Nephroid

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.

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. ==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. ====Parametric==== If the small circle has radius a , the fixed circle has midpoint (0,0) and radius 2a , the rolling angle of the small circle is 2\varphi and point (2a,0) the starting point (see diagram) then one gets the parametric representation: 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\ , \qquad 0\le \varphi The complex map z \to z^3 + 3z maps the unit circle to a nephroid. =====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 z around point 0 (origin) by an angle \varphi can be performed by the multiplication of point z (complex number) by e^{i\varphi} . Hence the rotation \Phi_3 around point 3a by angle 2\varphi is : z \mapsto 3a+(z-3a)e^{i2\varphi} , rotation \Phi_0 around point 0 by angle \varphi is :\quad z \mapsto ze^{i\varphi} . A point p(\varphi) of the nephroid is generated by the rotation of point 2a by \Phi_3 and the subsequent rotation with \Phi_0 : 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} . Herefrom one gets \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} (The formulae 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 were used. See trigonometric functions.) (x^2+y^2-4a^2)^3=108a^4y^2 shows that this equation is an implicit representation of the curve. =====Proof of the implicit representation===== With 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 one gets (x^2+y^2-4a^2)^3=(12a^2)^3\sin^6\varphi=108a^4(4a\sin^3\varphi)^2=108a^4y^2\ . x=3a\cos \varphi+a\cos3\varphi,\quad y=3a\sin \varphi+a\sin3\varphi). and the implicit one: (x^2+y^2-4a^2)^3=108a^4x^2. arclength is L= 24 a, area A= 12\pi a^2\ and radius of curvature is \rho=|3a\sin \varphi|. The proofs of these statements use suitable formulae on curves (arc length, area and radius of curvature) and the parametric representation above x(\varphi)=6a\cos\varphi-4a \cos^3\varphi \ , y(\varphi)= 4a\sin^3\varphi and their derivatives \dot x=-6a\sin\varphi(1 - 2\cos^2\varphi)\ ,\quad \ \ddot x= -6 a\cos \varphi(5-6\cos^2\varphi)\ , \dot y=12a\sin^2\varphi\cos\varphi \quad , \quad \quad \quad \quad \ddot y=12a\sin\varphi(3\cos^2\varphi-1)\ . Proof for the arc length: L=2\int_0^\pi{\sqrt{\dot x^2+\dot y^2}} \; d\varphi=\cdots =12a\int_0^\pi \sin\varphi\; d\varphi= 24a . Proof for the area: 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 . Proof for the radius of curvature: \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|. ==Construction== It can be generated by rolling a circle with radius a on the outside of a fixed circle with radius 2a . Hence, a nephroid is an epicycloid. ====Proof==== Let c_0 be the circle (2a\cos\varphi,2a\sin\varphi) with midpoint (0,0) and radius 2a . The diameter may lie on the x-axis (see diagram). The pencil of circles has equations: f(x,y,\varphi)=(x-2a\cos\varphi)^2+(y-2a\sin\varphi)^2-(2a\sin\varphi)^2=0 \ . The envelope condition is f_\varphi(x,y,\varphi)=2a(x\sin\varphi -y\cos\varphi-2a\cos\varphi\sin\varphi)=0\ . One can easily check that the point of the nephroid p(\varphi)=(6a\cos\varphi-4a \cos^3\varphi\; ,\; 4a\sin^3\varphi) is a solution of the system f(x,y,\varphi)=0, \; f_\varphi(x,y,\varphi)=0 and hence a point of the envelope of the pencil of circles. ====Proof==== The following consideration uses trigonometric formulae for \cos \alpha+\cos\beta,\ \sin \alpha+\sin\beta, \ \cos (\alpha+\beta), \ \cos2\alpha . 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 x=3\cos\varphi + \cos3\varphi,\; y=3\sin\varphi+\sin3\varphi : Herefrom one determines the normal vector \vec n=(\dot y , -\dot x)^T , at first. The equation of the tangent \dot y(\varphi)\cdot (x -x(\varphi)) - \dot x(\varphi)\cdot (y-y(\varphi))= 0 is: (\cos2\varphi\cdot x \ + \ \sin 2\varphi\cdot y)\cos \varphi = 4\cos^2 \varphi \ . For \varphi=\tfrac{\pi}{2},\tfrac{3\pi}{2} one gets the cusps of the nephroid, where there is no tangent. For \varphi\ne\tfrac{\pi}{2},\tfrac{3\pi}{2} one can divide by \cos\varphi to obtain \cos2\varphi \cdot x + \sin2\varphi \cdot y = 4 \cos\varphi \ . Equation of the chord: to the circle with midpoint (0,0) and radius 4 : The equation of the chord containing the two points (4\cos\theta, 4\sin\theta), \ (4\cos{\color{red}3}\theta, 4\sin{\color{red}3}\theta)) is: (\cos2\theta \cdot x + \sin2\theta \cdot y)\sin\theta = 4 \cos\theta\sin\theta \ . For \theta =0, \pi the chord degenerates to a point. For \theta \ne 0,\pi one can divide by \sin\theta and gets the equation of the chord: \cos2\theta \cdot x + \sin2\theta \cdot y = 4 \cos\theta \ . The two angles \varphi , \theta are defined differently ( \varphi is one half of the rolling angle, \theta is the parameter of the circle, whose chords are determined), for \varphi=\theta 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. ====Proof==== The circle may have the origin as midpoint (as in the previous section) and its radius is 4 . The circle has the parametric representation k(\varphi)=4(\cos\varphi,\sin\varphi) \ . The tangent at the circle point K:\ k(\varphi) has normal vector \vec n_t=(\cos\varphi,\sin\varphi)^T . The reflected ray has the normal vector (see diagram) \vec n_r=(\cos{\color{red}2}\varphi,\sin{\color{red}2}\varphi)^T and containing circle point K: \ 4(\cos\varphi,\sin\varphi) . Hence the reflected ray is part of the line with equation \cos{\color{red}2}\varphi\cdot x \ + \ \sin {\color{red}2}\varphi\cdot y = 4\cos\varphi \ , which is tangent to the nephroid of the previous section at point P:\ (3\cos\varphi + \cos3\varphi,3\sin\varphi+\sin3\varphi) (see above). The evolute of a curve is the locus of centers of curvature. In detail: For a curve \vec x=\vec c(s) with radius of curvature \rho(s) the evolute has the representation \vec x=\vec c(s) + \rho(s)\vec n(s). with \vec n(s) 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 x=3\cos\varphi + \cos3\varphi,\quad y=3\sin\varphi+\sin3\varphi \ , the unit normal vector pointing to the center of curvature \vec n(\varphi)=(-\cos 2\varphi,-\sin 2\varphi)^T (see section above) and the radius of curvature 3\cos \varphi (s. section on metric properties). Hence the evolute has the representation: x=3\cos\varphi + \cos3\varphi -3\cos\varphi\cdot\cos2\varphi=\cdots=3\cos\varphi-2\cos^3\varphi, y=3\sin\varphi+\sin3\varphi -3\cos\varphi\cdot\sin2\varphi\ =\cdots=2\sin^3\varphi \ , which is a nephroid half as large and rotated 90 degrees (see diagram and section above) (x^2+y^2-4a^2)^3=108a^4y^2 onto the curve of degree 6 with equation (4a^2-(x^2+y^2))^3=27a^2(x^2+y^2)y^2 (see diagram) . of the reflection of light off the inside of a cylinder.]]

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