1, polar coordinate equation

Horizontal direction: ρ=a( 1-cosθ) or ρ = a (1+cos θ) (a > 0)

Vertical direction: ρ=a( 1-sinθ) or ρ = a (1+sinθ) (a > 0)

2. Cartesian coordinate equation

The expressions of the plane rectangular coordinate system equation of the heart line are x 2+y 2+a * x = a * sqrt (x 2+y 2) and x 2+y 2-a * x = a * sqrt (x 2+y 2) respectively.

3. Parametric equation

x = a *(2 * cos(t)-cos(2 * t))y = a *(2 * sin(t)-sin(2 * t))

Extended data:

The enclosed area is 3/2 * pi * a 2, and the arc length is 8a.

Solutions for enclosed areas:

Take ρ=a( 1+cosθ) as an example.

Let the area element be dA, then da =1/2 * a ∧ 2 * (1+cos θ) ∧ 2 * d θ.

Using the integral method, the area of the upper half shaft can be obtained:

a =∫(π→0) 1/2 * a∧2 *( 1+cosθ)∧2 * dθ

=3/4*a∧2*π

So the area enclosed by the whole heart line is S=2A=3/2*a∧2*π.