## The Cardioid

The cardioid (Greek: "heart-like") is a familiar plane curve with some interesting properties. It is the locus of a point on the circumference of a circle rolling on a second equal circle of diameter a, and is, therefore, an epicycloid. If the origin is taken at the initial point of contact, the reference direction is to the left, and the moving circle rotates clockwise, the polar equation of the curve is r = a(1 - cos θ). The curve has a cusp at the origin, where the common tangent to the two circles is horizontal. It is easier to draw a cardioid by plotting from this equation than from the definition as an epicycloid. If you are not familiar with how a cardioid looks, make a good plot of one by this method. Polar coordinate paper can be used, or a protractor and straightedge, or rectangular coordinates can be calculated.

The derivation of the polar equation is shown at the right. It is a little trickier than might be expected. The first thing to notice is that the radius vector OP is parallel to the line of centres of the circles. This can be found from the equality of the arcs QP and QO, where Q is the moving point of contact (this is the condition of rolling without slipping). The chord of the arc QO or QP is of length (a/2)sin(θ/2). The angle between the chord and the common tangent is θ/2. The radius vector is twice the projection of the chord on the direction OP. Note that a is the diameter of one of the circles, not the radius.

The fact that OP is parallel to the line of centres of the circles provides an easy way to find a point P on the cardioid. In the diagram, let the circle with centre at A be the base circle. Draw an equal circle with centre B tangent to the base circle at any point. Then draw a line through O parallel to AB. The intersection of this line with circle B is a point on the cardioid. There are actually two intersections, one appropriate for a cardioid with cusp at O, and the other for a cardioid with a diametrically opposite cusp. The proper one to use is easily determined.

A pressure microphone has a sensitivity independent of direction, while a ribbon (velocity) microphone has a sensitivity proportional to cos θ. A properly proportioned combination of these two responses gives a response R = A(1 - cos θ), where θ = 0 corresponds to the backward direction, and θ = 180° to the forward direction. Such a microphone will not be as sensitive to sound coming from behind it as sound coming into the front, a very desirable characteristic. Such a microphone is said to have a cardioid response. The response is usually not an exact cardioid, however. It is usually specified as the ratio of front to back responses, expressed in dB.

A cardioid is also a special case of a Pascal Snail. A Pascal Snail is the path of a point M found by adding a fixed amount to the line OP from an origin O at the left of a diameter of a circle to a point P on a circle. For the cardioid, OM = OP + a, where a is the diameter of the circle. This gives another way of plotting a cardioid using only graphical computation. The polar equation follows at once from this construction.

The equation of the cardioid in rectangular coordinates is not enlightening: (x2 + y2)2 - 2ax(x2 + y2) = a2y2. If this is written in the form f(x,y,a) = 0, then we can easily find that the partials ∂f/∂x and ∂f/∂y are both zero at the origin, where the cusp is located.

The cardioid is the pedal curve of the circle with respect to a point on its circumference. The pedal is the locus of the foot of the perpendicular from a fixed point P on the tangent to the curve. The circle is its own pedal with respect to its centre (because the radius is always perpendicular to a tangent).

Ole Rømer studied the cardioid as a possible shape for gear teeth in 1674. I have not heard of cardioidal teeth, but cycloidal teeth are certainly practical.

### References

Bronshtein and Semendyayev, A Guide Book to Mathematics (Zürich: Harri Deutsch, 1973). p. 121.