The combination of rays is also sufficient in practice if the **cardioid** surface is replaced, by a spherical one.

The polar equation to the **cardioid** is r=a(1-}-cos 0).

The form of the limacon depends on the ratio of the two constants; if a be greater than b, the curve lies entirely outside the circle; if a equals b, it is known as a **cardioid**; if a is less than b, the curve has a node within the circle; the particular case when b= 2a is known as the trisectrix.

In the figure (1) is a limagon, (2) the **cardioid**, (3) the trisectrix.

The epicychid when the radii of the circles are equal is the **cardioid** (q.v), and the corresponding trochoidal curves are limacons.

This may be readily accomplished geometrically or analytically, and it will be found that the envelope is a **cardioid**, i.e.

The polar form is {(u+p) cos 26} a+{(u-p) sin 20) a = (2k)t, where p and k are the reciprocals of c and a, and u the reciprocal of the radius vector of any point on the caustic. When c =a or = oo the curve reduces to the **cardioid** or the two cusped epicycloid previously discussed.

It is a double mirror system, whose reflecting surfaces are a sphere a and a **cardioid** b.

**Cardioid** Condenser.

**CARDIOID**, a curve so named by G.