t. As a consequence of ( 4 ) the r i g h t number of ( 6 ) i s equal to - (ux, + vy, * wz, + d) • Therefore ( 6 ) reduces to 40. (7) axx,+ byy, + czz, * 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 equation of the po i n t of tangency of the plane (x, , y ( , z,) to the surface ( l ) . (1) 2. C o n d i t i o n that a P o i n t L i e s on the Surfaceg- Let the equation of the point on the surface be (8) r x + sy + t z - 1 = 0. Comparing equations (7) and (8) v;e have ax + hy * gz ( + u hx ( + by, + f z , + v _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ SX = gx * f y -5- cz, + w _/ux * vy, + ws, + dj . t " - 1 Put each f r a c t i o n equal to - A . Then ax, + hy, + gz, * u + A r = 0, hx, + by, + f z , * v + A s = 0, gx, * fy, + cz,.+ w + A t «= 0, ux, + vy, + v/z, + d - A = 0. We a l s o have rx, + sy, + tz, - 1 = 0. E l i m i n a t i n g x, , y, , z ( , A from the above equations, ne 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. Smith " S o l i d Geometry", p. 41. 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 which i s the same as ( 9 ) A r % Bs*"+ C t % 2Fst + 2Gtr + 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 determinant a h g u h b f v g f e w u v w d The 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 the p o i n t (8) l i e s on the surface ( l ) I n c i d e n t a l l y , ( 9 ) represents a c o n i c o i d i n the C a r t e s i a n system. 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 ) represents a c o n i c o i d i n the planar system of c o B r d i n a t e s . A proof that ( l ) represents a c o n i c o i d w i l l be given i n s e c t i o n 3, where no reference i s made, as above, to C a r t e s i a n c o o r d i n a t e s . (1) 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 Middle P o i n t s of a System of P a r a l l e l Chords:- L e t the equation of the surface be ( l ) , and l e t (8) be the equation of any p o i n t on t h i s e o n i c o i d ; r , s, t must s a t i s f y ( 9 ) . Let (10) Jt-x. + my + nz - 1 = 0 be the equation of a p o i n t on a l i n e whose d i r e c t i o n cosines are y\ , 4/» 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 ob t a i n a quadratic equation i n p, which shows that any given l i n e cuts the surface i n two p o i n t s . I t f o l l o w s that a l l s t r a i g h t l i n e s i n a plane cut the surface i n two p o i n t s , and ther e f o r e a l l plane sections of the surface are conic s e c t i o n s . This i s the d e f i n i t i o n of a e o n i c o i d . We have (12.) p""(A / f + B ^ + C S\ + 2C-«/A+ 2H/1/-) + 2p(AiM + Bm̂ _+ Cn F i i A + FmV + GlS + G-nA + H^A+ HmA •+ IT A + v>_ + ' WT/) * ( A ^ v * Bm% Cn%"2Fmn * 2Gn/+ 2Him + 2 * 2Vm + 2Wn + D) = 0, where A, B, C, ... , have the same values as i n s e c t i o n 2. I f (10) i s the equation of the middle point of the l i n e , the values of p obtained from (12) must be equal n u m e r i c a l l y but opposite 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. Hence (13) i ( A A + Eyu + C V ) + m(H A + B^A. + F-*/) * n(G A + Fyu. + 0/ ) * U A •«- Yyu- + f i / = 0, Therefore the plane whose p o l a r coBrdinates are given by (14) COS oC = COS y_? cos If where p R*~ * s t- QL R + R + S R *~ + U A + A A + EyU. + G V , H A" + B G A + F/U + c passes P Q R S through the p o i n t ( l O ) l 1 ^ But (14) represents a (1) C.f. equation (14), Chapter I . 4 4 . f i x e d plane when y\ , /b- , */ are f i x e d . Therefore the mid- p o i n t s of a l l p a r a l l e l chords whose d i r e c t i o n cosines are A * ^ * ^ H e In the plane (14). A plane which passes through the mid.-points of a system of 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 di a m e t r a l 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 , i t i s c a l l e d a p r i n c i p a l plane* 4. The P r i n c i p a l Plane;- I f the plane (14) i s pe r p e n d i c u l a r to the chords whose d i r e c t i o n cosines are A ,yu. , V , the d i r e c t i o n cosines of i t s p o l a r normal must he /\ %^u. , 4/* Therefore AA + H/c * GV H A + B/"- + i V G A + F>- + c V - \ - ~ ~ Put jT f o r the common value of each of these f r a c t i o n s ; then (A - J ) A + H/c + - 0, (16) H A + (B - 0, GA * F/c + (C - J )</ - 0. E l i m i n a t i n g X » ^ w e S e t [ A - / H G H B- J F 0 9 G F C- J which, when expanded, becomes the cubic ( 1 7 ) f - f! - & m where J = A + B + C, ^ = AB + BC + CA - F-G~- H*", and o© - ABC + 2 FGH - AF^~ EG*'- CH V. When. / i s determined, any two of the three r e l a t i o n s (16) w i l l g i v e the corresponding values of \ % ^ , t / . Since one r o o t o f a cub i c equation i s always r e a l , i t f o l l o w s that there i s always a t l e a s t one p r i n c i p a l plane. 5° Roots of ( I ? ) t - ^ 1 ) L e t £ , be any r o o t of (17) and l e t A o , yu0 , <t/0 (not a l l zero) be values of A * «S t h a t s a t i s f y (16) when f = f, • I f ̂  i s a complex number, A0 ,yu„ , may be complex. Let A° ~ A, + i A*., «/* - + i , where i = <fT and ^ ; , ̂ „ ̂  , ̂  , ̂  are r e a l . S u b s t i t u t e £ and these values of 'A 6 fyA-o » f o r I* » A J^A i n (16), m u l t i p l y the r e s u l t i n g equations by A, ~ iyA w ,yu, - i A v , V, - i <. , r e s p e c t i v e l y , and add. The r e s u l t i s ( A% AC *^ +*/s)fl - (A" * A v)A + 2( /t//\, + V v ^ W ) G + 2( /̂ y<*, + )H. The c o e f f i c i e n t of / / i s r e a l and d i f f e r e n t from zero, and ( l ) Snyder and Sisam, p. 79. the r i g h t member o f the equation i s also r e a l . Hence Jf i s r e a l . Since /; i s any r o o t of (17), a l l the r o o t s of (17) are r e a l . The c o n d i t i o n s that a l l the roots of (17) are zero are ABC + 2FGH - AF*'- BG^ - OH"" - 0, (18) AB . + " BC + CA - F*" - G1"- H « Q, A + B + C = 0. Square (18, 3 ) , i.e«, the t h i r d equation of (18), and sub- t r a c t twice .('18, 2),from i t . T h e ' r e s u l t i s A-" + B~'+ C" + 2 P V + 2G'"+ 2H""= 0. Sinee A, B, 0, e t c . , are assumed to be r e a l , i t f o l l o w s t h a t ' (19) A = B = C = F = G = H = 0. I f (19) i s t r u e , (9) reduces to 2IJr + 2Ts + 2Wt + D = 0. But. t h i s i s the, c o n d i t i o n that the plane whose p o l a r coor- dinates a r e 47 passes through the p o i n t whose equation i s r x + sy + t z - 1 = 0. Therefore the f i x e d plane (20) w i l l pass through a l l the p o i n t s on the c o n i c o i d , and hence the eonicoid reduces to a plane . This degenerate case i s obtained by l e t t i n g a l l the roots of the cubic be zero. Henceforth we s h a l l assume th a t a t l e a s t one root of the cubic i s d i f f e r e n t from zero. 6. E l i m i n a t i o n of the yz, zx, z terms Since a t l e a s t one o f the p r i n c i p a l planes i s not at 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 the system of reference so t h a t the new XY plane i s a p r i n c i p a l plane of the s u r f a c e 0 Let the equation of the 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 plane, the two p a r t s i n t o which the XY plane d i v i d e s the surf a c e must be e x a c t l y a l i k e . I f there i s a tangent plane ( x , , y ( , z ( ) at a point on one side of the XY plane, there must be a corresponding tangent plane (x ( , y( , - z ( ) at a p o i n t on the other s i d e . Sub- s t i t u t i n g each s e t of coordinates i n ( l ) , we o b t a i n a x ( v + by," + ez,1" + 2fy, z( + 2g^( x ( * 2hx,y, + 2ux, + 2vy( + 2wz( + d = 0 and ax" + by" * e z " - 2fy ( z, - 2gz (x + 2hx y( + 2ux ( ' + 2vy, - 2wz4 + d = 0« 48, Since these r e l a t i o n s are true f o r a l l tangent planes, i t f o l l o w s that f = g = w = 0. These r e s u l t s may be d e r i v e d i n a second way as f o l l o w s . Consider the three p o i n t s r ( x + s,y + t, z - 1 = 0, (£1) r_x + s^y + t u z - 1 = 0 , r x + s, y + t z - l = 0, on the surface and on one s i d e of the XY plane. Let these p o i n t s be considered as d i s t i n c t . L a t e r we s h a l l r e quire t h a t they approach coincidence. On the other s i d e of the XY plane we must have the corresponding p o i n t s r,x + s y - t z - l = 0, (22) r_x + s^y - t v z - 1 •= 0, r, x + s y - t , z «" 1 = 0. The coordinates (x, , JX , S ( ) o f the plane through the three p o i n t s (21) are given ( S e c t i o n 13, Chap. I) by 1 s, t, r ; a, t, r s t r 3 s 5 t 3 r, 1 t, r 1 t ^ r 3 1 \ 5. three p o i n t s (22) are g i v e n by Therefore (23) x ^ = Z i «* x, f y». = y, i Z L » - z , r s , 1 r v 1 S3 I A, ( I Z. » y*» 1 s, - t , 1 - t 1 s, r, - t r , r , i - t r 1 -\ 1 - t — A, i s L I i — 4, A, - J } -X 50 In the case where A 0, r e s u l t s s i m i l a r to (23) can he obtained by u s i n g p o l a r coBrdinates, Let the p o i n t s (21) approach coincidence; then the p o i n t s (22) w i l l do l i k e w i s e . At a l l steps i n t h i s process r e l a t i o n (23) holds f o r the.coordinates of the planes through the r e s p e c t i v e sets o f points,, I n the l i m i t , I .e„ where tangency occurs, 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 ( , yf , z ( ) at a poin t on one s i d e of the XY plane there must be a corresponding tangent plane (x ( , y( , - z ( ) a t a p o i n t on the other s ide. I f f = g = w •= 0, equation ( l ) becomes (24) ax 2"* by1"* cz^-t- 2hxy + 2ux + 2vy + d « 0. 7 . Reduction when d^-Q:- 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 whose equa- t i o n i s (24) becomes (a - u l ) x v + (b - v 1) y~ * cz*~+ 2/h - uv/xy + d = 0. The term i n xy 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 an angle 0 determined by |x I y 1 - 0 , 51 according to the r o t a t i o n formulae x - x'cos£ - y ' s i n 9 , y = x'sin6? + y'cos _? , Z = Z ' . Dropping primes, vje get an equation of the form a ( x v+ b, y v+ c, z^+ d = 0. Since d ^ 0, we can d i vide by -d and the r e s u l t i n g equation has the form ( 2 5 ) a 62c v+ ^ y 1 - * c s z " = 1. Hence f o r d^O, under a l l c o n d i t i o n s we can reduce equation ( l ) to the form (2jj) • 8. Reduction when d - 0;~ The equation to be considered i s ax*~+ b y % cz 1"* 2hxy + 2ux + 2vy = 0. ( i ) I f u = v = 0, by r o t a t i n g the 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 xy term. The r e s u l t i n g equation has the form (26) a Dx"+ b.yN- c . z ^ 0. ( i i ) I f v i s not zero 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 according t o the transformations 5 2 . vx'+ uy' y -J- V _ ; Z — 2 , and we o b t a i n an equation of the form a,x*+ b( e, z % 2h (xy + 2u (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 to the p o i n t whose equati on i s ~ a_ n, - 1 «= 0, 2u, u, we o b t a i n ( 2 7 ) b, y % c / z'1'* 2u,x = 0. Therefore equation ( I ) can be reduced to one o f the forms ( 2 3 ) , ( 2 6 ) s or ( 2 7 ) . 5. Center o f Gonicoid:- Consider equation ( 2 5 ) , namely ax*~+ by v+ c z v = 1. The center l i e s on the plane midway between p a r a l l e l tangents to the s u r f a c e . I f (x, y, z) i s tangent to the surface, (-x, -y, -2) i s a l s o tangent. Therefore the o r i g i n i s the center of t h i s type of conicoid® Consider equation ( 2 6 ) , namely a _ v * by%- c z v «= 0 e As before, the o r i g i n i s the c e n t e r . Suppose the c o n i c o i d reduces to by*+ cz 7"* 2ux = 0 . 53» L e t the p a r a l l e l planes (x , y; , zf ) and (kx, , ky; } } c 2 ) touch t h i s surface; t h a t i s by, v + c z ( v + 2 u x , « 0, bk^y^ * clc vs 2ukx, = 0 . I f xi = 0 t h i s eonicoid" i s a degenerate of ( 2 6 ) . I f u ^ 0 then k = 1, or e l s e the p a r a l l e l tangent planes are a l l a t i n f i n i t y . Therefore t he surface has no f i n i t e center* 1 0 . P o l a r P l a n e : - We s h a l l show t h a t the p o i n t s o f contact o f a l l tangent planes through a g i v e n p o i n t to a e o n i c o i d l i e on a plane. This plane i s c a l l e d the p o l a r plane of the poin t w i t h respect to the- e o n i c o i d , Conversely, the p o i n t i s c a l l e d the p o l a r p o i n t of. the plane w i t h respect t o the e o n i - c o i d . ( i ) Let the equation of the c on i c o i d be a x % by""+ cz' L =• 1. The equation of the tangent p o i n t of the plane (x,, yf , z ; ) i s given ( S e c t i o n 1, Chap. 2) by ( 2 8 ) axx, + byy( + czz, - 1 = 0 . Suppose the plane (x ( , y, , z,) passes through the p o i n t ( 2 9 ) r x + sy + t z - 1 = 0; then rx, * sy * tz, - 1 = 0 . The p o i n t ( 2 8 ) l i e s on the plane 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 of tan- gency a l l l i e on the plane (30) / r , s , t ) , \a b~ C / which must t h e r e f o r e be the p o l a r plane of the p o i n t (29) w i t h respect to the c o n i c o i d . ( i i ) L e t the equation of the c o n i c o i d be ax"+ by""+ cz ̂  - 0. The equation of the tangent p o i n t i s (31) axx, + byy, + czz, - 0« The p o i n t (31) l i e s on the plane (0, o„ o ) , and therefore a l l tangent p o i n t s are a t i n f i n i t y . This type o f c o n i c o i d w i l l be discussed l a t e r * ( i i i ) Let the equation of the c o n i c o i d be —• ' x. ' by + cz -s- 2uz 0. The equation of the tangent p o i n t i s (32) byy, + czz, + u(x + x ( ) =0. I f the plane ( z ( , y , z,) a l s o passes through the p o i n t whose equation i s (29) 9 the p o i n t (32) l i e s on the plane (33) /JL , us , u_t ) \ r br c r J * s i n c e i t s coordinates s a t i s f y (32). Therefore (33) i s the p o l a r plane of the p o i n t (29) w i t h respect to t h i s c o n i c o i d . 11* R e c t i l i n e a r Generators;- • L e t the equation of the surface be (34) a x + b y - c z » = l , 55. which may be w r i t t e n i n the form (ax + c s ) ( a x - cz) = ( l + b y ) ( l - by), or (35) ax -i- cz ^ 1 - by _ - 1 + by = ax - cs " t ! s a y * Then ax + cz «= 7|(l + by), (36) (ax - cz) 7j «• 1 - by . For every value of 1^ , these equations define a l i n e . Every poi n t l y i n g on the surf a c e (34) must s a t i s f y the r e l a t i o n ( S e c t i o n 2, Chap. 2 ) . (37) r * + s * _ t ~ a v b „ I f the point whose equation-is r ( x + s y + t ( z - l = 0, l i e s on the l i n e (36), i t f o l l o w s ( S e c t i o n 12, Chap, l ) t h a t r. = m — + m a n (38) s, = -m ,b + m, b } t ; = m, H. - m c ̂  ^ m + m, = 0. R e l a t i o n (37) holds when we re p l a c e r , s, t by r , ŝ  , t, r e s p e c t i v e l y . Therefore any poin t on the l i n e (36) a l s o l i e s on the surface (34), and (36) i s a system of r e c t i - l i n e a r generators of (34). Equat ion (34) may be wri t t e n 56 (39) ax * cz 1 + by ^ 1 - by = ax - cz = 9 s a y * Then ax + cz = _T (1 - by), (40) (ax - c z ) ^ - 1 + by, which i s a second system of r e c t i l i n e a r - generators of ( 3 4 ) . We can f i n d i n a s i m i l a r manner the equations of the generating l i n e s of the surface (41) b^y"" - c V = 2ux. The equations of the generators of one system are by - cs = 2 < r i , by + cz = JL. ; and o f the other system by + cz - 2 r x, by ™ cz = _ . 12. I n v a r i a n t s : - L e t the equation of the surface be ax*"+ by"+ c z " « 1. I f the axes'are r o t a t e d to new p o s i t i o n s according to equa- t i o n s (8) of Chapter I , the r e s u l t i n g equation i s of the form a ( x"+ bf y%- c( z % 2f ( yz + 2g ( zx + 2h( xy = 1, where a, = a j , v + 1J y^- + c % b ( = a /V„v + b + c 51. c, = a / ) 3 + b /U3 + e f", * a/1 x/l 3+ b/.y&4 + c < ^ 3 t g, = a/^A, + b / ^ , + c , Making use of the r e l a t i o n we obta i n A, A. 4/, a I h x g, a 0 o. h l a be 0 b ' 0 « , 0 0 e D. £ Therefore 2) i s unchanged by r o t a t i o n . In the same way i t can be shown that I = a + b + c, g - h J be . + ea + ab ~ i are unchanged by r o t a t i o n . I t can r e a d i l y be shown that these expressions are not 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 ge n e r a l e o n i c o i d i s (Section 1, Chap. 2) (42) A r " + Bs'" + Ct = 0, where A, B, C, ...... are the c o - f a c t o r s of a, b, e, , i n the determinant e e * e j?0. a B u h b f V S X c w XL V w d Let the axes be r o t a t e d to new p o s i t i o n s according to the formulae (8) of Chapter I , namely x = A, x' + Ax, y' + i z', y ~yu, -' + / - v y , + >fc» z', z = i / , x' + /f/v y' + 4/3 z'. The p o i n t whose equation r e f e r r e d to the o l d system i s - r x + sy + t z - 1 = 0, becomes ( r A + s>/ + t A/, •)_' * ( r ̂ _.+ s/«v + t̂ «C )y' + ( r /13 + s/^ + t V, )£' ~ 1 = 0, i n the new system; t h a t i s ' r ' = r A1 + Sju, + t , s' = r ^ v + s ^ + t , t' = r > j + s^k, + t *4 . The in v e r s e r e l a t i o n s are r e a d i l y found to be r = r M , + s' Ax.+ t'Aj s (43) s = 2?/*, + s'/<v + t > 3 , t = r 1 </, + s' + t V 3 . The degree of equation (42) w i l l be unaltered, as proved i n Chapter I , Section 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"1" + B s " + a t 1 " + 2Fst + 2Gtr + 2Hrs becomes changed i n t o A'r" + E'S" + C't1' + 2F'st + 2G'tr * 2H'rs; then, since s~+ t ^ Is unalte r e d by t h i s change of axes, (44) * Ar"+ B s n Ct^+ 2Fst + 2Gtr * 2Hrs - J (r"+ s"+ t~) w i l l be transformed i n t o (45) A' r ^ + B' s" + C t v + 2F'st + 2G-' t r + 2H'rs - J ( r % s % t " ) . The expressions (44) and (45) w i l l therefore be the product of l i n e a r f a c t o r s f o r the same values of J . The c o n d i t i o n t h a t (44) i s the product of l i n e a r f a c t o r s i s '« 0. A - J . H •"' G H B -J F G F C - t h a t i s , • J 3 - J*(A + B + C) + J ( B C + CA + AB - F*"- G V- H") ~ (ABC + 2FGH - AF^- EG**- GH V) = 0* The c o n d i t i o n that (45) i s the product of l i n e a r f a c t o r s i s s i m i l a r l y f3 - J V + B' + C' ) + | ( B ' C ' + C'A' + A'B' - F'- G'- H'1) '-(A' B'C' ^ F ^ ' E' - A' F' - B' G'"- C ' / ) - 0. Since the r o o t s o f the above cubic equations i n J are the same, the c o e f f i c i e n t s must be equal. 6o. Hence 3 = A + B + C, ^ = BC' + CA + AB - F"~ G7'- H", A H G H B F G F C , are 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 of axes to the p o i n t whose equation i s «T x + /5 y + y _ - 1 = 0 can he accomplished ( S e c t i o n 2j?, Chap. I) "by means of the formulae o< x'+ /9 y'+ #z' + 1 x ' + ^ y' + ' + 1 c< x' + /3y' + /z'+ 1 The p o i n t , whose equation r e f e r r e d to the o l d axes i s r x + sy + t z - 1 = 0, has the.e qua t i on rx' + sy' + t z ' - ( c< x' + /3 y '+ ^ z' + l ) = 0 r e f e r r e d to the new axes; t h a t i s r ' = r ~ c< , s ' «= s - /3 » t ' = t - r i therefore 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 not change any of the c o e f f i c i e n t s o f the second degree terms*, Therefore , j^. , are u n a l t e r e d by t r a n s l a t i o n of axes. Thus ^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 or r o t a t i o n , and are t h e r e f o r e i n v a r i a n t s . The proof that /\ 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 given f o r , The c o n d i t i o n that a p o i n t l i e s on a e o n i c o i d i s Ar + Bs + Ct + 2Fs"C + 2Gtr + 2Hrs + 2Ur + 2Ts + 2Wt + D=0. Let t h i s equation be transformed by a r o t a t i o n i n t o A ' r % B's^+ C' t^+ 2F'st + 2G' t r + 2H'rs + 2U'r + 2V's • + 2W't + "D'« 0. ,This r o t a t i o n transforms the expression A r % B s % Ct7'* 2Fst + 2Gtr + 2Hrs + 2IJr + 2Vs + 2Wt + D (48) - kCr^+'s"* 1) i n t o A' r % - B' s v+ C t L+ 2F'st + 2G-' t r + 2H' r s + 2tf'r + 27's (49) " + 2W't.- + D A- '3c(r"+ s~+ t*"+ 1 ) . The d i s c r i m i n a n t s of (48) and (4?) are, r e s p e c t i v e l y A - 3c H G TJ H B - 3c F 1 I and Or F C ~ 3c W JJ 1 W D - 62. A - k H 1 C-' -U' H' • B ' - k p' Y' G' F' C' - k W7 IT' Y' \7' B'-k The expressions (48) and (4?) are f a c t o r a b l e i n t o l i n e a r expressions f o r the same values of 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 tha t i t s d i s c r i m i n a n t equals zero. Hence, s i n c e the c o e f f i c i e n t of k* i n each case i s u n i t y , the constant terms of these d i s c r i m i n a n t s must be equal; that i s _\ = A 1 •» Hence, i s i n v a r i a n t under r o t a t i o n . I n order to prove th a t /_» i s i n v a r i a n t under tr a n s - l a t i o n , l e t the axes be t r a n s l a t e d to the p o i n t whose equa- t i o n i s The c o n d i t i o n t h a t the po i n t l i e s on the c o n i c o i d becomes Ar"+ B s % Ct w+ 2Fst + 2Gtr + 2Hrs + 2(A<* + K/i + G Y + u)r (50) + 2(H*L + B/3 + F r + V)s + 2(G K + F ̂ 3 + C * + W)t + D'- 0, where D* i s the l e f t member of (47) when r , s, t are replaced "by c< , ̂ 3 , y o The d i s c r i m i n a n t of (j?0) i s A H G A<^ +H/6+G y +TJ H B F H *<- +B Y +IT G F C G^+F/S+C flf+W A*< +Ĥ » +G r +u, H ,< +B̂ a +F r +v, G ̂  +F/3 +C ar +w, ~ i>' M u l t i p l y the f i r s t column by , the second by(3 , the t h i r d by Y » a n ( 3- s u b t r a c t t h e i r sum from the l a s t column» In the r e s u l t i n g determinant, m u l t i p l y the f i r s t row by o( the second by , the t h i r d by Y s an3. subtract t h e i r sum from the l a s t row. The r e s u l t i n g determinant i s /\ « Hence , so t h a t i s i n v a r i a n t under both t r a n s l a t i o n and r o t a t i o n . 64 CHAPTER H I C l a s s i f i c a t i o n of Surfaces 1. Review of Previous Work:- In Chapter I I we have seen that the c o n d i t i o n that a p o i n t whose equation i s (1) r x + sy + t z - 1 = 0 l i e s on the surface whose equation i s given by (2) ax'v'+ by1'-!- cz" + 2fyz + 2gzx + 2hxy + 2ux + 2vy + 2wz +d=0 Is (3) i r r + Bs"" + Ct" + 2Fst + 2Gtr + 2Hrs + 2Ur + 2?s + 2Wt +D«0 where A, B, C, .... , are the c o - f a c t o r s of a, b, c, .... , i n t he determinant 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 to (3) as the ••point-condition 8 8 equation. We have also seen that 3 = (A + B + C), j = (AB + BC + CA -F"-G""- H"), A H G- U H B F ¥ G F C W TJ Y W D V 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 . 5" = A H G H B F G F C and 2. The Sphere;- The sphere i s defined to he the locus of a p o i n t which moves so as to remain at a constant distance from a f i x e d p o i n t . This distance i s known as the radius and the f i x e d p o i n t i s the center of the sphere. Let the equation of the center he c^x + {2> y + z - 1 = 0, and l e t the radius he R; then we have / ( r -d. (s ~ 7 * r + ( t - * T - R, or ( r - . * 0 % (s - y 3 ) % ( t - y ) " - R.\ Therefore the general p o i n t - c o n d i t i o n equation 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 from z e r o . Conversely, any poin t r x + sy + t z k 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 sphere. The p o i n t - c o n d i t i o n equation of a sphere whose center i s the o r i g i n , i s seen to he T * S + t = R e The sphere may a l s o be"defined as the envelope of planes which move so as always to remain a t a constant d i s - tance from a f i x e d p o i n t . Thus y x N - yN- Z"1- 1 . oc x + /? y + y z-1 R t h a t i s (4) R \ x % y % 0 = ( ^ x + ^ y + y z - 1 ) " . The equation o f a sphere, center at the o r i g i n , i s seen to he R"^"* y~ + z"") « 1, or (5) ax v+ a y % as v+ d = 0. ( I f a and d have the same sign the sphere i s imaginary.) The p o i n t - c o n d i t i o n equation of the sphere (5) i s Ar% - BS^-J- Ct1'-*- 2Fst + 2Gtr + 2Hrs + 2Ur + 2Vs > 2Wt + S = 0, ' where A = a^~d, B a^d, C = a v 4 ( D = a 3 , and F = G = H - T T - V « = W = 0 . Therefore 3 - 3 a ld, ô ) = a'd 3, ^ = a fd J» 3 . The E l l i p s o i d ; - Consider the surface whose equation i s (6) a x + b y + c z = 1. The p o i n t - c o n d i t i o n equation of t h i s surface i s found to he ( 7 ) b c r + c a s + a b t = a b c . For a, b, c are a l l d i f f e r e n t from zero, and a, b, e i n 67. descending order of magnitude, we have a _ a1" _ a and Hence a p o i n t on the surface can not be at a distance from the o r i g i n greater than a nor l e s s than c. The surface i s therefore l i m i t e d i n every 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 that a l l plane s e c t i o n s of (6) are 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 surface i s c l e a r l y symmetrical w i t h respect to the three coordinate planes, the three coordinate axes, and the o r i g i n . The p o i n t s i n which i t cuts the axes are found by l e t t i n g s = t = 0, t = r - = 0, r = s = 0, r e s p e c t i v e l y , i n equation (7)• These p o i n t s are determined by the r e l a t i o n s r = ± a, ±ax - 1 = 0 , s = _r b, ± by - 1 = 0 , t = ± c, ±cz - 1 B 0, r e s p e c t i v e l y . Consider the system o f tangent planes through the p o i n t (8) mz - 1 = 0, on the Z a x i s . The coordinates of a l l planes through t h i s 68. point and touching the surface are (x, J, 1 ) where m a x + b y + e_ = i . - The p o l a r plane of the p o i n t (8) i s (0, 0, m_\ c. Translate c1- the o r i g i n to 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). The equation of (8) becomes ______-„ z - 1 = 0; m that i s , the coordinates of a l l planes through (8) w i l l be (V v m v » ^ * T ~y» Let these planes touch the surface whose new m - e equation i s a^x"1* b̂ y"*"* c" z*~ = QL. z + l ] ; so t h a t (9) a x + b y ^ = - v «= . — m ~ e ^ „ m " (1) Therefore we have an e l l i p s e , ^ ' For 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 imaginary. The r a t i o of the semi- axes remains constant, namely a : b. The major semi-axis i s equal to a / 1 - — , which i s seen t o be zero f o r m = c (1) Valgardsson "Line Coordinates"• and equal to a f o r m i n f i n i t e l y l a r g e * As m becomes i n d e f i - n i t e l y l a r g e the p o l a r plane (0, 0, m_ ) approaches c o i n c i - c dence w i t h the XY plane. I n the same way we could show that the s e c t i o n of the surface made by the YZ plane i s an e l l i p s e of semi axes b and c and t h a t the s e c t i o n made by the ZX plane i s am e l l i p s e of semi-axes c, a. We c a l l a, b, c the "semi-axes" of the e l l i p s o i d . I f a ~ b, the s e c t i o n s p a r a l l e l to the XY plane are c i r c l e s and the surface i s a surface of r e v o l u - t i o n . I f a = b = c we have a sphere. For the e l l i p s o i d Jr = - ( a b + b c + c a ; \ = (a"bV ) (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 equation (?) becomes a^b^t 1" = 0. I f a, b are d i f f e r e n t from zero, then t = 0. Hence f o r c = 0, the surface must l i e wholly i n the XY plane* In t h i s case S ~ - a"1" b \ }- °- P - o, A - o. 7 0 . I f b = c = 0, (6) becomes a X l e Hence the surface has degenerated i n t o the two points ax t1 = 0. Let a «= a ( A , b = h, A , c « e, A . Equation (6) then becomes (10) a^x^ b^ y w + ° Let A increase i n d e f i n i t e l y but l e t a, , b ( , c, remain f i x e d . I n the l i m i t vie have a ( x * b, y -s- e, z ^ 0. Hence t h i s equation i s the l i m i t i n g case of an 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 to be n o t i c e d t h a t t r a n s l a t i o n does not a f f e c t the l a t t e r equa- t i o n . Th.et only plane which i s tangent t o the surface i s the plane (0, 0, 0). The p o i n t - c o n d i t i o n equation of (10) I s b e _ c a, - a b v „ ' 1 r J L s / _J L t « a b e, . A- A" K In 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 equa- t i o n becomes Or'" + 0s*~ •* 0t*~ = a/ b", , whioh can be s a t i s f i e d only by p o i n t s at i n f i n i t y . I n t h i s case - ^ a ~ /S 0. 71 The Hyperboloid of One Sheet:- (11) Consider the surface whose equation i s a x + b y - c z = 1. The p o i n t - c o n d i t i o n equation of t h i s surface i s found to be (12) D c r + c a s - a b f = a b e. Let a, b, e be a l l d i f f e r e n t ' from zero. The s u r - face i s c l e a r l y symmetrical w i t h respect to the coordinate planes„ coordinate axes, 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 can show that the plane s e c t i o n s of the surface 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 values i n the XY plane s e c t i o n , and increase 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 plane. Thus i s the equation of the e l l i p s e when the plane passes through the p o i n t m The semi-axes are i n the r a t i o a : b and the semi-major a x i s has the value 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 zero. In the same way we f i n d that sections p a r a l l e l to the YZ plane are hyperbolas. In p a r t i c u l a r , i f we consider the s e c t i o n made by the p l a n e / — , 0, 0 ) , we o b t a i n the a x + b y = m + c —- ,2, 23 ** 1 — 0 & 72. equation b y - e z = m a*" This curve i s w e l l - d e f i n e d except f o r m = a, and t h i s i s seen to be the case where the plane i s a t a distance from the YZ plane equal to the semi-axis a of the e l l i p s e which i s formed by the i n t e r s e c t i o n of the surface by the XT plane. We can d i s c u s s t h i s case e a s i e r w i t h reference t o the p o i n t - condi t i o n equation which i s c a s - a b t = a b c - b c r . When r = a we have b c tha t i s £ _ + b_ t c The system of p o i n t s whose equations are ( 1 3 ) t y + t z - 1 » 0 c and (14) - A ty + tz - 1 = 0 c can be shown to define two l i n e s . For the d i r e c t i o n cosines of the l i n e j o i n i n g (13) to the 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 v + e1" cos y « o'" , j/b + which, are constant. In the same way we can show that (14) defines a line„ Therefore when ra = a, we have a p a i r of 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 , ^ = a v b " e v ( c v - a" - b " ) 9 Z_ = a b c » Suppose c = 0. This case has already been discussed under the e l l i p s o i d . I f b = 0 and a and e are d i f f e r e n t from zero, the equation becomes (14) a*-!1" - c^z'' = 1 (which i s Valgardssont s hyperbola i n l i n e coordina t e s ) . Itoen b = 0 $ '«= a^c , I f a •= b = 0, the surface i s imaginary. I f b = c = 0 we have the case a x — x, which represents a p a i r of p o i n t s , as we have already seen. i f we l e t a = a, A , b = b , A s c = c , A , then i t f o l l o w s t h a t a s + b, y - c, E « — The s e c t i o n of t h i s s urface made by a plane p a r a l l e l t o the XT plane has the equation a, x * b y = — , A where k depends only on the p o s i t i o n of the c u t t i n g plane. This i s an e l l i p s e whose semi-axes are A a, and A b, , k k both of which become i n f i n i t e as A becomes i n f i n i t e c In the same way we can show t h a t the major axes of the h y p e r b o l i c sections p a r a l l e l , to the other coordinate planes become i n f i n i t e as A becomes i n f i n i t e . In the l i m i t we have a^x^ * b/y" ~ e^z1" = Oo For t h i s l a s t equation _ J » j «*£ = A = 0. J5. The Hyperboloid of Two Sheets:- Consider the equation (15) a vx" - b^y"1"- c^z 1" = 1. The p o i n t - c o n d i t i o n equation f o r (lj?) i s (16) b"" <Tr1" - o"a s^ - a" b" t " = a" b 0 » 75 This surface i s symmetrical wi th respect t o the coordinate planes, coordinate axes, and the o r i g i n . As befor we can f i n d the sections made by planes p a r a l l e l to the coBrdinate planes.. The sections p a r a l l e l t o the XY and ZX planes are found to be hyperbolas, and the se c t i o n s by planes para 11 e l to the YZ' plane are e l l i p s e s . Suppose the plane p a r a l l e l to the YZ plane passes through the po i n t r x - 1 = 0. I t i s r e a d i l y seen t h a t the 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 degenerate i n t o p o i n t s on trie X a x i s . For t h i s s u r f a c e J)- = (GU&L~ + a1" - t^cT) ^ = S^\TG' (a"- b""- c") £ = - a'b'e* = - a ' b V . When b or c i s zero, cases are obtained which have been discussed a l r e a d y . Let us consider the case when the semi-axes become i n f i n i t e ; suppose the equation i s a x - b y - c z = — A Then there i s no p a r t of the surface between the planes p a r a l l e l to the YZ plane and passing through the p o i n t s ± a A x - 1 = 0. I f A approaches i n f i n i t y the distance between these points becomes i n f i n i t e . In the l i m i t we have the hyperboleid of 76, two sheets at i n f i n i t y . We have J = ^ ==<̂ -/\== 0. 6. She P a r a b o l o i d ; - Consider the surface defined by the equation ( 1 7 ) b^ y*" i- c's 1" + 2ux - 0 . The p o i n t - c o n d i t i o n equation of ( 1 7 ) i s ( 1 8 ) G^VL'S" * b ^ u ^ t + 2b"c"ur ~ 0. I f b, c, u are a l l d i f f e r e n t from aero, we may w r i t e , i n s t e a d of (18), . £ . + 0 . b" _ c"~ u The surface ( 1 7 ) I s symmetrical w i t h respect to the XY and ZX planes and the X a x i s . The p o l a r of the p o i n t ( 1 9 ) mx - 1 » 0 i s ( S e c t i o n 1 0 , Chap. I I ) the plane / , 0 , 0 j« Translate the o r i g i n to the p o i n t - mx - 1 « 0o Then the p o l a r plane w i l l be the new XY plane., Equations ( 1 9 ) and ( 1 7 ) , r e f e r r e d to the new axes, are r e s p e c t i v e l y 2mx - 1 = 0 , b^y" * ^ z 1 " - 2umx*~ + 2ux = 0 . l e t a l l the tangent planes pass through the p o i n t ( l ? ) ; t h a t i s x = 1 _ . Therefore we have 2m ^ u b y~ + c"2 ̂  B - 2m • Hence plane s e c t i o n s p a r a l l e l to the YZ plane are e l l i p s e s 77- of semi-axes J- 2m and c tt 2m « This e l l i p s e degenerat XL es to a p o i n t when m = 0; that i s , the YZ plane touches the surface at the o r i g i n . The e l l i p s e increases i n s i z e as the c u t t i n g plane i s moved f u r t h e r from the o r i g i n . I t i s to he noted that m and u must be opposite i n s i g n f o r r e a l e l l i p s e s . I f u I s p o s i t i v e the surface l i e s w h o l l y on the p o s i t i v e side of the YZ plane*, Consider any plane p a r a l l e l to the JZ plane, (0, m, 0 ) , say. Trans l a t e the o r i g i n to the p o i n t •L y - 1 « 0 ; m that i s the new XZ plane i s t h i s plane. Equation ( l 8 ) becomes (Chapter I I ) For any p o i n t i n the new XZ plane S = 0 . Therefore the p o i n t - c o n d i t i o n equation o f the plane s e c t i o n by the new JZ plane becomes ( 2 0 ) t 2r + 1 c XL m"b I t can eas i l y be shown the l i n e - c o n d i t i o n equation f o r a parabola has the same form as ( 2 0 ). Therefore the ( l ) This can be dose by a method s i m i l a r to that 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 plane i s a parabola,, I n the same way we can show that sections p a r a l l e l t o the XY plane y i e l d parabolas• We c a l l the surface whose equation i s (17) 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 - nate plane are e l l i p s e s and the sections p a r a l l e l t o the other two c oBrdinate planes are parabolas. In the same way we can i n v e s t i g a t e the surface whose equation i s (£1) b v y - c v i ' ' + 2ux = 0. Sections p a r a l l e l to the YZ plane y i e l d hyperbolas and sec- t i o n s p a r a l l e l to the other two coordinate planes y i e l d parabolas,. Therefore (21) represents an h y p e r b o l i c para- boloid® For ( I ? ) J - ~ u v ( b % - O , j - b ' o V ; &• » 0, /\ <= b' GC u 6 . For (21) J « - u"(b"- ou), ^ = - b ^ u * , ^ - 0, A - - b* c 4u" . I f u = 0, we have b y + c z =0 or b y - c s =0. The f i r s t i s a s p e c i a l case of the i n f i n i t e e l l i p s o i d , and the second represents a p a i r of 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 these two cases - ^ - = = 0o When e = 0, we have b^y " + 2ux = 0. This i s a parabola i n the XY p l a n e I n t h i s ease 3- = - b ? u u , jj< = = A •= 0. The p o i n t - c o n d i t i o n equation reduces to that i s , t = 0, and the p o i n t s a l l l i e i n the XY plane. (3) 7. I n v a r i a n t s f o r the Various Equations;- Equation A a vx"+ b > ^ c " z v = 1 - _ + a x + b y - c z = l + ? ? a x - b y ~ c z < = l - - ? ? b^y^- c^z'-s- 2ux « 0 - 0 + _ b^y"- c"z"l"+ 2ux = 0 + 0 ? (1) Valgardsson " l i n e GoBrdinates", Ch. I I I . (2) Valgardsson, Ch. I I , Sect. 4. (3) I t i s understood that a l l c o e f f i c i e n t s appearing i n the f o l l o w i n g t a b l e are d i f f e r e n t from zero. 8o, Equation A 3- 3 a x v + by"1' + ess1" = 0 0 0 0 0 a x v + b y " = 0 0 0 0 0 a Tx v+ 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 Since these are a l l the p o s s i b l e equations, we can say that when. £ ^ 0, £ 0 we have an e l l i p s o i d or an hyper- b o l o i d . of one or tw o sheets. If A ^ 0, «® = 0 we must have e i t h e r an e l l i p t i c or hype r b o l i c p a r a b o l o i d . I f ^ = =. jj - 0, and J? =f^ 0 we have a plane curve, which can be an e l l i p s e , parabola, or hyperbola. I f A - = ^ *= _9 ^ 0, the e quation represents two p o i n t s , or else may be s a t i s f i e d o nly by po i n t s a t i n f i n i t y . The o r i g i n a l equation represents two po 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 which a necessary 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 & vanish. 81 CHAPTER IT Reduction of the General Equation 1. General Statement In t h i s chapter we s h a l l consider the r e d u c t i o n o f the general equation when A G, that i s , when the equation represents an e l l i p s o i d , hyperboloid, or parab o l o i d * 2. Reduction of the P o i n t - C o n d i t i o n Equation;- Let the equation Ar"+ BS^-J- Ct"+ 2Fst + 2Gtr + 2Hrs * 2Ur + 2Ys (1) " ' + 2Wt *" D = 0 be the point-condi t l o n equat ion of the surface ax"* by"*'+ cz1'-!- 2fyz + 2gzx + 2hxy + 2ux + 2vy (2) " " * 2wz +"d « 0. We have seen ( S e c t i o n 4, Chap. I I ) that there i s at l e a s t one p r i n c i p a l plane. Take thi s plane f o r the XY plane i n a new system of co o r d i n a t e s . The degree of ( l ) w i l l be una l t e r e d by the tr 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 plane 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 t herefore i f r ( x + s ( y * t ( z - l = 0 be any p o i n t on the surface, the p o i n t r ( x * s y - t, z - 1 = 0 w i l l a l so be on the surface. From t h i s we see t h a t i n the transformed equation F = G- = W = 0. The reduced equation therefore i s A r % Bs" + Ct"+ 2Hrs + 2Ur + 27s + D = 0. l\Tow r o t a t e the X, Y axes through an angle O g i v e n by the r e l a t i o n tan 2& = 2H . A-vB according to the transformations (4j>) of Chapter I I , namely r = r'oos & + s ' s i n Q , s = - r' s i n 0 + s' cos 6 , t « t ' . ; Dropping primes, we get an equation of the form O ) A r % Bs w+ Ct v+ 2Ur + 2Ys + D = 0» ( i ) Let A, B, C be a i l f i n i t e and d i f f e r e n t from zero. We can then w r i t e equation (j>) i n the form A ( r * TJ ) \ B/s + T ) \ Ct"= U~4 ¥"~ DHD'. V A / V. B"7 A . B" Hence, by changing the o r i g i n to the p o i n t | x + J y - 1 - 0 by means o f formulae (46) of Chapter I I , we obta i n Ar"" + Bs" + C t " » 3)'. I f D1 be not zero we have 83. which we can w r i t e i n the form (4) a b c or (5) r " s ̂  „ t a b c or (6) r 1 " s"*- t E 1, ac cor ding as D; , D' , D' are a l l p o s i t i v e , two p o s i t i v e and A 1" U one negative, or one p o s i t i v e and two negative, r e s p e c t i v e l y * ( i f a l l three are negative the surface i s c l e a r l y imaginary.) I f 3' be aero, we have (7) A r " + Bs' u -s- C t " =0. ( i i ) Let A, any one o f the c o e f f i c i e n t s , he zero. Write the equation i n the form 2Ur + B^s + Y y C t V + D - Y v = 0. I f U be not zero, by changing 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 can reduce the equation to (8) B s " + C t " + 2Ur = 0, 8 4 . I f U = 0, we have the form (?) B s " + C t v + D' = 0, or, i f I)'= 0, the form (10) Bs" + Ct" = 0. ( i i i ) L e t B, C, two ©f the three c o e f f i c i e n t s , be zero. We then have A^r + ] J j \ ZJa * D' - I f we t r a n s l a t e the o r i g i n to the p o i n t U - 0, A the equation reduces to the form (11) r.*" = 23ts. I f , however, ¥ = 0, the equation i s equivalent to (12) r*" = 3c' © 3 o go Find the Equations of the Center of a Conicoid.:- I f the o r i g i n i s the center of the s u r f a c e , i t i s the middle p o i n t of a l l chords passing through i t; i f r , 2 c - t - s 1 y * t | z - l = 0 be any p o i n t on the surfac e , the point - r ( x - s ( y - t, z - 1 = 0 w i l l also 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, v + 2Fs, t, + 2Gt, r, + 2Hr, s, - 2Ur, the r e f o r e - 27a, -• 2Wt( + D « 0; TJr Vs ; + Wt, «= 0. Since t h i s r e l a t i o n holds f o r a l l p o i n t s on the surface, we must have U, V, W a l l zero. Hence, when the o r i g i n i s the center o f a c o n i c o i d , the c o e f f i c i e n t s of r , s, t are a l l zero. Let o r x + ^ y + y z - l « = 0 be the equation of the center of 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 of r , s, t i n the transformed equation w i l l a l l be zero. The transformed equation 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 + / ) x+ 2F(s + ̂  ) ( t i «f ) + 2G(t + y ) ( r + a<) + 2H(r + ^ " ) ( s +^ ) + 2tJ(r + <<) * 27(s + /3 ) + 2W(t + ) + D « 0. Hence"the equations g i v i n g the center are Ae(. * H ^ + G T + IT « 0, H<< + B /3 + F 2T + 7 = 0, G *c + Fy3 + C r + W •» 0. (13) 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 Q f The p o i n t - c o n d i t i o n equation of the e o n i c o i d when the center i s a t the o r i g i n i s (14) Ar"+ Bs v+ Ct"+ 2Fst + 2Gtr + 2Hrs + D ' = 0, where D ' i s obtained from (3) by p u t t i n g r = <*" , s -/3 , t = tY • M u l t i p l y equations (13) i n order by •< , ^ , % and s u b t r a c t the sum from D j then we have. (13) V = TJe< + Y/3 + W * + D* From (13) and (15) we have A H G TJ H B F Y I » 0; G F G W TJ:- Y w D-D'! 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 which may be w r i t t e n (17) v'c£> « A . I t i s seen that the equation of the center i s gi v e n by (18) x + V y + s - £> - 0, where e t c . , are the co-f a c t o r s of IT, Y, e t c , In Z\ 87. 4. The D i s c r i m i n a t i n g Cubic;- We have seen ( S e c t i o n 2) that by a proper choice of rec t a n g u l a r axes Ar*+ Bs^* Gt%- 2Fst + 2Gtr + 2Hrs can always be reduced to the form c< r v+ ^ s""+ y t - 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 without changing the o r i g i n , f o r the terms of second degree are not a l t e r e d by transforming to any p a r a l l e l axes. How r v + s^+ t *Is u n a l t e r e d by a change of rectangu- l a r axes through the same o r i g i n . Hence, when the axes are so changed t h a t A r v + Bs^+ C t % 2Fst + 2Gtr ->- 2Hrs becomes r^+ (h s^+ V t \ (19) Ar^+ Bs^+ GtN- 2Fst +' 2Gtr + 2Hrs - J ( r " + s" + t" ) w i l l become (20) ^ r v + ^3 s % f t * " - J'(rx'+ a*"* t . Both these expressions w i l l t herefore be the pro- duct of l i n e a r f a c t o r s f o r the same values of J • The c o n d i t i o n that (19) i s the product of l i n e a r f a c t o r s i s (21) | A - J H G II B - f F 1=0, G F C -jT But (20) i s the product o f l i n e a r f a c t o r s when J i s 88. equal to oC , ̂ , or , Hence oc , p s are the three r o o t s of (21). The equation when expanded i s f* - f (A + 3 + C) + f (AE + BC + CA - F*~- G-*"~ H*) - (ABC + 2FGH - A3?*"- SG^-CH") = 0, or a s ) y -&r*w - - This equation 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 £ 0:- From equation (18) we see tha t there i s a d e f i n i t e center a t a f i n i t e d i s t a n c e , unless «= 0. I f 0 and one o f ^ , <V , Q/0 Is d i f f e r e n t from zero ( i . e . A 0) there i s a d e f i n i t e center a t an i n f i n i t e d i s t a n c e . I f be not zero, change to p a r a l l e l axes through the center, and the equation becomes Ar"+ Bs^* C t % 2Fst + 2Gtr + 2Hrs + 35 ' « 0 S where D' i s found'as i n S e c t i o n 2.' How, keeping the o r i g i n f i x e d , change the axes i n such a manner that the equation i s reduced to the form oC r % fl> s % V t + 3)' = 0. Then, by Section 3, c*T , /3 , Y w i l l be the three r o o t s of 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 equation may be w r i t t e n i n the form JiU r % &/!> s"* ^ / t " + Z\ =0. I f the three q u a n t i t i e s "P* , 'QJJL. , °^J^ are 8?. a l l n e g a t i v e , the surface i s an e l l i p s o i d ; i f two of them are negative, the surface i s an hyperboloid of one sheet; i f one i s negative, the surface i s an hyperboloid of two sheets; and i f they are a l l p o s i t i v e , the surface i s an imaginary e l l i p s o i d . We have shown i n Chapter I I that the general equation can be reduced t o one of the three forms (23) azv+ by* + oz" - 1 = 0, (24) ax"+ by"-*- c s w ~ 0, (23) b y % cz*" + 2ux*= 0* We see from Section 7"of Chapter I I I that £ 0 always r e q u i r e s A -fc 0, which i s true only f o r (23)«. b. D i s c u s s i o n of the Case j9 = 0 : - When = 0, one root of the d i s c r i m i n a t i n g cubic must be zero. From Se c t i o n 4, Chapter I I , we see that one p r i n c i p a l plane must be the plane (0, 0, 0)« I f 0, vie must have tw0 f i n i t e p r i n c i p a l planes, and therefore the center 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 princ i p a l planes© I f $ - 0 and Z\ 0, equation (18) shows that the center i s a t i n f i n i t y . Since one root of the d i s c r i m i n a t i n g cubic i s zero, the equation can e a s i l y be solved; l e t the root s be 0, oC > ̂ 3 . Find the d i r e c t i o n cosines of the p r i n c i p a l a x i s by means of equations (16), Chapter I I , and take the 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, equation w i l l then become v+ (t> t* + 2U'r + 2V's + ZW't + JD = 0 , or, by s change* o f o r i g i n , «=<• s v + t % - 2U 'r = 0 . Hence we have the su r f a c e , which, expressed I n plane coordinates, i s ay v+ bs^* 2ux = O,^1^ since A ^ 0 . 7 . Summary;- Let us i n v e s t i g a t e the general equation of a e o n i c o i d . I f A ^ 0 and d 7̂  0 , i t f o l l o w s that JQ £ 0 and w e have an e l l i p s o i d , or hy p e r b o l o i d . I f & i s p o s i t i v e we have the hyperboloid of one sheet. I f & i s negative we d i s c o v e r the nature of the surface by s o l v i n g the d i s c r i m i n a t i n g cubic- three r o o t s w i t h the same sig n denote an e l l i p s o i d and r o o t s which d i f f e r i n s i g n denote an hyperboloid o f two sheets,, I f Z\ £ 0 but d = 0 , i t f o l l o w s thatj@ = 0 „ ^ This gives us an e l l i p t i c or h y p e r b o l i c p a r a b o l o i d according as A i s negative or p o s i t i v e , r e s p e c t i v e l y . The plane curves are found to be those surfaces f o r which a l l the i n v a r i a n t s except 3 vanish. I f d = 0 the plane curve i s a parabola. 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 or hyperbola according as J) i s negative or p o s i t i v e , r e s p e c t i v e l y * A p a i r o f p o i n t s i s g i v e n Mien = ^ ~ = £± ~ 0 provided t h a t the equation i s f a c t o r a b l e e Otherwise the equation represents an i n f i n i t e c o n i - c o i d or an i n f i n i t e conic * BIBLIOGRAPHY 1. Lambert " A n a l y t i c Geometry 1 3, The Macmillan Co., 1904. 2. Smith " S o l i d Geometry", Macmillan and Co., 188?o 3c Snyder and Sisam " A n a l y t i c Geometry of Space", Henry H o l t and Co.,1 1914. 4. Tanner and A l l e n " B r i e f Course i n A n a l y t i c Geometry",' American Book Co., 1911. 5. valgardsson "Line Coordinates", M.A. Thesis a t U n i v e r s i t y o f Manitoba. 6. Wilson " S o l i d Geometry and Conic S e c t i o n s " , Macmillan and Co., 1898.

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