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)

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.

parameter 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.

Extended data:

Math Story tells the love story of mathematician Descartes. Descartes 1596 was born in France and wandered to Sweden during the Black Death.

1649, 52-year-old Descartes met 18-year-old Swedish princess Christine on a street in Stockholm. A few days later, he received an unexpected notice that the king hired him as the little princess's math teacher. He followed to inform his guards to go to the palace and saw the girl he met in the street. From then on, he became the little princess's math teacher.

Under the careful guidance of Descartes, the little princess made rapid progress in mathematics. Descartes introduced himself to the princess.

Cartesian coordinate system is a new research field. Every day we are inseparable, so their love for each other, the princess's father, the king flew into a rage and ordered Descartes, the little princess Christine? Princess Christine begged the king to go into exile in France and put her father under house arrest.