What are the love axes and analytical formulas?

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.

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:

story

The Story of Mathematics tells the love story of mathematician Descartes. Descartes was born in France on 1596. When the Black Death broke out in continental Europe, he wandered to Sweden.

1649, on the street of Stockholm, Descartes, 52, met the Swedish princess Christine. 18. A few days later, he was unexpectedly informed that the king hired him as the little princess's math teacher. He came to the palace with the guards who came to inform him. He met the girl he met in the street. From then on, he became the little princess's math teacher.

The little princess's mathematics has advanced by leaps and bounds under the careful guidance of Descartes who introduced herself.

A new research field--rectangular coordinate system. Being inseparable every day makes them fall in love. When the princess's father found out, the king flew into a rage and ordered Descartes to be executed. After the little princess Christine begged, the king exiled her to France, and Princess Christine was placed under house arrest by her father.

Descartes fell seriously ill soon after he returned to France. He writes to the princess every day. Because she was intercepted by the king, Christine never received a letter from Descartes. Descartes died after sending the 13 letter to Christine. The letter 13 only contains a short formula: r=a( 1-sinθ).

The king couldn't understand that they didn't always talk about love. He called all the mathematicians in the city to the palace, but no one could work it out. He couldn't bear to see his beloved daughter moping all day, so he gave this letter to Christine who had been moping all the time.

As soon as the princess saw it, she immediately understood the intention of her lover. She immediately set out to draw a graph of the equation. She was very happy when she saw this number. She knows that her lover still loves her. The original figure of this equation is a heart. This is also the famous "heart line".

After the king died, Christine ascended the throne and immediately sent people to look for her sweetheart all over Europe. Unfortunately, she died and left before her, leaving her alone on the earth. ...

It is said that this world-famous alternative love letter is still preserved in the Descartes Memorial Hall in Europe.

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