Shape defined by equidistant points to an ellipse in $\Bbb{R}^3$

Question

Motivation:

I want to describe a rubber loop (see photo, the 3 violet parts are just decorative). Such a loop can be described as a torus but if you press the loop from the sides then its form resembles an ellipse, i.e. a deformed torus.

Equidistant points to a circle embedded in $\Bbb{R}^3$ form a torus. What is the name and equation of the shape formed by equidistant points to an ellipse embedded in $\Bbb{R}^3$? I could not find any source in the internet that deals with the description of such a shape.

In the literature we can find however an elliptic torus that is a different shape.

Edit:

A quite complicated parameterfree equation was given in an answer below. Is there another non-parametrized representation?

Answer 1

Topologically, your surface would still be considered a torus. (If it has a special name, I am not aware.)

In any event, you could use the Frenet-Serret equations to derive the following parametric equation:

$$\begin{align*} x&=a\cos u-\frac{b r \cos u\cos v}{\sqrt{b^2\cos^2 u+a^2\sin^2 u}}\\ y&=b\sin u-\frac{a r \sin u\cos v}{\sqrt{b^2\cos^2 u+a^2\sin^2 u}}\\ z&=r\sin v\end{align*}$$

where $a$ and $b$ are the ellipse's semiaxes, and $r$ is the radius of the circular cross-section. Here's a plot in Mathematica as an example:

With[{a = Sqrt[2], b = 1, r = 1/8}, 
     ParametricPlot3D[{a Cos[u], b Sin[u], 0} +
                      r {Cos[v], Sin[v]} . {{-b Cos[u], -a Sin[u], 0}/
                                            Sqrt[b^2 Cos[u]^2 + a^2 Sin[u]^2],
                                            {0, 0, 1}}, {u, 0, 2 π}, {v, 0, 2 π}]]

---

One can use Gröbner bases to derive an implicit Cartesian equation, but the result is long and uninformative:

$$\small a^8 \left(b^4-2 b^2 \left(r^2+y^2-z^2\right)+\left(-r^2+y^2+z^2\right)^2\right)+a^4 \left(b^8+2 b^6 \left(r^2+3 x^2-2 y^2-z^2\right)-2 b^4 \left(3 r^4+r^2 \left(4 \left(x^2+y^2\right)-6 z^2\right)-3 x^4-4 z^2 \left(x^2+y^2\right)+5 x^2 y^2-3 y^4+3 z^4\right)+2 b^2 \left(r^6+r^4 \left(2 x^2-4 y^2-3 z^2\right)+r^2 \left(-3 x^4-4 z^2 \left(x^2-2 y^2\right)-3 x^2 y^2+5 y^4+3 z^4\right)+3 x^4 \left(z^2-y^2\right)+x^2 \left(y^2+z^2\right) \left(y^2+2 z^2\right)-\left(y^2+z^2\right)^2 \left(2 y^2+z^2\right)\right)+\left(-r^2+y^2+z^2\right)^2 \left(-r^2+x^2+y^2+z^2\right)^2\right)+b^4 \left(-r^2+x^2+z^2\right)^2 \left(b^4+2 b^2 \left(-r^2+x^2-y^2+z^2\right)+\left(-r^2+x^2+y^2+z^2\right)^2\right)=2 a^6 \left(b^6+b^4 \left(-r^2+2 x^2-3 y^2+z^2\right)-b^2 \left(r^4+r^2 \left(3 x^2+2 (y-z) (y+z)\right)+3 x^2 (y-z) (y+z)-3 y^4-2 y^2 z^2+z^4\right)+\left(-r^2+y^2+z^2\right)^2 \left(r^2+x^2-y^2-z^2\right)\right)+2 a^2 \left(b^8 \left(r^2+x^2-z^2\right)-b^6 \left(r^4-z^2 \left(2 \left(r^2+x^2\right)+3 y^2\right)+2 r^2 x^2+3 r^2 y^2-3 x^4+3 x^2 y^2+z^4\right)+b^4 \left(-r^6+r^4 \left(4 x^2-2 y^2+3 z^2\right)+r^2 \left(-5 x^4+x^2 \left(3 y^2-8 z^2\right)+3 y^4+4 y^2 z^2-3 z^4\right)+2 x^6-x^4 \left(y^2-5 z^2\right)+x^2 \left(3 y^4-3 y^2 z^2+4 z^4\right)-3 y^4 z^2-2 y^2 z^4+z^6\right)+b^2 \left(-r^2+x^2+y^2+z^2\right)^2 \left(r^4-r^2 \left(x^2+y^2+2 z^2\right)+z^2 \left(x^2+y^2\right)-x^2 y^2+z^4\right)\right)$$

Answer 2

I assumed that equi-distance along a torus normal. Not sure if the assumption is correct.

Regarding nomenclature if you think physics, may be it is envelope of Huygen's wavelets from an ellipse wavefront ; in differential geometry we have the Bertrand parallel surfaces. So maybe can be called 2D Bertrand Parallel to an Elliptic Torus meridian.

The parallel surface can be found by Intrinsic equations linking arc length, curvature and tangent rotation as $ s, \kappa_g, \phi$ respectively.

The $(s,\kappa_g)$ relation is Cesàro, $(s, \phi)$ is Whewell, $ (\kappa_g,\phi) $ has afik no name, but I know an application example from the context of intrinsic definition of elliptic shells in Structural mechanics... Theory of Plates/Shells by S. Timoshenko, McGraw Hill. It is reproduced here.

There is no pretty analytical solution to the displaced profiles. Beyond a point one should avail the benefits of a numerical approach, or so I for one feel.

$$\kappa_g=\frac{d\phi}{ds}=\frac{ ((a \sin \phi)^2+(b \cos \phi)^2)^{\frac32}}{a^2 b^2} \tag 1 $$

Radii of curvatures at end of major, minor axes are respectively

$$ \frac{b^2}{a},\frac{a^2}{b}$$

The method can be applied to find a parallel curve/surface to any curve when its intrinsic equation is known.

Profile and Cartesian components are obtained straightaway by numerical integration.

$$ x'(s)= \cos \phi, y'(s)= \sin \phi \;;\tag2 $$

We can now take handle of $(x,y)$ above, add normal constant offset components + eccentric displacement of torus and later rotation in the usual way.. as for a circular section torus.

A Mathematica listing is given.

Typical non-elliptic parallel surface meridians are shown( An ellipse of $(a,b) = (5,4)$ with parallel displacement =1 , ( aka offset) , as well as swept (rotationally symmetric) toroid surface segment.

{a,b}={5.,4.};smax=28.5; ELLxpand={PH^[Prime](s)==((a sin(PH(s)))^2+(b cos(PH(s)))^2)^1.5/(a^2 b^2),PH(0)==[Pi]/2,Y^[Prime](s)==sin(PH(s)),X^[Prime](s)==cos(PH(s)),Y(0)==0,X(0)==a};NDSolve[ELLxpand,{PH,X,Y},{s,0,smax}]; {ph(t_),x(t_),y(t_)}={PH(t),X(t),Y(t)}/. First[%]; h=0;ell=ParametricPlot[h {sin(ph(s)),-cos(ph(s))}+{x(s),y(s)},{s,.0,smax},PlotStyle->{Blue,Thick}]; h=1;ellout=ParametricPlot[h {sin(ph(s)),-cos(ph(s))}+{x(s),y(s)},{s,.0,smax},PlotStyle->{Red,Thick}]; h=-1;ellin=ParametricPlot[h {sin(ph(s)),-cos(ph(s))}+{x(s),y(s)},{s,.0,smax},PlotStyle->{Green,Thick}]; Show[{ell,ellout,ellin},AspectRatio->0.75,PlotRange->All,GridLines->Automatic,PlotLabel->"BERTRAND__PARALLEL_(ELLIPSES?)"] u(s_)=h sin(ph(s))+x(s);v(s_)=y(s)-h cos(ph(s));c=7; ParametricPlot3D[{cos(t) (c+u(s)),v(s),sin(t) (c+u(s))},{s,.0,smax},{t,1,-3},Mesh->{22,29},PlotStyle->{Yellow,Thick},Boxed->False,Axes->False]