View source for Nephroid

{{short description|Plane curve; an epicycloid with radii differing by 1/2}}
{{more citations needed|date=May 2018}}
[[File:Nephroide-definition.svg|thumb|Nephroid: definition]]

In [[geometry]], a '''nephroid''' ({{ety|grc|''ὁ νεφρός'' (ho nephros)|[[kidney]]-shaped}}) 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.

==Name==
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.<ref>{{MathWorld|title=Nephroid|urlname=Nephroid}}</ref><ref>{{Cite web |title=Nephroid |url=https://mathshistory.st-andrews.ac.uk/Curves/Nephroid/ |access-date=2022-08-12 |website=Maths History |language=en}}</ref>

==Strict definition ==
A nephroid is 
* an [[algebraic curve]] of [[Degree of a polynomial|degree]] 6. 
* an [[epicycloid]] with  two [[Cusp (singularity)|cusps]] 
* a plane simple closed curve = a [[Jordan curve]]

=== Equations  ===
[[File:EpitrochoidOn2.gif|thumb|generation of a nephroid by a rolling circle]]

====Parametric====

If the small circle has radius <math>a</math>, the fixed circle has midpoint <math>(0,0)</math> and radius <math>2a</math>, the rolling angle of the small circle is <math>2\varphi</math> and point <math>(2a,0)</math> the starting point (see diagram) then one gets the [[Parametric equation|parametric representation]]:
:<math>x(\varphi) = 3a\cos\varphi- a\cos3\varphi=6a\cos\varphi-4a \cos^3\varphi \ ,</math>
:<math>y(\varphi) = 3a \sin\varphi - a\sin3\varphi =4a\sin^3\varphi\ , \qquad 0\le \varphi < 2\pi</math>

The complex map <math>z \to z^3 + 3z</math> maps the unit circle to a nephroid<ref>[https://www.math.uni-bonn.de/people/karcher/ATO%20URL%20Collection.pdf Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath]</ref>

=====Proof of the parametric representation=====
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 <math>z</math> around point <math>0</math> (origin) by an angle <math>\varphi</math> can be performed by the multiplication of point <math>z</math> (complex number) by <math> e^{i\varphi}</math>. Hence the 
:rotation <math>\Phi_3</math> around point <math>3a</math> by angle <math>2\varphi</math> is <math>: z \mapsto 3a+(z-3a)e^{i2\varphi}</math> ,
:rotation <math>\Phi_0</math> around point <math>0</math> by angle <math>\varphi</math> is <math>:\quad z \mapsto ze^{i\varphi}</math>.

A point <math> p(\varphi)</math> of the nephroid  is generated by the rotation of point <math>2a</math> by  <math>\Phi_3</math> and the subsequent rotation with <math>\Phi_0</math>:
:<math>p(\varphi)=\Phi_0(\Phi_3(2a))=\Phi_0(3a-ae^{i2\varphi})=(3a-ae^{i2\varphi})e^{i\varphi}=3ae^{i\varphi}-ae^{i3\varphi}</math>.
Herefrom one gets
:<math>
\begin{array}{cclcccc}
x(\varphi)&=&3a\cos\varphi-a\cos3\varphi &=& 6a\cos\varphi-4a \cos^3\varphi \ ,&& \\
y(\varphi)&=&3a\sin\varphi-a\sin3\varphi&=& 4a\sin^3\varphi &.&
\end{array}  </math> 
(The formulae <math>  e^{i\varphi}=\cos\varphi+ i\sin\varphi, \ \cos^2\varphi+ \sin^2\varphi=1, \ \cos3\varphi=4\cos^3\varphi-3\cos\varphi,\;\sin 3\varphi=3\sin\varphi -4\sin^3\varphi</math> were used. See [[trigonometric functions]].)

====Implicit ====
Inserting <math>x(\varphi)</math> and <math>y(\varphi)</math> into the equation
*<math>(x^2+y^2-4a^2)^3=108a^4y^2</math>
shows that this equation is an [[Implicit curve|implicit representation]] of the curve.

=====Proof of the implicit representation=====
With
:<math>x^2+y^2-4a^2=(3a\cos\varphi-a\cos3\varphi)^2+(3a\sin\varphi-a\sin3\varphi)^2 -4a^2=\cdots=6a^2(1-\cos2\varphi)=12a^2\sin^2\varphi</math>
one gets
:<math>(x^2+y^2-4a^2)^3=(12a^2)^3\sin^6\varphi=108a^4(4a\sin^3\varphi)^2=108a^4y^2\ .</math>

==Orientation ==
If the cusps are on the y-axis the parametric representation is 
:<math>x=3a\cos \varphi+a\cos3\varphi,\quad y=3a\sin \varphi+a\sin3\varphi).</math>
and the implicit one:
:<math>(x^2+y^2-4a^2)^3=108a^4x^2.</math>

== Metric properties ==
For the nephroid above the
*[[arclength]] is <math> L= 24 a, </math>
*[[area]]  <math> A= 12\pi a^2\ </math> and
*[[radius of curvature (mathematics)|radius of curvature]] is <math>\rho=|3a\sin \varphi|.</math>
The proofs of these statements use suitable formulae on curves ([[curve#length of a curve|arc length]], [[area#Areas of 2-dimensional figures|area]] and [[radius of curvature]]) and the parametric representation above
:<math>x(\varphi)=6a\cos\varphi-4a \cos^3\varphi \ , </math>
:<math>y(\varphi)= 4a\sin^3\varphi </math>
and  their derivatives 
:<math>\dot x=-6a\sin\varphi(1 - 2\cos^2\varphi)\ ,\quad \ \ddot x= -6 a\cos \varphi(5-6\cos^2\varphi)\ ,</math>
:<math>\dot y=12a\sin^2\varphi\cos\varphi \quad , \quad \quad \quad \quad 
\ddot y=12a\sin\varphi(3\cos^2\varphi-1)\ . </math>

;Proof for the arc length:
:<math>L=2\int_0^\pi{\sqrt{\dot x^2+\dot y^2}} \; d\varphi=\cdots =12a\int_0^\pi \sin\varphi\; d\varphi= 24a</math> .

;Proof for the area:
:<math> A=2\cdot \tfrac{1}{2}|\int_0^\pi[x \dot y-y \dot x]\; d\varphi|=\cdots= 24a^2\int_0^\pi\sin^2\varphi\; d\varphi= 12\pi a^2</math> .

;Proof for the radius of curvature:
:<math>\rho = \left|\frac {\left({\dot{x}^2 + \dot{y}^2}\right)^\frac32}{\dot {x}\ddot{y} - \dot{y}\ddot{x}}\right|=\cdots= |3a\sin \varphi|.</math>

[[File:Nephroide-kreise.svg|thumb|Nephroid as envelope of a pencil of circles]]
==Construction==
* It can be generated by rolling a circle with radius <math>a</math> on the outside of a fixed circle with radius <math>2a</math>. Hence, a nephroid is an [[epicycloid]].

=== Nephroid as envelope of a pencil of circles ===
*Let be <math>c_0</math> a circle and <math>D_1,D_2</math> points of a diameter <math>d_{12}</math>, then the envelope of the pencil of circles, which have midpoints on <math>c_0</math> and are touching <math>d_{12}</math> is a ''nephroid'' with cusps <math>D_1,D_2</math>.

====Proof====
Let <math>c_0</math> be the circle <math>(2a\cos\varphi,2a\sin\varphi)</math> with midpoint <math>(0,0)</math> and radius <math>2a</math>. The diameter may lie on the x-axis (see diagram). The pencil of circles has equations:
:<math> f(x,y,\varphi)=(x-2a\cos\varphi)^2+(y-2a\sin\varphi)^2-(2a\sin\varphi)^2=0 \ .</math>
The envelope condition is
:<math>f_\varphi(x,y,\varphi)=2a(x\sin\varphi -y\cos\varphi-2a\cos\varphi\sin\varphi)=0\ . </math>
One can easily check that the point of the nephroid <math>p(\varphi)=(6a\cos\varphi-4a \cos^3\varphi\; ,\; 4a\sin^3\varphi)</math> is a solution of the system <math>f(x,y,\varphi)=0, \; f_\varphi(x,y,\varphi)=0</math> and hence a point of the envelope of the pencil of circles.

=== Nephroid as envelope of a pencil of lines ===
[[File:Nephroide-sek-tang-prinzip.svg|thumb|nephroid: tangents as chords of a circle, principle]]
[[File:Nephroide-sek-tang.svg|thumb|nephroid: tangents as chords of a circle]]
Similar to the generation of a [[cardioid]] as envelope of a pencil of lines the following procedure holds:
# Draw a circle, divide its perimeter into equal spaced parts with <math>3N</math> points (see diagram) and number them consecutively.
# Draw the chords: <math>(1,3), (2,6), ...., (n,3n),...., (N,3N), (N+1,3), (N+2,6), ...., </math>. (i.e.: The second point is moved by threefold velocity.)
# The ''envelope'' of these chords is a nephroid.

====Proof====
The following consideration uses [[trigonometric formulae]] for
<math> \cos \alpha+\cos\beta,\ \sin \alpha+\sin\beta, \ \cos (\alpha+\beta), \ \cos2\alpha</math>. In order to keep the calculations simple, the proof is given for the nephroid with cusps on the y-axis.
''Equation of the tangent'': for the nephroid with parametric representation 
:<math>x=3\cos\varphi + \cos3\varphi,\; y=3\sin\varphi+\sin3\varphi</math>: 
Herefrom one determines the normal vector <math>\vec n=(\dot y , -\dot x)^T  </math>, at first.
<br />
The equation of the tangent <math>\dot y(\varphi)\cdot (x -x(\varphi)) - \dot x(\varphi)\cdot (y-y(\varphi))= 0</math> is:
:<math>(\cos2\varphi\cdot x \ + \ \sin 2\varphi\cdot y)\cos \varphi = 4\cos^2 \varphi \ .</math>
For <math> \varphi=\tfrac{\pi}{2},\tfrac{3\pi}{2}</math> one gets the cusps of the nephroid, where there is no tangent. For  <math> \varphi\ne\tfrac{\pi}{2},\tfrac{3\pi}{2}</math> one can divide by <math>\cos\varphi</math> to obtain
*<math>\cos2\varphi \cdot x +   \sin2\varphi \cdot y = 4 \cos\varphi \ .</math>

''Equation of the chord'': to the circle with midpoint <math>(0,0)</math> and radius <math>4</math>: The equation of the chord containing the two points <math>(4\cos\theta, 4\sin\theta), \ (4\cos{\color{red}3}\theta, 4\sin{\color{red}3}\theta)) </math> is:
:<math>(\cos2\theta \cdot x +   \sin2\theta \cdot y)\sin\theta = 4 \cos\theta\sin\theta \ .</math>
For <math>\theta =0, \pi</math> the chord degenerates to a point. For <math>\theta \ne 0,\pi</math> one can divide by <math>\sin\theta</math> and gets the equation of the chord:
*<math>\cos2\theta \cdot x +   \sin2\theta \cdot y = 4 \cos\theta \ .</math>
The two angles <math>\varphi , \theta</math> are defined differently (<math>\varphi</math> is one half of the rolling angle, <math>\theta</math> is the parameter of the circle, whose chords are determined), for <math>\varphi=\theta </math> one gets the same line. Hence any chord from the circle above is tangent to the nephroid and 
* ''the nephroid is the envelope of the chords of the circle.''

=== Nephroid as caustic of one half of a circle {{anchor|Caustic}} ===
[[File:Nephroide-kaustik-prinzip.svg|thumb|nephroid as caustic of a circle: principle]]
[[File:Nephroide-kaustik.svg|thumb|nephroide as caustic of one half of a circle]]

The considerations made in the previous section give a proof for the fact, that the [[Caustic (mathematics)|caustic]] of one half of a circle is a nephroid.

* If in the plane parallel light rays meet a reflecting half of a circle (see diagram), then the reflected rays are tangent to a nephroid.

====Proof====
The circle may have the origin as midpoint (as in the previous section) and its 
radius is <math>4</math>. The circle has the parametric representation 
:<math>k(\varphi)=4(\cos\varphi,\sin\varphi) \ .</math>
The tangent at the circle point <math>K:\  k(\varphi)</math> has normal vector  <math>\vec n_t=(\cos\varphi,\sin\varphi)^T</math>. The reflected ray has the normal vector (see diagram) <math>\vec n_r=(\cos{\color{red}2}\varphi,\sin{\color{red}2}\varphi)^T</math> and containing circle point <math>K: \ 4(\cos\varphi,\sin\varphi) </math>. Hence the reflected ray is part of the line with equation 
:<math>\cos{\color{red}2}\varphi\cdot x \ + \ \sin {\color{red}2}\varphi\cdot y = 4\cos\varphi \ ,</math>
which is tangent to the nephroid of the previous section at point 
:<math>P:\ (3\cos\varphi + \cos3\varphi,3\sin\varphi+\sin3\varphi)</math> (see above).

[[File:Caustic00.jpg|thumb|right|Nephroid caustic at bottom of tea cup]]

== The evolute and involute of a nephroid ==
[[File:Nephroide-evol.svg|300px|thumb|nephroid and its evolute<br />
magenta: point with osculating circle and center of curvature]]

=== Evolute ===
The [[evolute]] of a curve is the locus of centers of curvature. In detail: For a curve <math>\vec x=\vec c(s)</math> with radius of curvature <math>\rho(s)</math> the evolute has the representation
:<math>\vec x=\vec c(s) + \rho(s)\vec n(s).</math>
with <math>\vec n(s)</math> the suitably oriented unit normal.

For a nephroid one gets: 
*The ''evolute'' of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram).

====Proof====
The nephroid as shown in the picture has the parametric representation
:<math>x=3\cos\varphi + \cos3\varphi,\quad y=3\sin\varphi+\sin3\varphi \ ,</math>
the unit normal vector pointing to the center of curvature 
:<math>\vec n(\varphi)=(-\cos 2\varphi,-\sin 2\varphi)^T</math> (see section above)
and the radius of curvature <math>3\cos \varphi</math> (s. section on metric properties).
Hence the evolute has the representation:
:<math>x=3\cos\varphi + \cos3\varphi  -3\cos\varphi\cdot\cos2\varphi=\cdots=3\cos\varphi-2\cos^3\varphi,</math>
:<math>y=3\sin\varphi+\sin3\varphi -3\cos\varphi\cdot\sin2\varphi\ =\cdots=2\sin^3\varphi \ ,</math> 
which is a nephroid  half as large and rotated 90 degrees (see diagram and section {{Section link||Equations}} above)

=== Involute ===
Because the evolute of a nephroid is another nephroid, the [[involute]] of the nephroid is also another nephroid. The original nephroid in the image is the involute of the smaller nephroid.

[[File:Nephroide-inv.svg|thumb|inversion (green) of a nephroid (red) across the blue circle]]

== Inversion of a nephroid ==
The [[Inversive geometry|inversion]] 
:<math> x \mapsto \frac{4a^2x}{x^2+y^2}, \quad y\mapsto \frac{4a^2y}{x^2+y^2}  </math> 
across the circle with midpoint <math>(0,0)</math> and radius <math>2a</math> maps the nephroid with equation 
:<math>(x^2+y^2-4a^2)^3=108a^4y^2</math> 
onto the curve of degree 6 with equation 
: <math>(4a^2-(x^2+y^2))^3=27a^2(x^2+y^2)y^2</math> (see diagram) .

[[File:Brennlinie.GIF|framed|A nephroid in daily life: a [[Caustic (optics)|caustic]] of the reflection of light off the inside of a cylinder.]]

== References ==
{{Reflist}}
*Arganbright, D., ''Practical Handbook of Spreadsheet Curves and Geometric Constructions'', CRC Press, 1939, {{ISBN|0-8493-8938-0}},  p.&nbsp;54.
* Borceux, F., ''A Differential Approach to Geometry: Geometric Trilogy III'', Springer, 2014, {{ISBN|978-3-319-01735-8}}, p.&nbsp;148.
*Lockwood, E. H., ''A Book of Curves,'' Cambridge University Press, 1961, {{ISBN|978-0-521-0-5585-7}}, p.&nbsp;7.

== External links ==
{{Commonscat|Nephroid}}
* [http://mathworld.wolfram.com/Nephroid.html Mathworld: nephroid]
* [http://xahlee.info/SpecialPlaneCurves_dir/Nephroid_dir/nephroid.html Xahlee: nephroid]

[[Category:Roulettes (curve)]]
[[Category:Sextic curves]]