Paths, Cardioid

Cardioid

The cardioid has a simple geometric construction.  Place a circle of radius 1 in the plane, with center at -1,0.  This is the fixed circle.  Let a second circle of radius 1 roll around the first circle.  When the rolling circle is to the right of the fixed circle, the common point of tangency is called p.  Note that p coincides with the origin.  As the circle rolls around, p traces the cardioid.  Roll the circle up just a bit and watch what happens.  Draw a line connecting the two centers, then draw a radius in the fixed circle connecting the center to the origin, then draw a radius in the roling circle connecting the center to p.  Finally draw a segment from the origin to p.  This quadrilateral is a perfect trapezoid.  The base angle is θ, hence the top of the trapezoid, i.e. the shorter side, is 2(1-cos(θ)).  This is the radial distance to p, from the origin, when the angle is θ.  This reproduces the formula given above.

Returning to arc length, the length of the cardioid is given by the following integral, as θ runs from 0 to 2π.

∫ sqrt(r² + r′²)

Replace r with 2-cos(θ) and get this.

∫ sqrt(8-8cos(θ))

Pull 2 out, and replace θ with 2u.  Integrate from 0 to π.

∫ 2sqrt(2-2cos(2u))

Use the double angle formula to get this.

∫ 2sqrt(4sin²(u))

∫ 4sin(u)

-4cos(u)

Evaluate from 0 to π to get an arc length of 8.

To find the area, integrate r from 0 to 2-2cos(θ), giving 2(1-cos(θ))².  Integrate this with respect to θ and get the following.

3θ - 4sin(θ) + sin(2θ)/2