Cardioid proofs
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- This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article Cardioid. This article is currently an experimental vehicle to see how we might be able to provide proofs and details for math articles without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for current discussion. This article is "experimental" in that it is a proposal for one way that we might be able to deal with expressing proofs.
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[edit] Theorem
The parametric equations are
The same shape can be defined in polar coordinates by the equation
[edit] Proof
Start from r = 1 + cosθ. From Polar_coordinates#Cartesian_and_cylindrical and Trigonometric_identity#Double-angle_formulas we see that we get the cartesian coordinates:
[edit] Another proof
Equations (1) and (2) define a cardioid whose cuspidal point is (−1/2, 0). To convert to polar, the cusp should preferably be at the origin, so add 1/2 to the abscissa:
The polar radius ρ(θ) is given by
Expand:
Simplify by noticing that
Thus,
Then, since
it follows that
[edit] Area derivation
The objective is to integrate the area of the cardioid
- r = 1 − cosθ.
The integral is
- .
Integration with respect to dr yields
Distribute the integral among the three terms, and integrate the first two, to obtain
The second term vanishes, and integrating the third term yields
The last term within brackets vanishes, so that
Cardioids of any size are all similar to each other, so increasing the cardioid's linear size by a factor of a increases the cardioid's areal size by a factor of a2, Q.E.D. (return to article)