NephroidNephroid"a special plane curve"
A nephroid
is a special plane curve that is derived from the epicyloid. The
epicycloid is a figure traced out by the circle of radius b that rolls
on the outside of the circle of radius a. Epicycloids are defined
by these parametric equations:
x = (a + b) cos(t) - b cos((a/b +1)t),
y = (a + b) sin(t) - b sin((a/b +1)t)
When
a=2b a nephroid is obtained. Nephroids can be defined as the trace
of a point fixed on a circle of radius 1/2 r that rolls around a fixed
circle with radius r. Another way to say it is the Nephroid is a
trace of a fixed point on a circle rolling around a fixed circle that has
a radius two times as large as the moving circle's.
The parametric equations that define a nephroid are:
x = a(3 cos(t) - cos(3t)),
y = a(3 sin(t) - sin(3t))
This is a nephroid with a=1.
This is a nephroid with a=4. Notice that the only change is in
the size of the nephroid.
In 1678 Huygens showed that the nephroid is the catacaustic of a circle
when the light source is at infinity. However, the name nephroid
wasn't used until R.A. Proctor, the English mathematician, used the term
in The Geometry of Cycloids. Nephroid, which means kidney-shape,
was used by Proctor to describe a two cusped epicycloid. A catacaustic
is the trace of rays from a given point reflecting off a curve. In
this case the point is a light source. If you were to take a light
source and reflect of a circle you would get a nephroid. A nephroid
is also the catacaustic of a cardiod if the light source is emitted from
the cusp of the cardiod.
This link provided much information about nephroids: http://www.best.com/~xah/SpecialPlaneCurves_dir/Nephroid_dir/nephroid.html
You can find more information about Huygens here:
http://www.maths.tcd.ie/pub/HistMath/People/Huygens/RouseBall/RB_Huygens.html
More on nephroids was found here:
http://www.astro.virginia.edu/~eww6n/math/Nephroid.html