| e | 1 | ||
| r = | — | cosx + | — ; |
| 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, measured from the circumference of the circle. The point O is the node of the limaçon; 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 = p / (e cosx). Hence the equation of its inverse is r = (e cosx) / p, which is the equation of the base circle of the limaçon.
If the conic which we invert be an ellipse, the point O will be an acnode on the Limaçon; 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 inverse curve is called the Cardioid.
The limaçon 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 describe a limaçon of which A B will be the base circle.
The pedal of a circle with respect to any point is a limaçon. 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 limaçon.
The base circle is the locus of the instantaneous centre for all points on the limaçon. Let B O P be a line cutting a circle in B and Q. Let the line revolve about B, Q following the circle; the point P will trace a limaçon.
Now, for any instant, the instantaneous center will be the same whether Q be following the circle or the tangent at the point where the line cuts the circle. Therefore the instantaneous center for the point P is found by erecting a perpendicular to the line P B, through B, and a normal to the circle at Q. (Williamson’s Diff. Cal. Art. 294). The intersection (C) of these two lines is the instantaneous center for the curve at the point P. But by elementary geometry C is on the circle. Now as the line P B revolves through 360° around B, the line B C which is always perpendicular to it also makes a complete revolution and the instantaneous center C moves once round the base circle.
Below we give a list of theorems obtained by inverting the corresponding theorems respecting a conic. In these theorems any circle through the pole is called a nodal circle, any chord through the pole is called a nodal chord, and the line through the pole perpendicular to the axis of the curve is called the latus rectum. The letters e and p signify respectively the eccentricity and half the latus rectum of the inverted conic.