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Polar line coordinates and their application to pedal curves Heaslip, Leonard William 1928

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U . B . C . L I B R A R Y CAT. »r> Lki 69. lit* ft<s- rijPfe ACC. WO. ^ ^ POLAR LINE COORDINATES and THEIR APPLICATION to PEDAL CURVES by Leonard William Heaslip. * * * A Thesis submitted for the Degree of MASTER OF ARTS in the Department of MATHEMATICS CONTENTS Section Page 1. Introduction . 1 2. Rectangular Line coordinater 1 3. Polar Line Coordinates 3 4. Transformations 4 5. First Form of Equation of a Point. . . . 5 6 Second Form of Equation of a Point . . . 5 7. The Circle 6 Q. Translation of Axis 7 9. A Curve and Its First Pedal Curves. Polar Coordinates 7 10. A Curve and Its First Pedal Curves. Rectangular Coordinates 8 11. First Positive Pedal Curves of Conies . . 9 12. First Negative Pedal Curves of Conies . . 10 13. A Conie Its Own Pedal 11 1. Introduction. This paper introduces a system of polar line coordinates, applied to plane analytic geometry, and develops a general method of finding the pedal curves of any given curve. Transformations are worked out which enable one to pass from the equations of a system of curves in rectangular line coordinates to the equations of their first positive pedal curves in rectangular point coordinates and in the opposite direction for the first negative pedal curves. * 2. Rectangular Line Coordinates* In this paper are assumed the relations of plane analytic geometry expressed in terms of a system of rectangular line coordinates defined as follows, •"•he coordinates(x,y) of a line are the reciprocals of the intercepts of the line on a set of rectangular axes. In this system a point whose rectangular point coordinates are (r,s) has the equation rx+sy-1^0 (l) A conic is defined as the envelope of a line which moves-so that the product of its distances from two fixed points called the foci is a constant k. The line joining the foci is called the principal axis. If the x axis is taken as the principal axis and the origin as one focus and the other focus a distance 2c from the origin the general equation of any conic becomec + • k k ( 2 ) A treatment of rectangular line coordinates developing. . the usual relations of plane analytic geometry is given by valgardsson in a paper on record in the university o f Manitoba.The results in #2 are mainly from that paper. If k is positive and equal to b the conic is an ellipse and its equation is * 1 2cx x + y , „ _ . . t . (3) If a - o the foci coincide ana we have the circle with its equation JL bL ' (4) x+y - • z If k is negative and equal to -b the conic is an hyperbola and its equation becomes X f y r - + - , [5) b b If k is positive but b and c each approach infinity and 2. b the limit of -- equals p, a finite quantity, the conic is a c parabola with its equation * 2x " " "p • (6) The formulae for translation of axes to the point rx + sy-J. ^  0 are x — ^ — and y = 7l " It rx, + sy» Urx, + sy. (7) and for rotation through an angle (9 x^. xposfl-y.sin a and y =: xsin 6 r^cos 6 r (8) If these transformatioiir.are applied to equation(2) the degree remains unchanged. Eence every conic has a second degree equation. Under rotation y~is invariant. Hence reference to equation (2) shows that if a conic has a focus at the origi n its equation has no xy term and it has equal coefficients for x and y. Complete information concerning the general second degree equation AxV2Hxy f Byfrt 2Gx +2Fy -)-G -0 is summarized in the table below in which. A H Cr <S> = H B F G F C - C(Af B)-(Pf &} , A - H-AD, <S>*o C ^ o Ellipse if J < 0 andC^ tcSfe-gree in sign. Imaginary locus ifj| > 0 and € and CS>agree in sign. Hyperbola if G and c8>differ in sign, while P has any value. c Parabola. ^ is always negative. O^o c * 0 Two real points i f ^ ^ O . Two imaginary points <3>- 0 c = 0 Two distinct real points, one of which is infinitely distant, and the other is in a finite region of the plane. <3 ^ o C r O s Two infinitely distant real points i f A ^ O . Two coincident and infinitely distant points if^ 0 Two imaginary pointson the line at infinityif^ 0 3 Polar Line Coordinates. The position of a line is determined by reference to a fixed line called the polar axis and a fixed point on it called the pole. The coordinateflsof a line are where f is the perpendicular distance from the pole to the line and 6 is the vectorial angle between the polar axis and this perpendicular where the usual conventions regarding positive and negative angles and line segments are followed. o \ X Example. "Let 0 be the pole and OX the polar axis. Then the line ABis defined by the coordinates {rf &) > /^being the perpendicular distance OP and Q the angle between OX and OP measured counter clockwise. A Transformations. Let the origin ana x axis or a rectangular line coordinate system coincide with the pole and polar axis respectively of a polar line system. Then the coordinates (XY) and (fid) of any line in the two systems are seen to be related as follows, y Hence to change from a rectangular to a polar line coordinate system the transformations are cos ft * sin & (9) and from polar to rectangular, (10) 5. First fform of Bquation of a $olnt. Lst P be the point whose distance from 0 is d and for which the angle between OXand OP i'a <=< . Draw any line {f, $) through P. From the diagram /o cos > hence the required equation of the point P is d cos (<9—<) . (11) 6. Second Form of Equation of a Point. Let P be the point whose equation is required and r and s the7 lengths of the projections of OP on OX and on a line perpendicular to OX respectively. Through P draw any line (/?#). Then it is readily seen from the diagram that r co s 0 -f s sln£, (12) This will be called the standard equation of a point and will be the form chiefly used in further work.It is especially useful since the distances r and s correspond to the r and s of the rectangular line equation (l) of a point. That these equations represent the same point is apparent for on applying the transformations of (9} to rx + sy - 1» 0 we obtain r cos & s sin 6 , _ r» + 7° 1 ~ ° which upon simplification is f° z. r co s & + s s in & . 7. The Circle. Definition.- The circle is the envelope of a line which is at a constant distance from a fixed point called the centre. Let the line {f3f B) envelope the circle whose radius is a and whose centre is the point c r cos ^  -j- s sin£. Itis evident from the figure that the equation of the circle Is f* = r cos 0 -f. s sin£)-|-a (13) If the centre Is on the polar axis r=±A, s=0 and the equation reduces to a(l± cos £ ). ( 1 4 ) Again if the centre is at the pole r = s = 0 and the equation further reduces to (15) If^in (13) a=O ywe obtain the equation of a point, 7 8. Translation of Axis. If in the diagram of Section 7 the pointer: r cos^+a sin£ becomes the pole and the polar axis is parallel to OX.then the new/2* called ^ 'will equal a or f°' ~ cos & -+ s sin 9). That is the transformation necessary to change the origin from © to the point f* - r cos b s sin 6 is to replace by /^-y r cos 6 -f s sin B , (l6) 9. A Curve and Its First Pedal Curves. Polar Coordinates. Consider the circle (14) f* - a(l cos 0). Let 0P(2/^) be the perpendicular from the pole 0 on a tangent to the circle. The locus of P is called the first positive pedal curve of the circle with respect to the point 0, while the circle is the first negative pedal curve of 'the locus of P. Since the coordinates of the line (/f d) are identical with the coordinates which define P in a polar point coordinate system it is obvious that in this latter system the equation of the locus of P is the cardiod a(1i cos B) By similar reasoning it is seen that the first positive pedal curve of any curve whose equation in polar line 8 coordinates is is given in polar point coordinates by the same equation ttfe), o. In general any curve G, defined in polar line coordinates has a curve Gas its firstpositive pedal curve in polar point coordinates with the same equation, and the curve G^in polar point coordinates has the curve C; in polar line coordinates as its first negative pedal curve. 10. A Curve and Its First Pedal Curves. Rectangular Coordinates. The transformation formulae from rectangular to polar line coordinates are cos 0 sin 6 x * ' and rectangular to polar point coordinates x ?^qos ^  y^/^siXX B Hence for line coordinates cos fa fsin& y , ? T whereas for point coordinates / ^ s l y p 2 - " x'ty* > ~ x "if t. z. -Reversing the process, for point coordinates P sin & and for line coordinates . z. i° = --x- sin e . y P x*-*1' P xVy t. Therefore if we have an equation of a curve in rectangular line coordinates we can pass to the equationof its first positive pedal in rectangular point coordinates by replacing x by --Z--+ . y by ^ ' (17) J x t y and x +y Similarly if we have an equation of a curve in rectangular point coordinates we can pass to the equation of its first negative pedal in rectangular line coordinates by replacing exactly in the same way, x by and y by • x + y 1 xNy 2- ( l 8 ) 11. First Positive Pedal Curves of Conies. The general second degree equation of a conic, real or degenerate in rectangular line coordinates is A x % 2Hxy + By2-t2Gx *2Fy-f-C (l9) The corresponding first positive pedal curve equation in point coordinates obtained by substituting from (17) in (l$) is Ax2_^ 2Hxy-t-By2f2(Gx-y F y ) ( x 2 f y 2 H C(x2 + y 2 ) = 0 (20) For a central conic C ^ O so the first positive curve is in general of the fourth degree. If a focus is at the origin A ^ B and H - 0 and the equation of the pedal curve reduces to the circle A f2(Gx-^ Fy) + C(x2f y 2) c 0 (2l) For a parabola C2 0 so the equation of the pedal curve in point coordinates is in general of the third degree. 10 If, however, the focus is at the origin the equation becomes A. + 2 (G-x t ).- 0 (22) which represents a straight line. 12. First Negative Pedal Curves of Conies. The general second degree equation of a conic, real or degenerate, in rectangular point coordinates is Ax 2t 2Hxy + By2-+ 2 Gx -t2Fy -HC - 0 (23) The corresponding first negative pedal curve equation in line coordinates obtained by substituting from (18) in (23) is Ax 2+ 2Hxy + By2 + 2(Gxt Fy)(x2+ y 2) +C(x2+ y )^0. (24) If the general conic does not pass through the origin C^ 0 so the firstnegative pedal curve is of the fourth degree. If A ;B and H = 0/in which case the given conic is a circle^ the equation of the first negative pedal reduces to A+ 2(3x+ Fy)t C(x24 y 2)= 0 (25) which in general represents a conic in line coordinates with a focus at the origin. For C^0,in which case the conic(23) passes through the origin,the equation of the first negative pedal is in general of th® third degree. if, however,the given conic is a circle passing through the Qrigin then A = B and H - C -0 and the equation of the first negative pedal reduces to A^2(Gx^-Fy)-0 (26) which represents a point. 11 13. A Conic Ita Own Pedal. The question arises whether any conic can have itself as a pedal curve. By reference to (21) it is seen that under the substitutions employed here the only conic transforming into a conic is one with a focus at the origin, in which case it transforms into a circle. Hence a cir±le with centre at the origin is the only conic which can have Itself as a pedal curve. That a circle so situated does have itself as pedal curve is seen upon applying the transformations (17) or (13) ^ 2 . to either the line equation(4), x2+ y - gg > or the point equation, x2-f-y2_za2 , of the circle. 


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