Nephroid: Difference between revisions
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{{short description|Plane curve; an epicycloid with radii differing by 1/2}} |
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{{more citations needed|date=May 2018}} |
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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. |
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==Name== |
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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> |
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==Strict definition == |
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A nephroid is |
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* an [[algebraic curve]] of [[Degree of a polynomial|degree]] 6. |
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* an [[epicycloid]] with two [[Cusp (singularity)|cusps]] |
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* a plane simple closed curve = a [[Jordan curve]] |
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[[File:EpitrochoidOn2.gif|thumb|generation of a nephroid by a rolling circle]] |
[[File:EpitrochoidOn2.gif|thumb|generation of a nephroid by a rolling circle]] |
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The '''nephroid''' (from the [[Greek language|Greek]] ὁ νεφρός ''ho nephros'') is a [[plane curve]] whose name means 'kidney-shaped' (compare ''[[nephrology]]''). Although the term ''nephroid'' was used to describe other curves, it was applied to the curve in this article by Proctor in 1878. This curve looks like a peanut. <ref>{{MathWorld|title=Nephroid|urlname=Nephroid}}</ref> |
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====Parametric==== |
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⚫ | 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]]: |
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:<math>x(\varphi) = 3a\cos\varphi- a\cos3\varphi=6a\cos\varphi-4a \cos^3\varphi \ ,</math> |
:<math>x(\varphi) = 3a\cos\varphi- a\cos3\varphi=6a\cos\varphi-4a \cos^3\varphi \ ,</math> |
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:<math>y(\varphi) = 3a \sin\varphi - a\sin3\varphi =4a\sin^3\varphi\ , \qquad 0\le \varphi < 2\pi</math> |
:<math>y(\varphi) = 3a \sin\varphi - a\sin3\varphi =4a\sin^3\varphi\ , \qquad 0\le \varphi < 2\pi</math> |
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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> |
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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 |
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 |
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: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_3</math> around point <math>3a</math> by angle <math>2\varphi</math> is <math>: z \mapsto 3a+(z-3a)e^{i2\varphi}</math> , |
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\end{array} </math> |
\end{array} </math> |
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(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]].) |
(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]].) |
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;proof of the implicit representation: |
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====Implicit ==== |
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With |
With |
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:<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> |
:<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> |
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one gets |
one gets |
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:<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> |
:<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> |
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==Orientation == |
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;other orientation: |
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If the cusps are on the y-axis the parametric representation is |
If the cusps are on the y-axis the parametric representation is |
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:<math>x=3a\cos \varphi+a\cos3\varphi,\quad y=3a\sin \varphi+a\sin3\varphi).</math> |
:<math>x=3a\cos \varphi+a\cos3\varphi,\quad y=3a\sin \varphi+a\sin3\varphi).</math> |
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\ddot y=12a\sin\varphi(3\cos^2\varphi-1)\ . </math> |
\ddot y=12a\sin\varphi(3\cos^2\varphi-1)\ . </math> |
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; |
;Proof for the arc length: |
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:<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> . |
:<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> . |
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; |
;Proof for the area: |
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:<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> . |
:<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> . |
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; |
;Proof for the radius of curvature: |
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:<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> |
:<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> |
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[[File:Nephroide-kreise.svg|thumb|Nephroid as envelope of a pencil of circles]] |
[[File:Nephroide-kreise.svg|thumb|Nephroid as envelope of a pencil of circles]] |
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==Construction== |
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== Nephroid as envelope of a pencil of circles == |
=== Nephroid as envelope of a pencil of circles === |
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*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>. |
*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>. |
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====Proof==== |
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;proof: |
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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: |
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: |
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:<math> f(x,y,\varphi)=(x-2a\cos\varphi)^2+(y-2a\sin\varphi)^2-(2a\sin\varphi)^2=0 \ .</math> |
:<math> f(x,y,\varphi)=(x-2a\cos\varphi)^2+(y-2a\sin\varphi)^2-(2a\sin\varphi)^2=0 \ .</math> |
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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. |
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. |
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== Nephroid as envelope of a pencil of lines == |
=== Nephroid as envelope of a pencil of lines === |
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[[File:Nephroide-sek-tang-prinzip.svg|thumb|nephroid: tangents as chords of a circle, principle]] |
[[File:Nephroide-sek-tang-prinzip.svg|thumb|nephroid: tangents as chords of a circle, principle]] |
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[[File:Nephroide-sek-tang.svg|thumb|nephroid: tangents as chords of a circle]] |
[[File:Nephroide-sek-tang.svg|thumb|nephroid: tangents as chords of a circle]] |
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# The ''envelope'' of these chords is a nephroid. |
# The ''envelope'' of these chords is a nephroid. |
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====Proof==== |
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;proof: |
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The following consideration uses [[trigonometric formulae]] for |
The following consideration uses [[trigonometric formulae]] for |
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<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. |
<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. |
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''Equation of the tangent'': for the nephroid with parametric representation |
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:<math>x=3\cos\varphi + \cos3\varphi,\; y=3\sin\varphi+\sin3\varphi</math>: |
:<math>x=3\cos\varphi + \cos3\varphi,\; y=3\sin\varphi+\sin3\varphi</math>: |
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Herefrom one determines the normal vector <math>\vec n=(\dot y , -\dot x)^T </math>, at first. |
Herefrom one determines the normal vector <math>\vec n=(\dot y , -\dot x)^T </math>, at first. |
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*<math>\cos2\varphi \cdot x + \sin2\varphi \cdot y = 4 \cos\varphi \ .</math> |
*<math>\cos2\varphi \cdot x + \sin2\varphi \cdot y = 4 \cos\varphi \ .</math> |
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''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: |
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:<math>(\cos2\theta \cdot x + \sin2\theta \cdot y)\sin\theta = 4 \cos\theta\sin\theta \ .</math> |
:<math>(\cos2\theta \cdot x + \sin2\theta \cdot y)\sin\theta = 4 \cos\theta\sin\theta \ .</math> |
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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: |
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: |
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* ''the nephroid is the envelope of the chords of the circle.'' |
* ''the nephroid is the envelope of the chords of the circle.'' |
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[[File:Nephroide-kaustik-prinzip.svg|thumb|nephroid as caustic of a circle: principle]] |
[[File:Nephroide-kaustik-prinzip.svg|thumb|nephroid as caustic of a circle: principle]] |
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[[File:Nephroide-kaustik.svg|thumb|nephroide as caustic of one half of a circle]] |
[[File:Nephroide-kaustik.svg|thumb|nephroide as caustic of one half of a circle]] |
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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. |
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. |
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* 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. |
* 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. |
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====Proof==== |
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;proof: |
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The circle may have the origin as midpoint (as in the previous section) and its |
The circle may have the origin as midpoint (as in the previous section) and its |
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radius is <math>4</math>. The circle has the parametric representation |
radius is <math>4</math>. The circle has the parametric representation |
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which is tangent to the nephroid of the previous section at point |
which is tangent to the nephroid of the previous section at point |
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:<math>P:\ (3\cos\varphi + \cos3\varphi,3\sin\varphi+\sin3\varphi)</math> (see above). |
:<math>P:\ (3\cos\varphi + \cos3\varphi,3\sin\varphi+\sin3\varphi)</math> (see above). |
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[[File:Caustic00.jpg|thumb|right|Nephroid caustic at bottom of tea cup]] |
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== The evolute and involute of a nephroid == |
== The evolute and involute of a nephroid == |
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magenta: point with osculating circle and center of curvature]] |
magenta: point with osculating circle and center of curvature]] |
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=== |
=== Evolute === |
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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 |
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 |
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:<math>\vec x=\vec c(s) + \rho(s)\vec n(s).</math> |
:<math>\vec x=\vec c(s) + \rho(s)\vec n(s).</math> |
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with <math>\vec n(s)</math> the suitably oriented unit normal. |
with <math>\vec n(s)</math> the suitably oriented unit normal. |
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For a |
For a nephroid one gets: |
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*The ''evolute'' of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram). |
*The ''evolute'' of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram). |
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====Proof==== |
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;proof: |
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The nephroid as shown in the picture has the parametric representation |
The nephroid as shown in the picture has the parametric representation |
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:<math>x=3\cos\varphi + \cos3\varphi,\quad y=3\sin\varphi+\sin3\varphi \ ,</math> |
:<math>x=3\cos\varphi + \cos3\varphi,\quad y=3\sin\varphi+\sin3\varphi \ ,</math> |
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:<math>x=3\cos\varphi + \cos3\varphi -3\cos\varphi\cdot\cos2\varphi=\cdots=3\cos\varphi-2\cos^3\varphi,</math> |
:<math>x=3\cos\varphi + \cos3\varphi -3\cos\varphi\cdot\cos2\varphi=\cdots=3\cos\varphi-2\cos^3\varphi,</math> |
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:<math>y=3\sin\varphi+\sin3\varphi -3\cos\varphi\cdot\sin2\varphi\ =\cdots=2\sin^3\varphi \ ,</math> |
:<math>y=3\sin\varphi+\sin3\varphi -3\cos\varphi\cdot\sin2\varphi\ =\cdots=2\sin^3\varphi \ ,</math> |
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which is a nephroid half as large and rotated 90 degrees (see diagram and section |
which is a nephroid half as large and rotated 90 degrees (see diagram and section {{Section link||Equations}} above) |
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=== |
=== Involute === |
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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. |
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. |
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* [http://xahlee.info/SpecialPlaneCurves_dir/Nephroid_dir/nephroid.html Xahlee: nephroid] |
* [http://xahlee.info/SpecialPlaneCurves_dir/Nephroid_dir/nephroid.html Xahlee: nephroid] |
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[[Category: |
[[Category:Roulettes (curve)]] |
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[[Category:Sextic curves]] |
Latest revision as of 20:07, 11 July 2023
This article needs additional citations for verification. (May 2018) |
In geometry, a nephroid (from Ancient Greek 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
[edit]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.[1][2]
Strict definition
[edit]A nephroid is
- an algebraic curve of degree 6.
- an epicycloid with two cusps
- a plane simple closed curve = a Jordan curve
Equations
[edit]Parametric
[edit]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[3]
Proof of the parametric representation
[edit]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.)
Implicit
[edit]Inserting and into the equation
shows that this equation is an implicit representation of the curve.
Proof of the implicit representation
[edit]With
one gets
Orientation
[edit]If the cusps are on the y-axis the parametric representation is
and the implicit one:
Metric properties
[edit]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
Bauwesen
[edit]- 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.
Nephroid as envelope of a pencil of circles
[edit]- 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 .
Proof
[edit]Let be the circle with midpoint and radius . The diameter may lie on the x-axis (see diagram). The pencil of circles has equations:
The envelope condition is
One can easily check that the point of the nephroid is a solution of the system and hence a point of the envelope of the pencil of circles.
Nephroid as envelope of a pencil of lines
[edit]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 points (see diagram) and number them consecutively.
- Draw the chords: . (i.e.: The second point is moved by threefold velocity.)
- The envelope of these chords is a nephroid.
Proof
[edit]The following consideration uses trigonometric formulae for . 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
- :
Herefrom one determines the normal vector , at first.
The equation of the tangent is:
For one gets the cusps of the nephroid, where there is no tangent. For one can divide by to obtain
Equation of the chord: to the circle with midpoint and radius : The equation of the chord containing the two points is:
For the chord degenerates to a point. For one can divide by and gets the equation of the chord:
The two angles are defined differently ( is one half of the rolling angle, is the parameter of the circle, whose chords are determined), for 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
[edit]The considerations made in the previous section give a proof for the fact, that the 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
[edit]The circle may have the origin as midpoint (as in the previous section) and its radius is . The circle has the parametric representation
The tangent at the circle point has normal vector . The reflected ray has the normal vector (see diagram) and containing circle point . Hence the reflected ray is part of the line with equation
which is tangent to the nephroid of the previous section at point
- (see above).
The evolute and involute of a nephroid
[edit]Evolute
[edit]The evolute of a curve is the locus of centers of curvature. In detail: For a curve with radius of curvature the evolute has the representation
with 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
[edit]The nephroid as shown in the picture has the parametric representation
the unit normal vector pointing to the center of curvature
- (see section above)
and the radius of curvature (s. section on metric properties). Hence the evolute has the representation:
which is a nephroid half as large and rotated 90 degrees (see diagram and section § Equations above)
Involute
[edit]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.
Inversion of a nephroid
[edit]The inversion
across the circle with midpoint and radius maps the nephroid with equation
onto the curve of degree 6 with equation
- (see diagram) .
References
[edit]- ^ Weisstein, Eric W. "Nephroid". MathWorld.
- ^ "Nephroid". Maths History. Retrieved 2022-08-12.
- ^ Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath
- Arganbright, D., Practical Handbook of Spreadsheet Curves and Geometric Constructions, CRC Press, 1939, ISBN 0-8493-8938-0, p. 54.
- Borceux, F., A Differential Approach to Geometry: Geometric Trilogy III, Springer, 2014, ISBN 978-3-319-01735-8, p. 148.
- Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961, ISBN 978-0-521-0-5585-7, p. 7.