Elliptic cyclides

An elliptic cyclide can be represented parametrically by the following formulas (see section Cyclide as channel surface):

${\displaystyle x={\frac {d(c-a\cos u\cos v)+b^{2}\cos u}{a-c\cos u\cos v}}\ ,}$

${\displaystyle y={\frac {b\sin u(a-d\cos v)}{a-c\cos u\cos v}}\ ,}$

${\displaystyle z={\frac {b\sin v(c\cos u-d)}{a-c\cos u\cos v}}\ ,}$

${\displaystyle 0\leq u,v<2\pi \ .}$

The numbers ${\displaystyle a,b,c,d}$ are the semi major and semi minor axes and ${\displaystyle c}$ the linear eccentricity of the ellipse:

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,z=0\ .}$

The hyperbola ${\displaystyle {\frac {x^{2}}{c^{2}}}-{\frac {z^{2}}{b^{2}}}=1,y=0}$ is the focal conic to the ellipse. That means: The foci/vertices of the ellipse are the vertices/foci of the hyperbola. The two conics form the two degenerated focal surfaces of the cyclide.

${\displaystyle d}$ can be considered as the average radius of the generating spheres.

For ${\displaystyle u=const}$ , ${\displaystyle v=const}$ respectively one gets the curvature lines (circles) of the surface.

The corresponding implicit representation is:

${\displaystyle (x^{2}+y^{2}+z^{2}+b^{2}-d^{2})^{2}-4(ax-cd)^{2}-4b^{2}y^{2}=0\ .}$

In case of ${\displaystyle a=b}$ one gets ${\displaystyle c=0}$, i. e. the ellipse is a circle and the hyperbola degenerates to a line. The corresponding cyclides are tori of revolution.

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(ellipt.) Dupin cyclides for designparameters a,b,c,d
${\displaystyle d=0}$	${\displaystyle 0<d<c}$	${\displaystyle d=c}$	${\displaystyle c<d}$	${\displaystyle d=a}$	${\displaystyle a<d}$
symm. horn cyclide	horn cyclide	horn cyclide	ring cyclide	ring cyclide	spindle cyclide

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More intuitive design parameters are the intersections of the cyclide with the x-axis. See section Cyclide through 4 points on the x-axis.

Parabolic cyclides

A parabolic cyclide can be represented by the following parametric representation (see section Cyclide as channel surface):

${\displaystyle x={\frac {p}{2}}\,{\frac {2v^{2}+k(1-u^{2}-v^{2})}{1+u^{2}+v^{2}}}\ ,}$

${\displaystyle y=pu\,{\frac {v^{2}+k}{1+u^{2}+v^{2}}}\ ,}$

${\displaystyle z=pv\,{\frac {1+u^{2}-k}{1+u^{2}+v^{2}}}\ ,}$

${\displaystyle -\infty <u,v<\infty \ .}$

The number ${\displaystyle p}$ determines the shape of both the parabolas, which are focal conics:

${\displaystyle y^{2}=p^{2}-2px,\ z=0\ }$ and ${\displaystyle \ z^{2}=2px,\ y=0\ .}$

${\displaystyle k}$ determines the relation of the diameters of the two holes (see diagram). ${\displaystyle k=0.5}$ means: both diameters are equal. For the diagram is ${\displaystyle k=0.7}$.

A corresponding implicit representation is

${\displaystyle \left(x+\left({\frac {k}{2}}-1\right)p\right)\left(x^{2}+y^{2}+z^{2}-{\frac {k^{2}p^{2}}{4}}\right)+pz^{2}=0\ .}$

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parabolic Dupin cyclides for designparameters p=1, k
${\displaystyle k=0.5}$	${\displaystyle k=1}$	${\displaystyle k=1.5}$
ring cyclide	horn cyclide	horn cyclide

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Remark: By displaying the circles there appear gaps which are caused by the necessary restriction of the parameters ${\displaystyle u,v}$.

Ellipse as directrix

In the x-y-plane the directrix is the ellipse with equation

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\ ,\ z=0\quad }$ and ${\displaystyle \ a>b}$.

It has the parametric representation

${\displaystyle \ x=a\cos \varphi \;,\ y=b\sin \varphi \;,\ z=0\ .}$

${\displaystyle a}$ is the semi major and ${\displaystyle b}$ the semi minor axis. ${\displaystyle c}$ is the linear eccentricity of the ellipse. Hence: ${\displaystyle c^{2}=a^{2}-b^{2}}$. The radii of the generating spheres are

${\displaystyle r(\varphi )=d-c\cos \varphi \ .}$

${\displaystyle d}$ is a design parameter. It can be seen as the average of the radii of the spheres. In case of ${\displaystyle c=0}$ the ellipse is a circle and the cyclide a torus of revolution with ${\displaystyle d}$ the radius of the generating circle (generatrix).

In the diagram: ${\displaystyle \;a=1,\;b=0.99,\ d=0.25\;}$.

Maxwell property

The following simple relation between the actual sphere center (ellipse point) and the corresponding sphere radius is due to Maxwell:^[5]

• The difference/sum of the sphere's radius and the distance of the sphere's center (ellipse point) from one (but fixed) of the foci is constant.

Proof

The foci of the ellipse ${\displaystyle \ E=(a\cos \varphi ,b\sin \varphi ,0)\ }$ are ${\displaystyle \ F_{i}=(\pm c,0,0)\ }$. If one chooses ${\displaystyle \ F_{1}=(c,0,0)\ }$ and calculates the distance ${\displaystyle \ |EF_{1}|\ }$, one gets ${\displaystyle \ |EF_{1}|=a-c\cos \varphi \ }$. Together with the radius of the actual sphere (see above) one gets ${\displaystyle \ |EF_{1}|-r=a-d\ }$.
Choosing the other focus yields: ${\displaystyle \ |EF_{2}|+r=a+d\ .}$

Hence:

In the x-y-plane the envelopes of the circles of the spheres are two circles with the foci of the ellipse as centers and the radii ${\displaystyle a\pm d}$ (see diagram).

Cyclide through 4 points on the x-axis

The Maxwell-property gives reason for determining a ring cyclide by prescribing its intersections with the x-axis:

Given: Four points ${\displaystyle x_{1}>x_{2}>x_{3}>x_{4}}$ on the x-axis (see diagram).

Wanted: Center ${\displaystyle m_{0}}$, semiaxes ${\displaystyle a,b}$, linear eccentricity ${\displaystyle c}$ and foci of the directrix ellipse and the parameter ${\displaystyle d}$ of the corresponding ring cyclide.

From the Maxwell-property one derives

${\displaystyle 2(a+d)=x_{1}-x_{4}\;,\quad \ 2(a-d)=x_{2}-x_{3}\ .}$

Solving for ${\displaystyle a,c}$ yields

${\displaystyle a={\frac {1}{4}}\left(x_{1}+x_{2}-x_{3}-x_{4}\right),\quad }$ ${\displaystyle d={\frac {1}{4}}\left(x_{1}-x_{2}+x_{3}-x_{4}\right)\ .}$

The foci (on the x-axis) are

${\displaystyle f_{1}={\frac {1}{2}}(x_{2}+x_{3}),\quad f_{2}={\frac {1}{2}}(x_{1}+x_{4})}$ and hence

${\displaystyle c={\frac {1}{4}}\left(-x_{1}+x_{2}+x_{3}-x_{4}\right)\;,\quad \ b={\sqrt {a^{2}-c^{2}}}\ }$

The center of the focal conics (ellipse and hyperbola) has the x-coordinate

${\displaystyle m_{0}={\frac {1}{4}}\left(x_{1}+x_{2}+x_{3}+x_{4}\right)\ .}$

If one wants to display the cyclide with help of the parametric representation above one has to consider the shift ${\displaystyle m_{0}}$ of the center !

Meaning of the order of the numbers ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$

(The calculation above presumes ${\displaystyle x_{1}>x_{2}>x_{3}>x_{4}}$, see diagram.)

(H) Swapping ${\displaystyle x_{1},x_{2}}$ generates a horn cyclide.
(S) Swapping ${\displaystyle x_{2},x_{3}}$, generates a spindel cyclide.
(H1) For ${\displaystyle x_{1}=x_{2}=2}$ one gets a 1-horn cyclide.
(R) For ${\displaystyle x_{2}=x_{3}=0}$ one gets a ring cyclide touching itself at the origin.

Parallel surfaces

By increasing or decreasing parameter ${\displaystyle d}$, such that the type does not change, one gets parallel surfaces (similar to parallel curves) of the same type (see diagram).

Hyperbola as directrix

The second way to generate the ring cyclide as channel surface uses the focal hyperbola as directrix. It has the equation

${\displaystyle {\frac {x^{2}}{c^{2}}}-{\frac {z^{2}}{b^{2}}}=1\ ,\ y=0\ .}$

In this case the spheres touch the cyclide from outside at the second family of circles (curvature lines). To each arm of the hyperbola belongs a subfamily of circles. The spheres of one family enclose the cyclide (in diagram: purple). Spheres of the other family are touched from outside by the cyclide (blue).

Parametric representation of the hyperbola:

${\displaystyle (\pm c\cosh \psi ,0,b\sinh \psi )\ .}$

The radii of the corresponding spheres are

${\displaystyle R(\psi )=a\cosh \psi \mp d\ .}$

In case of a torus (${\displaystyle c=0}$) the hyperbola degenerates into the axis of the torus.

Maxwell-property (hyperbola case)

The foci of the hyperbola ${\displaystyle \;H_{\pm }(\psi )=(\pm c\cosh \psi ,0,b\sinh \psi )\;}$ are ${\displaystyle \;F_{i}=(\pm a,0,0)\;}$. The distance of hyperbola point ${\displaystyle \;H_{+}(\psi )=(c\cosh \psi ,0,b\sinh \psi )\;}$ to the focus ${\displaystyle F_{1}}$ is ${\displaystyle \;|H_{+}F_{1}|=a\cosh \psi -c\;}$ and together with the sphere radius ${\displaystyle R_{+}(\psi )=a\cosh \psi -d}$ one gets ${\displaystyle \;|H_{+}F_{1}|-R_{+}=d-c\;}$. Analogously one gets ${\displaystyle \;|H_{+}F_{2}|-R_{+}=d+c\;}$ . For a point on the second arm of the hyperbola one derives the equations: ${\displaystyle \;R_{-}-|H_{-}F_{1}|=d-c\;,\ R_{-}-|H_{-}F_{2}|=d+c\;.}$

Hence:

In the x-z-plane the circles of the spheres with centers ${\displaystyle H_{\pm }(\psi )}$ and radii ${\displaystyle R_{\pm }(\psi )}$ have the two circles (in diagram grey) with centers ${\displaystyle F_{1/2}=(\pm a,0,0)}$ and radii ${\displaystyle d\mp c}$ as envelopes.

Derivation of the parametric representation

Example cylinder

a) Because lines, which do not contain the origin, are mapped by an inversion at a sphere (in picture: magenta) on circles containing the origin the image of the cylinder is a ring cyclide with mutually touching circles at the origin. As the images of the line segments, shown in the picture, there appear on line circle segments as images. The spheres which touch the cylinder on the inner side are mapped on a first pencil of spheres which generate the cyclide as a canal surface. The images of the tangent planes of the cylinder become the second pencil of spheres touching the cyclide. The latter ones pass through the origin.
b) The second example inverses a cylinder that contains the origin. Lines passing the origin are mapped onto themselves. Hence the surface is unbounded and a parabolic cyclide.

Example cone

The lines generating the cone are mapped on circles, which intersect at the origin and the image of the cone's vertex. The image of the cone is a double horn cyclide. The picture shows the images of the line segments (of the cone), which are circles segments, actually.

Example torus

Both the pencils of circles on the torus (shown in the picture) are mapped on the corresponding pencils of circles on the cyclide. In case of a self-intersecting torus one would get a spindle cyclide.

Villarceau circles

Because Dupin ring-cyclides can be seen as images of tori via suitable inversions and an inversion maps a circle onto a circle or line, the images of the Villarceau circles form further two families of circles on a cyclide (see diagram).

Determining the designparameters ${\displaystyle a,b,c,d}$

The formula of the inversion of a parametric surface (see above) provides a parametric representation of a cyclide (as inversion of a torus) with circles as parametric curves. But the points of a parametric net are not well distributed. So it is better to calculate the design parameters ${\displaystyle a,b,c,d}$ and to use the parametric representation above:

Given: A torus, which is shifted out of the standard position along the x-axis. Let be ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$ the intersections of the torus with the x-axis (see diagram). All not zero. Otherwise the inversion of the torus would not be a ring-cyclide.
Wanted: semi-axes ${\displaystyle a,b}$ and linear eccentricity ${\displaystyle c}$ of the ellipse (directrix) and parameter ${\displaystyle d}$ of the ring-cyclide, which is the image of the torus under the inversion at the unitsphere.

The inversion maps ${\displaystyle x_{i}}$ onto ${\displaystyle {\tfrac {1}{x_{i}}}}$, which are the x-coordinates of 4 points of the ring-cyclide (see diagram). From section Cyclide through 4 points on the x-axis one gets

${\displaystyle a={\frac {1}{4}}\left({\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}-{\frac {1}{x_{3}}}-{\frac {1}{x_{4}}}\right),\quad }$ ${\displaystyle d={\frac {1}{4}}\left(-{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}-{\frac {1}{x_{3}}}+{\frac {1}{x_{4}}}\right)\ }$ and

${\displaystyle c={\frac {1}{4}}\left({\frac {1}{x_{1}}}-{\frac {1}{x_{2}}}-{\frac {1}{x_{3}}}+{\frac {1}{x_{4}}}\right)\;,\quad \ b={\sqrt {a^{2}-c^{2}}}\ }$

The center of the focal conics has the x-ccordinate

${\displaystyle m_{0}={\frac {1}{4}}\left({\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+{\frac {1}{x_{3}}}+{\frac {1}{x_{4}}}\right)\ .}$