CodePudding user response:

CodePudding user response:

Heart line, is a fixed point on a circle in it around the tangent and the radius of the same with another circle formed by rolling when trajectory, named for its shaped like hearts,

CodePudding user response:

Polar equation
Horizontal direction: rho=a (1 - cos theta) or rho=a (1 + cosine theta) (a> 0)
Vertical direction: rho=a (1 - sine theta) or rho=a (1 + sine theta) (a> 0)
Rectangular coordinate equation of
Cardioid expression plane rectangular coordinate system of equations respectively x ^ 2 + x ^ 2 + a * y=a * SQRT (x ^ 2 + y ^ 2) and x ^ 2 + y ^ 2 - a * x=a * SQRT (x ^ 2 + y ^ 2)
Parameter equation
- pi<=t<=PI or 0 & lt;=t<=2 * PI
X=a * (2 * cos (t) - cos (2 * t))
Y=a * (2 * sin (t) - sin (2 * t))
Enclosed area of 3/2 a ^ 2 * PI * and formation of the arc length is 8 a
Enclosed area: the application of minimal polyomial to rho=(1 + cosine theta) as an example a
That area is dA yuan,
DA=1/2 * a Sunday afternoon 2 * (1 + cosine theta) Sunday afternoon 2 * d theta.
Using the integral method to the area of the upper shaft
A=?(PI -> 0) A Sunday afternoon 2 1/2 * * (1 + cosine theta) Sunday afternoon 2 * d theta.
=3/4 * a Sunday afternoon 2 * PI
So the heart-shaped line formed by the area S=2 a=3/2 Sunday afternoon 2 * PI * a