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.
Nephroid: definition
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
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A nephroid is
Equations
edit
generation of a nephroid by a rolling circle Parametric
edit
If the small circle has radius
a
{\displaystyle a}
(
0
,
0
)
{\displaystyle (0,0)}
2
a
{\displaystyle 2a}
2
φ
{\displaystyle 2\varphi }
(
2
a
,
0
)
{\displaystyle (2a,0)}
parametric representation :
x
(
φ
)
=
3
a
cos
φ
−
a
cos
3
φ
=
6
a
cos
φ
−
4
a
cos
3
φ
,
{\displaystyle x(\varphi )=3a\cos \varphi -a\cos 3\varphi =6a\cos \varphi -4a\cos ^{3}\varphi \ ,}
y
(
φ
)
=
3
a
sin
φ
−
a
sin
3
φ
=
4
a
sin
3
φ
,
0
≤
φ
<
2
π
{\displaystyle y(\varphi )=3a\sin \varphi -a\sin 3\varphi =4a\sin ^{3}\varphi \ ,\qquad 0\leq \varphi <2\pi }
The complex map
z
→
z
3
+
3
z
{\displaystyle z\to z^{3}+3z}
[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
z
{\displaystyle z}
0
{\displaystyle 0}
φ
{\displaystyle \varphi }
z
{\displaystyle z}
e
i
φ
{\displaystyle e^{i\varphi }}
rotation
Φ
3
{\displaystyle \Phi _{3}}
3
a
{\displaystyle 3a}
2
φ
{\displaystyle 2\varphi }
:
z
↦
3
a
+
(
z
−
3
a
)
e
i
2
φ
{\displaystyle :z\mapsto 3a+(z-3a)e^{i2\varphi }}
rotation
Φ
0
{\displaystyle \Phi _{0}}
0
{\displaystyle 0}
φ
{\displaystyle \varphi }
:
z
↦
z
e
i
φ
{\displaystyle :\quad z\mapsto ze^{i\varphi }}
A point
p
(
φ
)
{\displaystyle p(\varphi )}
2
a
{\displaystyle 2a}
Φ
3
{\displaystyle \Phi _{3}}
Φ
0
{\displaystyle \Phi _{0}}
p
(
φ
)
=
Φ
0
(
Φ
3
(
2
a
)
)
=
Φ
0
(
3
a
−
a
e
i
2
φ
)
=
(
3
a
−
a
e
i
2
φ
)
e
i
φ
=
3
a
e
i
φ
−
a
e
i
3
φ
{\displaystyle 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
x
(
φ
)
=
3
a
cos
φ
−
a
cos
3
φ
=
6
a
cos
φ
−
4
a
cos
3
φ
,
y
(
φ
)
=
3
a
sin
φ
−
a
sin
3
φ
=
4
a
sin
3
φ
.
{\displaystyle {\begin{array}{cclcccc}x(\varphi )&=&3a\cos \varphi -a\cos 3\varphi &=&6a\cos \varphi -4a\cos ^{3}\varphi \ ,&&\\y(\varphi )&=&3a\sin \varphi -a\sin 3\varphi &=&4a\sin ^{3}\varphi &.&\end{array}}}
(The formulae
e
i
φ
=
cos
φ
+
i
sin
φ
,
cos
2
φ
+
sin
2
φ
=
1
,
cos
3
φ
=
4
cos
3
φ
−
3
cos
φ
,
sin
3
φ
=
3
sin
φ
−
4
sin
3
φ
{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,\ \cos ^{2}\varphi +\sin ^{2}\varphi =1,\ \cos 3\varphi =4\cos ^{3}\varphi -3\cos \varphi ,\;\sin 3\varphi =3\sin \varphi -4\sin ^{3}\varphi }
trigonometric functions .)
Implicit
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Inserting
x
(
φ
)
{\displaystyle x(\varphi )}
y
(
φ
)
{\displaystyle y(\varphi )}
(
x
2
+
y
2
−
4
a
2
)
3
=
108
a
4
y
2
{\displaystyle (x^{2}+y^{2}-4a^{2})^{3}=108a^{4}y^{2}}
shows that this equation is an implicit representation of the curve.
Proof of the implicit representation
edit
With
x
2
+
y
2
−
4
a
2
=
(
3
a
cos
φ
−
a
cos
3
φ
)
2
+
(
3
a
sin
φ
−
a
sin
3
φ
)
2
−
4
a
2
=
⋯
=
6
a
2
(
1
−
cos
2
φ
)
=
12
a
2
sin
2
φ
{\displaystyle x^{2}+y^{2}-4a^{2}=(3a\cos \varphi -a\cos 3\varphi )^{2}+(3a\sin \varphi -a\sin 3\varphi )^{2}-4a^{2}=\cdots =6a^{2}(1-\cos 2\varphi )=12a^{2}\sin ^{2}\varphi }
one gets
(
x
2
+
y
2
−
4
a
2
)
3
=
(
12
a
2
)
3
sin
6
φ
=
108
a
4
(
4
a
sin
3
φ
)
2
=
108
a
4
y
2
.
{\displaystyle (x^{2}+y^{2}-4a^{2})^{3}=(12a^{2})^{3}\sin ^{6}\varphi =108a^{4}(4a\sin ^{3}\varphi )^{2}=108a^{4}y^{2}\ .}
Orientation
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If the cusps are on the y-axis the parametric representation is
x
=
3
a
cos
φ
+
a
cos
3
φ
,
y
=
3
a
sin
φ
+
a
sin
3
φ
)
.
{\displaystyle x=3a\cos \varphi +a\cos 3\varphi ,\quad y=3a\sin \varphi +a\sin 3\varphi ).}
and the implicit one:
(
x
2
+
y
2
−
4
a
2
)
3
=
108
a
4
x
2
.
{\displaystyle (x^{2}+y^{2}-4a^{2})^{3}=108a^{4}x^{2}.}
Metric properties
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For the nephroid above the
arclength is
L
=
24
a
,
{\displaystyle L=24a,}
area
A
=
12
π
a
2
{\displaystyle A=12\pi a^{2}\ }
radius of curvature is
ρ
=
|
3
a
sin
φ
|
.
{\displaystyle \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
(
φ
)
=
6
a
cos
φ
−
4
a
cos
3
φ
,
{\displaystyle x(\varphi )=6a\cos \varphi -4a\cos ^{3}\varphi \ ,}
y
(
φ
)
=
4
a
sin
3
φ
{\displaystyle y(\varphi )=4a\sin ^{3}\varphi }
and their derivatives
x
˙
=
−
6
a
sin
φ
(
1
−
2
cos
2
φ
)
,
x
¨
=
−
6
a
cos
φ
(
5
−
6
cos
2
φ
)
,
{\displaystyle {\dot {x}}=-6a\sin \varphi (1-2\cos ^{2}\varphi )\ ,\quad \ {\ddot {x}}=-6a\cos \varphi (5-6\cos ^{2}\varphi )\ ,}
y
˙
=
12
a
sin
2
φ
cos
φ
,
y
¨
=
12
a
sin
φ
(
3
cos
2
φ
−
1
)
.
{\displaystyle {\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
∫
0
π
x
˙
2
+
y
˙
2
d
φ
=
⋯
=
12
a
∫
0
π
sin
φ
d
φ
=
24
a
{\displaystyle 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
⋅
1
2
|
∫
0
π
[
x
y
˙
−
y
x
˙
]
d
φ
|
=
⋯
=
24
a
2
∫
0
π
sin
2
φ
d
φ
=
12
π
a
2
{\displaystyle 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
ρ
=
|
(
x
˙
2
+
y
˙
2
)
3
2
x
˙
y
¨
−
y
˙
x
¨
|
=
⋯
=
|
3
a
sin
φ
|
.
{\displaystyle \rho =\left|{\frac {\left({{\dot {x}}^{2}+{\dot {y}}^{2}}\right)^{\frac {3}{2}}}{{\dot {x}}{\ddot {y}}-{\dot {y}}{\ddot {x}}}}\right|=\cdots =|3a\sin \varphi |.}
Nephroid as envelope of a pencil of circles Construction
edit
It can be generated by rolling a circle with radius
a
{\displaystyle a}
2
a
{\displaystyle 2a}
epicycloid . Nephroid as envelope of a pencil of circles
edit
Let be
c
0
{\displaystyle c_{0}}
D
1
,
D
2
{\displaystyle D_{1},D_{2}}
d
12
{\displaystyle d_{12}}
c
0
{\displaystyle c_{0}}
d
12
{\displaystyle d_{12}}
nephroid with cusps
D
1
,
D
2
{\displaystyle D_{1},D_{2}}
Let
c
0
{\displaystyle c_{0}}
(
2
a
cos
φ
,
2
a
sin
φ
)
{\displaystyle (2a\cos \varphi ,2a\sin \varphi )}
(
0
,
0
)
{\displaystyle (0,0)}
2
a
{\displaystyle 2a}
f
(
x
,
y
,
φ
)
=
(
x
−
2
a
cos
φ
)
2
+
(
y
−
2
a
sin
φ
)
2
−
(
2
a
sin
φ
)
2
=
0
.
{\displaystyle f(x,y,\varphi )=(x-2a\cos \varphi )^{2}+(y-2a\sin \varphi )^{2}-(2a\sin \varphi )^{2}=0\ .}
The envelope condition is
f
φ
(
x
,
y
,
φ
)
=
2
a
(
x
sin
φ
−
y
cos
φ
−
2
a
cos
φ
sin
φ
)
=
0
.
{\displaystyle 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
(
φ
)
=
(
6
a
cos
φ
−
4
a
cos
3
φ
,
4
a
sin
3
φ
)
{\displaystyle p(\varphi )=(6a\cos \varphi -4a\cos ^{3}\varphi \;,\;4a\sin ^{3}\varphi )}
f
(
x
,
y
,
φ
)
=
0
,
f
φ
(
x
,
y
,
φ
)
=
0
{\displaystyle f(x,y,\varphi )=0,\;f_{\varphi }(x,y,\varphi )=0}
Nephroid as envelope of a pencil of lines
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nephroid: tangents as chords of a circle, principle 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
3
N
{\displaystyle 3N}
Draw the chords:
(
1
,
3
)
,
(
2
,
6
)
,
.
.
.
.
,
(
n
,
3
n
)
,
.
.
.
.
,
(
N
,
3
N
)
,
(
N
+
1
,
3
)
,
(
N
+
2
,
6
)
,
.
.
.
.
,
{\displaystyle (1,3),(2,6),....,(n,3n),....,(N,3N),(N+1,3),(N+2,6),....,}
The envelope of these chords is a nephroid. The following consideration uses trigonometric formulae for
cos
α
+
cos
β
,
sin
α
+
sin
β
,
cos
(
α
+
β
)
,
cos
2
α
{\displaystyle \cos \alpha +\cos \beta ,\ \sin \alpha +\sin \beta ,\ \cos(\alpha +\beta ),\ \cos 2\alpha }
Equation of the tangent : for the nephroid with parametric representation
x
=
3
cos
φ
+
cos
3
φ
,
y
=
3
sin
φ
+
sin
3
φ
{\displaystyle x=3\cos \varphi +\cos 3\varphi ,\;y=3\sin \varphi +\sin 3\varphi }
Herefrom one determines the normal vector
n
→
=
(
y
˙
,
−
x
˙
)
T
{\displaystyle {\vec {n}}=({\dot {y}},-{\dot {x}})^{T}}
y
˙
(
φ
)
⋅
(
x
−
x
(
φ
)
)
−
x
˙
(
φ
)
⋅
(
y
−
y
(
φ
)
)
=
0
{\displaystyle {\dot {y}}(\varphi )\cdot (x-x(\varphi ))-{\dot {x}}(\varphi )\cdot (y-y(\varphi ))=0}
(
cos
2
φ
⋅
x
+
sin
2
φ
⋅
y
)
cos
φ
=
4
cos
2
φ
.
{\displaystyle (\cos 2\varphi \cdot x\ +\ \sin 2\varphi \cdot y)\cos \varphi =4\cos ^{2}\varphi \ .}
For
φ
=
π
2
,
3
π
2
{\displaystyle \varphi ={\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}}
φ
≠
π
2
,
3
π
2
{\displaystyle \varphi \neq {\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}}
cos
φ
{\displaystyle \cos \varphi }
cos
2
φ
⋅
x
+
sin
2
φ
⋅
y
=
4
cos
φ
.
{\displaystyle \cos 2\varphi \cdot x+\sin 2\varphi \cdot y=4\cos \varphi \ .}
Equation of the chord : to the circle with midpoint
(
0
,
0
)
{\displaystyle (0,0)}
4
{\displaystyle 4}
(
4
cos
θ
,
4
sin
θ
)
,
(
4
cos
3
θ
,
4
sin
3
θ
)
)
{\displaystyle (4\cos \theta ,4\sin \theta ),\ (4\cos {\color {red}3}\theta ,4\sin {\color {red}3}\theta ))}
(
cos
2
θ
⋅
x
+
sin
2
θ
⋅
y
)
sin
θ
=
4
cos
θ
sin
θ
.
{\displaystyle (\cos 2\theta \cdot x+\sin 2\theta \cdot y)\sin \theta =4\cos \theta \sin \theta \ .}
For
θ
=
0
,
π
{\displaystyle \theta =0,\pi }
θ
≠
0
,
π
{\displaystyle \theta \neq 0,\pi }
sin
θ
{\displaystyle \sin \theta }
cos
2
θ
⋅
x
+
sin
2
θ
⋅
y
=
4
cos
θ
.
{\displaystyle \cos 2\theta \cdot x+\sin 2\theta \cdot y=4\cos \theta \ .}
The two angles
φ
,
θ
{\displaystyle \varphi ,\theta }
φ
{\displaystyle \varphi }
θ
{\displaystyle \theta }
φ
=
θ
{\displaystyle \varphi =\theta }
the nephroid is the envelope of the chords of the circle. Nephroid as caustic of one half of a circle
edit
nephroid as caustic of a circle: principle 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 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. The circle may have the origin as midpoint (as in the previous section) and its
radius is
4
{\displaystyle 4}
k
(
φ
)
=
4
(
cos
φ
,
sin
φ
)
.
{\displaystyle k(\varphi )=4(\cos \varphi ,\sin \varphi )\ .}
The tangent at the circle point
K
:
k
(
φ
)
{\displaystyle K:\ k(\varphi )}
n
→
t
=
(
cos
φ
,
sin
φ
)
T
{\displaystyle {\vec {n}}_{t}=(\cos \varphi ,\sin \varphi )^{T}}
n
→
r
=
(
cos
2
φ
,
sin
2
φ
)
T
{\displaystyle {\vec {n}}_{r}=(\cos {\color {red}2}\varphi ,\sin {\color {red}2}\varphi )^{T}}
K
:
4
(
cos
φ
,
sin
φ
)
{\displaystyle K:\ 4(\cos \varphi ,\sin \varphi )}
cos
2
φ
⋅
x
+
sin
2
φ
⋅
y
=
4
cos
φ
,
{\displaystyle \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
φ
+
cos
3
φ
,
3
sin
φ
+
sin
3
φ
)
{\displaystyle P:\ (3\cos \varphi +\cos 3\varphi ,3\sin \varphi +\sin 3\varphi )}
Nephroid caustic at bottom of tea cup The evolute and involute of a nephroid
edit
nephroid and its evolute The evolute of a curve is the locus of centers of curvature. In detail: For a curve
x
→
=
c
→
(
s
)
{\displaystyle {\vec {x}}={\vec {c}}(s)}
ρ
(
s
)
{\displaystyle \rho (s)}
x
→
=
c
→
(
s
)
+
ρ
(
s
)
n
→
(
s
)
.
{\displaystyle {\vec {x}}={\vec {c}}(s)+\rho (s){\vec {n}}(s).}
with
n
→
(
s
)
{\displaystyle {\vec {n}}(s)}
For a nephroid one gets:
The evolute of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram). The nephroid as shown in the picture has the parametric representation
x
=
3
cos
φ
+
cos
3
φ
,
y
=
3
sin
φ
+
sin
3
φ
,
{\displaystyle x=3\cos \varphi +\cos 3\varphi ,\quad y=3\sin \varphi +\sin 3\varphi \ ,}
the unit normal vector pointing to the center of curvature
n
→
(
φ
)
=
(
−
cos
2
φ
,
−
sin
2
φ
)
T
{\displaystyle {\vec {n}}(\varphi )=(-\cos 2\varphi ,-\sin 2\varphi )^{T}}
and the radius of curvature
3
cos
φ
{\displaystyle 3\cos \varphi }
x
=
3
cos
φ
+
cos
3
φ
−
3
cos
φ
⋅
cos
2
φ
=
⋯
=
3
cos
φ
−
2
cos
3
φ
,
{\displaystyle x=3\cos \varphi +\cos 3\varphi -3\cos \varphi \cdot \cos 2\varphi =\cdots =3\cos \varphi -2\cos ^{3}\varphi ,}
y
=
3
sin
φ
+
sin
3
φ
−
3
cos
φ
⋅
sin
2
φ
=
⋯
=
2
sin
3
φ
,
{\displaystyle y=3\sin \varphi +\sin 3\varphi -3\cos \varphi \cdot \sin 2\varphi \ =\cdots =2\sin ^{3}\varphi \ ,}
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 (green) of a nephroid (red) across the blue circle Inversion of a nephroid
edit
References
edit
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. External links
edit
Wikimedia Commons has media related to
Nephroid .
![{\displaystyle 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de13ba263785d0122ca3e5cefc4e65addd140efa)
