On Pascal's Limacon and the Cardioid. 



BY H. C. RIGGS. 



The inverse of a conic with respect to a focus is a curve called 

 Pascal's Limacon. From the polar equation of a conic, the focus 

 being the pole, it is evident that the polar equation of the lima9on 

 may be written in the form: 



r=: — cos:c H : 



P P 



where e and p are constants, being respectively the eccentricity and 

 semi-latus rectum of the conic. 



From the above equation it is readily seen that the curve may be 

 traced by drawing from a fixed point O on a circle any number of 

 chords and laying off a constant length on each of these lines, meas- 

 ured from the circumference of the circle. The point O is the node 

 of the limacon; and the fixed circle, which I shall call the base circle, 

 is the inverse of the directrix of the conic. This is readily shown as 



follows: — the polar equation of the directrix is r= . Hence the 



ecoso: 



ecoStX 

 equation of its inverse is r= , which is the equation of the base 



. . P 



circle of the limacon. 



If the conic which we invert be an ellipse, the point O will be an 

 acnode on the Limacon; if the conic be a hyperbola, the point O is 

 a crunode. If the conic be a parabola, O is then a cusp and the in- 

 verse curve is called the Cardioid. 



The lima9on may also be traced as a roulette. 



Let the circle A C have a diameter just twice that of the circle A B. 

 Then a given diameter of A C will always pass through a fixed point 

 Q on the circle A B, (Williamson's Diff. Cal. Art. 286) and will have its 

 middle point on the circle A B. Now any point P on the diameter of 

 A C will always be at a fixed distance from C and will therefore de- 

 scribe a limacon of which A B will be the base circle. 



The pedal of a circle with respect to any point is a limacon. This 

 may be inferred from the general theorem that the pedal of a curve is 

 the inverse of its polar reciprocal, (Salmon's H. P. C. Art. 122). For 

 the polar reciprocal of a conic from its focus is a circle and hence its 

 pedal is a lima9on. 



(89) KAN, UNIV. QIJAK., VOL. I., NO. 2, OCT., lS9i. 



