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

-pi & lt； = t & lt=pi or 0

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

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

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

Solution of closed area: Take ρ=a( 1+cosθ) as an example.

Let the area element be dA, then

dA = 1/2 * a∧2 *( 1+cosθ)∧2 * dθ

The area of the upper half shaft is obtained by integral method.

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.

Extended data:

Descartes and Christina did have a friendship in history. But Descartes came to Sweden at the invitation of Christina on 16491October 4, when Christina had become the queen of Christina. Descartes and Christina are mainly discussing philosophical issues rather than mathematics. It is recorded that Descartes can only discuss philosophy with Queen Christina at five o'clock in the morning because of her tight schedule. Descartes' real cause of death was pneumonia caused by cold weather and overwork, not the Black Death.

References:

Baidu Baike Heart Line