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Concept
Nephroid
Summary
In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger one by a factor of one-half.
Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Richard A. Proctor in 1878.
A nephroid is
an algebraic curve of degree 6.
an epicycloid with two cusps
a plane simple closed curve = a Jordan curve
If the small circle has radius , the fixed circle has midpoint and radius , the rolling angle of the small circle is and point the starting point (see diagram) then one gets the parametric representation:
The complex map maps the unit circle to a nephroid
The proof of the parametric representation is easily done by using complex numbers and their representation as complex plane. The movement of the small circle can be split into two rotations. In the complex plane a rotation of a point around point (origin) by an angle can be performed by the multiplication of point (complex number) by . Hence the
rotation around point by angle is ,
rotation around point by angle is .
A point of the nephroid is generated by the rotation of point by and the subsequent rotation with :
Herefrom one gets
(The formulae were used. See trigonometric functions.)
Inserting and into the equation
shows that this equation is an implicit representation of the curve.
With
one gets
If the cusps are on the y-axis the parametric representation is
and the implicit one:
For the nephroid above the
arclength is
area and
radius of curvature is
The proofs of these statements use suitable formulae on curves (arc length, area and radius of curvature) and the parametric representation above
and their derivatives
Proof for the arc length
Proof for the area
Proof for the radius of curvature
It can be generated by rolling a circle with radius on the outside of a fixed circle with radius . Hence, a nephroid is an epicycloid.
Let be a circle and points of a diameter , then the envelope of the pencil of circles, which have midpoints on and are touching is a nephroid with cusps .
Official source
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