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Tangential co-ordinates James, Ralph Duncan 1930-12-31

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TANGENTIAL CO-ORDIITATES "by Ralph Duncan James  A Thesis submitted for the  Degree of  MASTER OP ARTS  in the  Department of  MATHEMATICS  The University of British Columbia APRIL, 1930  TABLE OF GOUTSITS  Section  Page  1. p  1 Important Relations from Earlier Theses .  1  rr O  Points of Contact  .  '±  4.  The Condition that a Point Shall Lie on .  5  .  a Conic  .  .  .  .  .  .  .  .  .  .  ro.  Diameters.  6.  The ~:;es of the C-eueral Conic  7.  Properties of Diameters  8.  The Evolute of a Curve  9.  The Family of Curves  10.  .  .  .  .  .  6  Conjugate Diameters . . . .  9 11 -Jyt^fx,y) = 0 .  Loci of Points  . . . .  13  Transformations Between Point and Line Co-ordinates  18  12.  Polar Reciprocals  13.  Geometric Interpretation of Certain  .  .  .  .  .  .  .  .  Integrals in Line Co-ordinates . 14. 15.  8  Envelopes of One-parameter Families of Curves.  11.  .  The Tr^isf:rmation  x=  x  x f r  ,"  7 , i.M-y1-  P3  1.  TANGENTIAL GO-ORDINATES.  1.  Introduction,  This paper outlines certain developments  in the theory of Analytic Geometry, employing tangential or line co-ordinates. It supplements the work of Master's Theses written by Yalgardsson of the University of Manitoba, and 1. W. Heaslip of the University of British Columbia.  In  particular it treats of diameters of conics", evolutes, envelopes, loci of points, transformations between point and line co-ordinates, polar reciprocals of curves, and applications of the Calculus to line co-ordinate Geometry.  2.  Important Relations From Earlier Theses.  The relations  listed below, (a) to(k) from Yalgardsson's thesis, and (1) to (n) from Heaslip'a, are required in the development of this . paper. (a)  The co-ordinates of a line in rectangular line  co-  ordinates are defined as the reciprocals of the intercepts of the line on a set of rectangular axes. (b)  A point, other than the origin, whose point co-ordinates  are (Z, Y), has in line co-ordinates the equation, Zx Note.  +  Yj  -  1  =  0  Throughout this paper large letters are used to  (1)  denote point oo-ordinates, and small  letters, line co-  ordinates. (c)  The equation of the point of intersection of tv/o  lines, fx, ,y, ) and (x x ,y t ) is, y - y> _ fd)  y.  The angle 9 , between the negative extension of the  y - axis and the line joining the origin to a point P, is called the vectorial angle of the point P, and tan 6  - - I  »  d£  Y (e)  dx  (3)  The equation of the midpoint of the line Joining two  points Pj and P 2 whose equations are,  and  Z, x  +  Y, y  -  -\x  +  Yry  -  1 —  0  1 = 0 ,  respectively, is, (Z,x + Y, y (f)  1) +  (Z z x +  Two lines (x, ,y, ) and (x2 ,y3L)  (i) parallel if  x, y t  fii) perpendicular if (g)  -  x,xx  xjLyl +  =  1) = 0  (4)  are 0  yf y  s=.  0  (5)  A curve may "be considered as the envelope of a  variable line, and its equation in line co-ordinates is called its line or tangential equation. fh)  The equation of the point of intersection of the  tangents at the points where the line (x,,y,) cuts the conic, ax% is  2hxy + by*"+ 2gx + 2fy + c  =  0  fax,thy ( +g)x 4- (hx,+ by ( + f)y + g x ^ f y ^ c = 0  (6)  fi)  The equation of the pole of the line fx, ,y.) with  respect to the circle x* + is  y* -  1  x ( x + y, y - 1  fj)  a  0 0  (V)  The general equation of the second degree represents a  conic section. The nature of the conic is shown in the following table, where 0  0 C £ 0  &  =  I a h g| h b fl , f s c ( a + b ) - (f%- g x ), and A s lg f c|  Ellipse, if 0 and c and © a g r e e in sign Imaginary locus if f > 0 and c and © agree in sign Hyperbola, if c and ® differ in sign, while $ has any value.  #0  c = o  Parabola,  e  Two real points, if f<0. Two imaginary points if f > 0.  =o  c Q  =0  c  =0  1 e  ~ o  c = o ? (k)  h*—ab  =0  ^  is always negative.  Two distinct real points, one of which is infinitely distant, and the other is in a finite region of the plane. Two Two if Two if  infinitely distant real points if A > 0 coincident and infinitely distant points A s O imaginary points on the line at infinity 6 < 0  The equation of the centre of a central conic given by  the general equation is, gx  (1) are  f  fy +  c  o  (8)  The co-ordinates of a line in polar line co-ordinates (  6 ) , where n  is the perpendicular distance from the  4. pole to the line, and Q  is the vectorial angle between  the polar axis and this perpendicular. (m)  Any curve, whose equation in polar line co-ordinates  is  f ( f>, 6 ) a  0  has as its first positive pedal the curve given by the same equation  f (P,G)  -  0  in polar point co-ordinates. (n)  If the equation of a curve is given in rectangular  line co-ordinates, the equation of its first positive pedal in point co-ordinates may be found by replacing x  2  by X  Y  , and y by  + i'1  .  Z*- -t- Y x  Similarly, if the equation of the curve is given in rectangular point co-ordinates, the equation of its first negative pedal in line co-ordinates may be found by replacing  3.  X by  x «• t x + y  " 1 by , and  y x  L  , + y*-  .  Points of Contact,  let (x,,y,) be the co-ordinates of the tangent touching the curve  y = f fx) at the point P.  let f  ,  be a  5. neighboring tangent touching the curve at Q. The equation of the point of intersection R of the lines (x, ,y( ) and (Xj+A*,  ), by equation (2), is,  }  y  -  y. _ 4 L Ax, and  As Q, approaches P along the curve,  approach  zero. In the limit H and Q, coincide with P.  Therefore  y - y. ,  o) V d x y,  x - x,  is the equation of the point of contact , P*, of the tangent ( x . ,y, ) to the cur.ve y e f ( x ) . The equation may also be written in the form '£i x P, x , - 5!  4.  -  1 P,x, -  y 3f  -  l  =  o  (10)  The Condition that a Point Shall Lie on a Conic. Let the  equation's  of the point and conic be Xx  and  +  Yy  -  1  = 0  (11/  a x % 2hxy -fby%2gx-f 2fy-+ c  = 0  (12)  respectively. Substituting for y from (11) in (12), we obtain  (aY-2hXY-»-bX x )x+2(hY-&X + gY x -fXY)* + (b + 2fY -fcY* ) =  (13)  0  The point lies on the conic if and only if (13) has equal roots. The condition for equal roots is the vanishing of the discriminant  (hY-bX -f-gY-fXY)*"- (aY-2hXY+bX *) (b+2fY+cY*).  Accordingly (f  b c) X*"-f 2 (ch -f g) XY + (gl- ac) Y 1  +2(fh-bg)X-fS(gh-af)Y+hVab  =. 0  (14)  6. is the condition that the point shall lie on the conic.  5.  Diameters.  (i)  The diameter of a central conic is defined as the line  Conjugate Diameters.  passing through the points of contact of parallel tangents to the conic.  Let fx, ,y, ) and fkx, ,ky() he the conic  parallel lines tangent to  ax*+2hxy-+by4-2gx + 2 f y + c =  0  at the points P and Q respectively. The equations of P and Q, are  (ax,+hy,+ I •g )wx'+ (hx.+by,-!.f)y4 • I • v| • —gx.-f • V fy-f>—Ic• =— v 0•  and  (kfax. + hjj)-* gjx 4^kfhx,-4-by() -4 f|y-f k(gx,4-f^) + c -  —  w  0  respectively. On solving, we obtain the co-ordinates of the diameter through P and Q, viz., gx.4fy.-f c, x'- -  + fyJ G« ax,4hy,4 g,  hx,4 by, + f k(hx,4by,)4f hxr4by^ f  k( ax,4-hy.) -f g,  k(hx,4bjj)+f  hy( t g, y =  -  k(ax, + hy.) 4- R.  ( Ch-fg) X,4 (be - f*")y (af-gh)x,4 (fh- bg)y  gx, + fy^4 c k(gx.4- fy.)4c hx.4-by 4 f  > ( ax, 4 hy,) 4 g,  !g ~a c ? ! " .chly* (af-gh)x,4-(fh - bg) y,  k(hx,+ byt) +f  (15) On trial it is seen that gx' +  fy« -f- c  =  0  7. and'thus, equation (8), every diameter passes through the centre. The diameter passing through the points of contact of tangents parallel to (x',y') is said to he conjugate to the diameter  (x',y').  Let the co-ordinates of the two tangents parallel to (x',y') be (k, x'tkfy')  and (kfcx',kxy') respectively. Let  the points of contact of the tangents he R and S respectively. The respective equations of these  points  are  |k, (ax'+hy) + gj x 4- jk, fhx'+hy') •+• f)y-H:, ( g x V fy) + c a  and  I k t (ax'+h/) + gj x + |k x (hxV by) + fjy+k^ (gxV fy) + c s? 0  0  Solving, we obtain the co-ordinates of the diameter conjugate to (x',y'),  viz.,  »  _ ~  ( ch — f g ) x * (be —• f *) y 1 (af- gh)x'+(fh — bg) y'  ..»  „  (g*- ac) x  ~  ( a f - g h ) x ' - f - ( f h - b g ) y»  y  (16)  (fg — oh) y f  Substituting x ' and y ' from equation (15), these reduce to X*  =  CX  f  ,  y"  -  2Z-  (I?)  It is seen, equation (5 i), that the diameter (x",y") is parallel to the tangent (x,,y ), gx ' -f- fy" -/- c  a  and by trial,  0  Dividing x " by y* in equati ns (16), we obtain x" _ y" or  (ch- f g)x *•*(be — f *)y' (g*- —ac)x'-f-(fg —ch)y'  (g\- ac) x' x"f ( f g - c h ) (x'y + x ' V ) + ff*"- b o ) y ' y » 0 (18)  8. which is the condition that two diameters shall be conjugate. In the case of the ellipse i 2 "I. ax b y  1.  —  1  —  o  this reduces to in x x r r fii)  , 4 _ b =  The parabola may be considered as the limiting case  of a central conic, given by the general equation of the second degree, as the parameter c  approaches zero. Prom this  point of view, the diameter of the general parabola may be defined a^s the limiting position of the line passing through the points of contact of parallel  tangents (x, ,y, )  and (kx,,ky,) to the general conic as c and k simultaneously approach zero. Accordingly, by reference to the equations (15) obtained under (i) of this section, the co-ordinates of the diameter passing through £he point of contact of a tangent (x, ,y, ) to the general parabola are as follows, '  7  ~  f(gg,-»fy,) (af — gh)x,+(3Fh— bg)y,  '-  ( +f (af — ghfx/H f h — b g ) y,  6. The Axes of the General Conic. diameters are the axes of the conic.  (19)  Perpendicular conjugate Let the co-ordinates  of conjugate diameters be (x',y') and (x"y").  Por  perpendicularity, equation (5 ii), we have  (§')($ = ~1  x" Replacing =„ by its value from equation (18) , we obtain  9. (fg -oh)x'V(f*- b o - g V ac)x'y'+(ch-fg)y's 0 Since a diameter passes through the centre of the conic, we have  gx' -+• fy' -h  c  =  0  On solving these two equations we obtain the co-ordinates of the axes, viz., » ~ y  7.  =  -2 g ( f g- ch) - f (f -g*) - c f (a-b )± tfft f-efi + o( a-b )(j4 (fg-ch f 2 [h ( f-g^ +£g (a-b) j  "  2f ( fg-ch) -g( f-Pfj -cg(a-b )?gj(( -4-0 (a-b )ft4 (fg-ch)* 2 j h ( f - ^ + f g (a-b) \ 1 1 (20)  Properties of Pi meters.  the c oni c  let the chord (-<t,ya) cut  ax •* 2hxy + by*-+ 2gx -+ 2 fy-+ c  =  0  at the point P whose equation is Xx  +  Yy +  1  =  0  Since the point P lies on the conic and also on the line, we must have the relations, (equation (14) ), (f-bc) Xf-2 (ch-fg) XY+( g-ac) Y+2 (fh-bg)X+2 (gh-af) Y i h*"— ab and  Xxt  - 0  -+- Yy t —  (21) 1  =  0  (22)  Substituting for Y from (22) in (21), we obtain £(g*-ac) x* -2 (ch-f g) x* y x + (f" -bc) jf ] X ** -+Z ( (eh-fg) yv - (g x -ac) xx+( fh-bg) y*- (gh-af) x ^ + { (g*-ac) + 2 (gh-af  h l -ab)  J  =  0  X  (23)  and similarly, [ ( - a c ) x* -2 ( ch-f g) x z y x + ! f 1 -b c) y ^ Y * -f-2 f ( ch-fg) xx-( f*-bc) y r - ( fh-bg) x ^ - f ( gh-af j x*] Y -f-|(f*-bc)-f- 2(fh-bg)x t +(h l -ab)x* j -  0  (24)  10 Let the roots of equations (23) and (24) be  and Y, , x%  respectively. The equations of the points in which the line (z .y ) cuts the conic are therefore + and  ^y  -»-  -  1  ti  -  1  = 0  0  (25) (26)  The equation of the midpoint of the line joining these two points,(equation (4) ), is (X,+ X t ) x +  (Y.+Yjy  - 2 = 0  which, because of equations (23) and (24), becomes |(ch-fg) yt -fgl-ac)xJt+( fh-bg) 3^*"-(gh-af) xxyjj x + {(ch-fg) x ^ f * - b e )  (fh-bg) xtyk+( gh-af) x*] y  H- f (g 1 -ac)xj-2(ch-fg)x 1 y i +(f l -bc)y 1> l ]a 0 Let t'.-e line (x x ,y z ) be parallel to (x, , y, ) , ir =  kx, uid j x =  ' (27)  'V. jh. cace  ky, . Equation (27) thus becomes  [(ch-fg) y( - (g -ac) x,+ (fh-bg) ky,"1- f gh-af) kx, y§"J x 4- £ (ch-fg) x, - (f% -be) y( - (fh-bg) kx, v-f (gh-af) kx* ^ y -+ { (g x -ac)kx*-2 (ch-fg) kx, v + (f x -bc) ky.^0 By trial we see that the co-ordinates (x',y') of the diamet passing through the point of contact of the tangent (x, ,y ) satisfy this equation. Hence the diameter bisects all chord parallel to the tangent (x, , y, ) . The equation of the point of intersection of the tangen to the conic at the extremities of the chord (kx, ,ky, ), ^which is parallel to (x, ,y, j), by equation (6), is £ k(ax(-f hv) f g^x-f^k (hx^bjj) + ijy +  kfgx.-ff^+c  11. On trial it is seen that the co-ordinates of the diameter (x',y') satisfy this equation. Therefore the tangents at the extremities of any chord parallel to a tangent (x, ,y, } intersect on the, diameter passing through the point of contact of the tangent fx,,y,). The co-ordinates of the axes of the general parabola are obtained by putting c = 0 x  =  in equations (SO). h(f%g4+fg(a-b>  " -  They are  '  (28)  sj =  0, ,  h(fN-g^ + fg(a-b) •  Comparing equations (19) and (£8), we see that the diameters of a parabola are parallel to the axis. 8.  The Evolute of a Curve.  The evolute of a curve is  defined a s the envelope of the normals to the curve. Let the equation of the curve be y  =. f(x)  (29)  Let (x,,y() be any tangent to the curve and (x z ,y r ) the corresponding normal. Siu-ce the normal is perpendicular  to  the tangent and passes through its point of contact we have the relations x.i. xx. -h 71 t-1. « and Solving for  x,  =• 0  —-y, P.*. -y.  - 1 =  0  and y x and dropping the subscripts, we have  12. It follows that the equation of the evolute is obtained on eliminating x, said y, between equations (29) and (30).  If  the equation of the curve is given in the form f(x,y) = then  ^  =  _ Is fy X -  ,  and equations (30) nay be written  y. fx, + Y, ) v _ V f - V f • y<>»  Example.  0  x,(x. f>.±v, fr. ) w f > . ol  X, Xy(  y, -  X, fy.(  To find the equation of the evolute of the ellipse a x - f - b y fX( =  —  2a l x,  1 ,  =  0  f v = 2b x y, .  Therefore, equation (31), V - y. (2a^x,X-t- 2b* y.*) _ " (2aix,y, - 2b*x, v ) y  and  -  1 (a*- b-)x,  _ x, (2a*~x^-f- 2 __ - " U a ' x . y , - 2b«-x, yt ) ~ a  ,  b  1 (a1-- b*)y,  _ a x fb y  =1  (a-"bx )*x The equation of the evolute is b x + a y 9.  = (a -b ) x y  .  The Family of Curves Af, (x,y) -J- /tf^fx.y) = f, (x,yl f t (x,y)  = =  be the tangential equations of  0.  let  0  (32)  0  (33) two curves. Then  A f, (x,y)+ ytuf„ (x,y) ^  0  (34)  is the equation of any curve touching the common tang-en• s of the curves (32) and (33). This follows, since equation (34) is satisfied by the values of x and y for which both (32) and (33) vanish, and so is satisfied by the co-ordinates of the tangents, real or imaginary, which are common to the  13. curves (32) and (33).  10.  Envelopes of One-parameter Families of Curves. Loci  of Points.  The equation f(x,y,ot) =  0  (35)  defines a one-parameter family of curves or points. Consecutive curves or points of such a family are defined as curves which correspond to two consecutive values of the parameter. The envelope of a family of curves given "by equation (35) is defined as the envelope of the limiting position of common tangents to consecutive curves. Let oC and ot + Ac<he two consecutive values of the parameter. Then f (x, y, oC ) and  =  f(x, y, ot+A*)  0 =  (36) 0  (37)  are consecutive curves, and f(x, y,  A oC  - f(xt y,  )  _  0  is the equation of a curve touching the common tangents of  -v. (37). ., . <_> — <J 1 the curves (36) and Accordingly, L  J f (x. y, o( + A<X) - f(x, y, 0 I A <X a  fecU, y,<* ) =  )( J  0  (33)  is an equation satisfied "by the co-ordinates of the common tangent to (36) and (37) in its limiting position. It follows that the equation of the envelope is obtained on eliminating oC between equations (35) and (38) .  14. The equation of a point involves in general two parameters.  A point, however, may be restricted in position,  through being constrained to satisfy some geometric condition. In that case the totality of points satisfying the condition constitutes a locus.  Any condition imposed upon the point  may be expressed as a relation between the parameters.  It  is thus possible to eliminate one of the parameters from the equation of the point.  The problem of finding the equation  of the locus of a. point, satisfying a certain geometric condition, therefore reduces to that of finding the equation of the locus of a point whose equation involves a single parameter.  Such an equation is linear in x and y and has  the form (35) above.  For a fixed value of oL ,  f(x, y, * ) and  -  0  f (x,y, d + M ) =  0  (39)  are consecutive points, Pi and P z of the locus. f(x, y, * +  - f (x, y,* ) _  A o<.  The equation  0  is the equation of a point on the line passing through P x and P z .  Accordingly L  [f(x, I  - f(x, y,*) A*  = f*(x, y , * )  J =  0  J f40)  is an equation which is satisfied by the coordinates of the limiting position of the line P t P 2 .  The locus is clearly  the envelope of these limiting lines determined by letting oC vary.  The co-ordinates of any such line satisfy both  15. (39) ana (40), and so must satisfy the equation obtained on eliminating c{ between (39) and (40). is, accordingly, that of the locus.  This resultant equation It is thus seen that  th$ method of determining the locus of a point is identical with that of finding the envelope of a one-parameter family of curves. A simpler treatment is possible when the equation of the point,, for fixed values of x and y, is algebraic in terms of the parameter (i)  .  Consider the equation of a point oC f, (x, y)  f^fx, y) —  0  (41)  where o( is a variable parameter, and f t (x, y) =  0,  f-jjx, y) — 0 are linear in x and y and independent of  .  Equation (41) is satisfied by the co-ordinates of the line passing through the two points whose equations are f, (x, y) - o f^(x, y) - 0, respectively.  In other words, the locus of a point is a  straight line when the equation of the point contains a single variable parameter which enters to the first degree only. (ii) Consider the equation of a point / f , (x, y)+<XfJx, y R f 3 ( x , y) = 0 where c( , f, , f ,  f 3 have meanings as above.  (42)  For a fixed  line, (x, , y, ) there are two values of ^ for which (42) is satisfied.  If (x,,y,) is tangent to the locus of (42), the  16. two values "become coincident.  The values of x and y for  which (42) has equal roots are given by equating the discriminant to zero.  Thus  j f ^ x , y)j - 4 jf, (x, y)J(f 3 (x, y ) ] = 0 is the equation satisfied by all values of x and y which are the co-ordinates of tangents to (42) .  It is therefore the  equation of the locus, and since it is of the second degree, the locus must be a conic section, (iii)  Consider the equation of a point  ffx, y,o04= off, fx, y)-f  -j-f^, fx, y) = 0  f43)  The equation of the locus of (43) is obtained by equating to zero the discriminant of f43) considered as an equation in o(.  This discriminant is found by eliminating c* between  (43) and f^fx, y, cX. ) =  0, which is precisely the method  given in a previous paragraph for finding the equation of an envelope. Example 1.  The locus of the point whose equation is oCk - <*y - 1 - 0  is clearly Example 2.  y* = 4x. To find the locus of a point which moves so that  its distance from a fixed point is always a constant times its distance from a fixed straight line. 35  A  iY  X '  Let the equation of the point ?  be  17. Xx +Yy  - 1 = o  (44)  Take the fixed point at the origin, and let AB, (-c, 0), be the fixed straight line.  From  )op|= e ,| PIvlJ  fe, a constant), we have  X*>YV= ev(I4cf Y"= (ev- 1)X H- ae'-cX-fe^c51  or  Substituting in equation (44) ,  (45)  we obtain  £( e —1) X % 'Ze^oX-h e v o^j  (1 - Xx) 1  This equation cannot be put in the form of equation (42) where f, ( f t ' and f3 are independent of X and linear in x and y, and thus'we use the general method. Differentiating equation (45) with respect ot X, we T  _  -  have  y'e^ ± x e x - x -y  Making this substitution, equation (45) reduces to e*c *(xVy*" )~h 2e v ex  -  (1 - e v ) = 0  (46)  The method of obtaining this equation shows that every point on the locus lies on the curve. (46). We must now show that every point on the curve (46) is on the locus. 3u pose that the point P  is o£ the curve (46), but not on the locus.  Thus we have  X* f  and  e* (IC+o f ( i\ J,l|  From equation (47) it follows that ± J s f 7 ? « e (X l + c) whence  |?i  in contradiction to our hypothesis. Accordingly, every point on the curve (46) is on the loeus.  (47)  18. 11. Transformations Between Point and Line Co-ordinates.  Let (xt the curve  ) "be the co-ordinates of the tangent AB to y = f(x)  at the point P. Let the point co-  ordinates of P "be (Xj.Yj). The equation of P is X, x -f Y, y -  1  =  0  However, since P is the point of contact of the tangent (x, ,y4 ) its equation is El x P, x, -y, * where  1 T p, x, -y, if  1  =  0  =  Comparing these equations, it is seen that P, x, -y, » ^ •  The equation of the line AB  Y  =  -  1  p, x, -y,  in point co-ordinates is  x, X -f- y, Y -  1  =  0  Since AB is tangent to the curve, its equation is 1 - Y, ~ P,(X - X, ) where P, . (g) ( • This may "be written  whence we have x  ~ P, X, -Y, ' Vs*  1 P(  -Y, '  Hence, dropping primes, the equations v — rr-r P 1 —— P x -y  Ysr  (48),  1  v  p x -y  "" P X -Y  x = ~  _  (49)  L _  y ~  p X -Y  give the transformations from point to line co-ordinates, and from line to point co-ordinates respectively. Let the tangential equation of a curve be y = f(x)  (50)  The substitutions' of equations (48) give I , ( P ) = 1  PX - Y or  P(X, Y, P) =  (PX - Y / 0  (51)  In general, P appears in this equation to a degree higher than the first, and thus for any given  X  and  Y,  P  will  not be single-valued. However, to every tangent (x, y) satisfying equation (50) there is a point of contact (X, Y) and one and only one corresponding value of the slope P, such that  X,  Y,  and  values of  X  and  roots for  P  are those satisfying the P-discriminant  relation.  This relation contains the singular solution of  Y  P  satisfy equation (51).  for which equation (51) gives equal  the differential equation (51).1  1  See D. A. Murray,  Those  Thus we see that the  Differential Equations,  §33,  page 42.  20. equation of the curve (50) in point co-ordinates is the singular solution of (51), that is, the envelope of the general solution of (51). If (50) is an algebraic equation, than on clearing of fractions, equation (51) will be a polynomial in (PX-Y), with coefficients functions of form  P. Its solutions are of the  PX - Y — f(P)  (52)  This equation is a Clairaut equation, 1 GX - Y = f(C)  and has for solution *  (53)  which represents a family of straight lines. The singular solution of equation (52) of equation (53).  That is,  is the G - discriminant relation the required equation in point  co-ordinates is the envelope of the family of straight lines (53) . Obviously the above argument and indicated procedure also applies,  on a change of notation, when passing from  point to line co-ordinates. Example 1.  An ellipse, with centre at the origin,  and axes coinciding with the co-ordinate axes, has in line co-ordinates the equation Z  2- , 2.  a x + b y where a  and  b  =  1,  are the semi-major and semi-minor axes  respectively. Applying the transformations of equations (49), we obtain  (a* -X)p l + 2XYP + (b«-Y) =  0  The P - discriminant relation is 1  See:  D. A. Hurray,  Differential Equations, § 33.page 42  21. { & - X)(fe2- Y) = 0 X 1 , Y* =, 1  or  a-  +  which is the equation of the ellipse in point co-ordinates. Example 2.  The equation of the astroid in point  co-ordinates is X*+  = af  Applying the transformations (48), we have p* -f- 1 - a"5(px - y p  <  (54)  Differentiating with respect to p, we obtain p or  5  = a x(px - y) 3  a z px J s px - y  Making this substitution, equation (54) becomes i 1 azxi - 1  (55)  Eliminating p between equations (54) and (55), we obtain l i z. a. a . x -f- y =  a x y,  .  (5b)  the tangential equation of the astroid. The astroid may be defined as the envelope of a line, the sum of the squares of whose intercepts on the axes is a constant. Accordingly, its tangential equation is  or  x  2.  -f y  I  1 2. 1 = a x y  The transformations (49') give (P*+l) (PX-Yf =  a2"P*  M57)  22. Differentiating with respect to P, we have 2 f p ' + 1)(PX - Y)X+2P(PX - Yj^^a*-?  (53)  From equation (57) we obtain p4- D  .2.  (3?x  Y) =  " p^FT On making this substitution, equation (58) becomes P  + 1 =  a* fx  (59)  Eliminating P between equations (58) and (59), we have X* + y^ = a T , the point equation of the astroid.  12.  Polar Reciprocals.  Let the point equation of a  curve be f(I, Y) - 0  (GO)  The question arises, what locus lots 2 U , y) = 0 represent in line co-ordinates?  (61) Evidently to every point  (X,,Y,) satisfying equation (60) there corresponds a tangent (x, = X, , y = Y,) satisfying equation (61). Since 1 1 1 1 this tangent- has intercepts t = — ~ = v » i t s P0*3^ >-. a i y, i, equation is X, X + Y , Y - 1 = 0 This is clearly the equation of the polar of the point (X,, Y, ) with respect to the circle x* +  y" - 1 - 0  Thus we see that the curve (61) is the envelope of the  23. polars of points on the curve (60). That is, it is the 1 polar reciprocal  of (60).  Therefore a curve in rect-  angular point co-ordinates and its polar reciprocal in rectangular line co-ordinates have identical equations. By reference to the footnote it is easily seen that a curve in rectangular line co-ordinates and its polar reciprocal in point co-ordinates have identical equations.  I-Ience the  transformations (48) and (49) when applied to the equation of any curve give its polar reciprocal,. the equation of the curve and that of its polar reciprocal "being in the same system of co-ordinates.  13.  Geometric Interpretation of Pertain Integrals.  An interpretation can how be given for any integral in line co-ordinates.  For, from the previous section, it is seen  that the usual interpretation applies to the polar reciprocal of the given curve and not to the curve itself.  Thus  I -j- fly Y . dj, gives the length of arc of a curve in point co-ordinates, while 1  / J I + ^  y.Ax gives the  The polar reciprocal of a curve, with respect to the  circle IC1 -f- Y*" - 1, may he defined as the envelope of the polars of points on the curve, or, as the locus of the poles of tangents to the curve.  24. length of arc of the polar reciprocal of a curve whose equation is in line co-ordinates.  The integral  J  ~Y. c( X  gives the area under a curve in point co-ordinates, while fydx  gives the area under the polar reciprocal of a curve  "whose equation is in line co-ordinates. Again, (§2, (m) ) , any curve with equation f{{>, 0 ) =• 0 in polar line co-ordinates has as its first positive pedal the curve given by the same equation f f P ^ © ) =• 0 in polar point co-ordinates.  Hence the interpretation of an integral  in polar line co-ordinates is applied to the first positive pedal of the curve.  14.  Pedal Curves.  curve be f(x r y) =  Let the tangential equation of a 0.  The equation of its polar reciprocal  is found in point co-ordinates by replacing x by X and y by Y.  The equation of its first positive pedal is found in  X point co-ordinates by replacing x by — — —  and y by —  V  ,  Thus we see that the equation of the first positive pedal of any curve in point co-ordinates may be obtained from the equation of the polar reciprocal of the curve in the same V system of co-ordinates by replacing X by X * ^ y-'-and Y by y V f t y*' Accordingly the first positive pedal of any curve is the inverse of the polar reciprocal of the curve. Similarly, the tangential equation of the polar reciprocal of any curve in point co-ordinates is obtained by replacing X by x and Y by y.  25. The equation of the first positive pedal of the polar reciprocal is found by replacing x by  X  x*1 y  jSnd y by J.X.  * -f Y  r  Therefore, the first positive pedal of the polar reciprocal of any curve is the inverse of the curve.  15.  The Transformation  x -  x  x r  x  > y=  y*  By means of the transformation  Y= we r+i1 X V Y: may write the equation of the inverse of any curve. The question arises whether the transformation x=  has a geometrical significance,  x+y x'+y' ^s the figure shows, let  AB be a tangent (x, y) to the curve ffx, y) = 0.  Let A' 3'  be the line  be the  ^x1 -+ y1' xa  perpendiculars respectively.  X  0  f/ yr  OA CA1 X  Hence  ?  and  A' 3'  ^sT" X  A  y  /  so that  and 0P' = r  £  A3 is parallel to A B  or  and OP-^o  from the origin on the lines AB  £  have  y1/  l  x  since v  OP _ OP'" f'  y  -  y x'^y1-  = 1 ff and p' are inverse points.  V,hence we  From this it follows  that J± b' is the polar of the point  P.  Therefore, the  envelope of A / B / is the polar reciprocal of the locus of  P,  that is, of the first positive pedal of the curve f(x,y) » 0. Hence the transformations  when applied to the tangential equation of a curve, give the equation of the polar reciprocal of the first positive pedal of the curve. Applying (62) Xx  to the equation of a point P +  Yy -  1  =  0  (63)  =  0  (64)  we have y*"- Xx - Yy which,  § 2 (j) ,  is the equation of a parabola. By means  of equation (14), we see that the inverse point of P X X^+Vx  Y X*+V  y  =  o  lies on the parabola, and also, equations (28), lies on the axis of the parabola. It is therefore the vertex.  It  follows that the equations (62) transform a point into a parabola whose vertex is the inverse point of th~ given point.  

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