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.

parameter equation

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.

polar 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)

Extended data

Christine was a princess in Sweden in the17th century. She is beautiful, kind and intelligent, and especially likes math. One day, she put on casual clothes and went outside the palace. On the way, she saw many beggars, one of whom was very special. He didn't take the initiative to ask passers-by for charity, but quietly squatted on the ground and concentrated on studying math problems.

The man didn't know that the lady standing in front of him was the princess, but he was surprised by the mathematical talent revealed in the lady's words, so he was happy to talk to Christine. Princess Christine realized that he was originally a mathematician, but unfortunately, for some reason, he was frustrated in doing mathematics in France and was down and out, and finally he wandered to Sweden.

So Princess Christine invited mathematicians to the palace to be her math teacher, and they discussed math problems together and talked about everything. For a long time, they were immersed in the world of mathematics and love, and they were very happy.