Is there a name for the set defined by the Minkowski sum of circles in orthogonal planes?

Question

Recently, I started thinking about the set of points defined by the Minkowski sum of 1D circles in orthogonal planes. The reason for this is to extend the well known result that 1D linear/harmonic oscillators are equivalent to uniform circular motions in the 2D plane, projected onto a line. Generalizing this, it is possible to think of n-dimensional harmonic oscillators as projections of the sum of uniform motions on n circles in n orthogonal planes in an (n+1)-dimensional Euclidean space with a common vertical axis. To illustrate this construction in the n=2 case, I have created an animated GIF showing two unit circles centered at the origin in the x-z and y-z planes:

In this animation, the view rotates to show the same trajectory from two different viewpoints. The second view visually shows the linear projection onto the gray plane. Purple square shows the sum of the projections of the red and blue balls onto this (x-y) plane. (There is a random phase shift between the red and blue balls.) More images and animations of this type are posted on my Twitter feed.

The alignment of the gray ball and the purple square in the projection view visually indicates that 2D linear oscillator motions (purple square) are indistinguishable from projected motions on the 3D surface (gray ball). This is an instance of the property that sums of linear projections of vectors are equivalent to linear projections of sums of vectors, by linearity of the projection operator. This particular instance of this simple property may be relevant due to the importance of linear/harmonic oscillators in mathematical physics.

By extension, n-dimensional oscillators can be thought of as projections of sums of uniform motions on n circles on n orthogonal planes in (n+1) dimensions. Although unit circles and identical velocities for the uniform circular motions are shown in this animation, the example could easily be generalized to circles with arbitrary radii and points moving with arbitrary fixed velocities.

My questions are:

1. Does the set of points defined by the Minkowski sum of this type of arrangement of n circles in (n+1)-dimensions have a name?
2. If so, are there any references or discussions about its properties and potential relevance to physics?

For the sake of clarity, below is some Matlab code to compute a cloud of 3D points representing a finite sample of points in this set, using uniform samples of the two circles:

function circle = circle_sum( Npoint )

if nargin < 1 || isempty( Npoint ) 
  Npoint = 4 * 1e3 ; % default number of points per circle
end

circle.x = zeros( Npoint, 3 ) ;
circle.y = zeros( Npoint, 3 ) ;

for kt = 1 : Npoint % generate the 3D coordinates of two 1D circles in orthogonal planes

  tk    = 2 * pi * (kt - 1) / Npoint ; % circle position parameter
  costk = cos( tk ) ; % projection onto the x-y plane
  sintk = sin( tk ) ; % vertical coordinates

  circle.x( kt, : ) = [ costk 0 sintk ] ; % point on circle in the x-z plane 
  circle.y( kt, : ) = [ 0 costk sintk ] ; % point on circle in the y-z plane 

end

circle.sum = zeros( Npoint, Npoint, 3 ) ;

for ks = 1 : Npoint  % points on the Minkowski sum of the circles
  for kt = 1 : Npoint
    circle.sum( kt, ks, : ) = circle.x( ks, : ) + circle.y( kt, : ) ;
  end
end

Also, here is some Matlab code to plot this surface from its parametric form:

syms s t
x = cos(s) ; y = cos(t) ; z = sin(s) + sin(t) ;
fsurf(x, y, z, [0 2*pi 0 2*pi])
axis square
xlim([-2 2]), ylim([-2 2])
camlight
figSize = 480 ;
az = -35 ;
el = 15 ;
view( az, el ) ;
set( gcf, 'position', [10 10 figSize figSize] )
set( gca, 'xtick', [], 'ytick', [], 'ztick', [] )

Resulting image:

Answer 1

The Minkowski sum of these $2$ circles can be described analytically as the sum :

$$\begin{pmatrix}\cos(a)\\\sin(a)\\0\end{pmatrix}+\begin{pmatrix}0\\\sin(b)\\\cos(b)\end{pmatrix}=\begin{pmatrix}x=\cos(a)\\y=\sin(a)+\sin(b)\\z=\cos(b)\end{pmatrix}$$

where $a,b$ take all values in $[0,2\pi]$.

We can transform this parametric equation into a cartesian equation

$$x^2+y^2+z^2=2+2 \sin(a) \sin(b)$$

$$x^2+y^2+z^2-2 = \pm 2\sqrt{1-\cos^2(a)}\sqrt{1-\cos^2(b)}$$

$$x^2+y^2+z^2-2 = \pm 2\sqrt{1-x^2}\sqrt{1-z^2}$$

yielding finally the equivalent quartic equation :

$$(x^2+y^2+z^2-2)^2=4(1-x^2)(1-z^2)$$

Expanded :

$$x^4 + y^4 + z^4 + 2(x^2y^2 - x^2z^2 + y^2z^2) - 4y^2 = 0.$$

Here is a Matlab program that takes profit of this implicit equation using the powerful "isosurface" function (see figure below, of course similar to yours) :

clear all;close all;axis equal off
set(gcf,'color','w');a=1;b=2;axis([-a,a,-a,a,-b,b]);
I=-a:0.02:a;J=-b:0.02:b;[x,y,z]=meshgrid(I,I,J);
X=x.^2;Y=y.^2;Z=z.^2;
e=X.^2+Y.^2+Z.^2+2*X.*Y-2*X.*Z+2*Y.*Z-4*Y;
[faces,verts,colors] = isosurface(x,y,z,e,0,z);
   patch('Vertices',verts,'Faces',faces,'FaceVertexCData',colors,'FaceColor','interp','EdgeColor','none');
alpha(0.2);
view([130,30]);