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EleNTorus™ software computes a parametric representation of the n-dimensional sphere, cylinder, ellipsoid, and torus. It uses a recursive formula over dimensions. Also, EleNTorus computes Jacobian and Hessian matrices, containing tangents and curvatures.

The parametric representation of the unit 1-sphere (circle) is

```
y
```^{0} = cos x^{0},

y^{1} = sin x^{0}.

If this circle is rotated about an axis passing through the center, a sphere results. The unit 2-sphere is

```
y
```^{0} = cos x^{0},

y^{1} = sin x^{0} cos x^{1},

y^{2} = sin x^{0} sin x^{1}.

This representation is constructed from the previous by an addition formula and suffix factors. Likewise, if the 2-sphere is rotated in a new dimension, a 3-sphere results. That is, each point on the 2-sphere is carried around in a circle in the new dimension. The 3-sphere is near impossible to visualize. Nonetheless, we know all its properties. The unit 3-sphere is

```
y
```^{0} = cos x^{0},

y^{1} = sin x^{0} cos x^{1},

y^{2} = sin x^{0} sin x^{1} cos x^{2},

y^{3} = sin x^{0} sin x^{1} sin x^{2}.

Again, this representation is constructed from the previous by an addition formula and suffix factors. It is clear that the same construction can be applied recursively, giving a n-sphere. EleNTorus software uses this formula recursion to compute a n-sphere. It also computes tangent and curvature at every point as described below.

Besides being an interesting curiosity, what is the importance of the n-sphere?
Consider a point in a plane.
The circle centered on this point describes the points an equal distance in all directions.
Likewise, a 2-sphere describes those points in ordinary 3-space.
A n-sphere plays a similar role in (*n*+1)-space.
Also, a n-sphere has constant curvature in all directions at every point.

The 1-sphere (circle) formula can be generalized to the 1-ellipsoid (ellipse) with semiaxis radius *a _{r}*.
It is

```
y
```^{0} = a_{0} cos x^{0},

y^{1} = a_{1} sin x^{0}.

The 2-ellipsoid is

```
y
```^{0} = a_{0} cos x^{0},

y^{1} = a_{1} sin x^{0} cos x^{1},

y^{2} = a_{2} sin x^{0} sin x^{1}.

The 3-ellipsoid is

```
y
```^{0} = a_{0} cos x^{0},

y^{1} = a_{1} sin x^{0} cos x^{1},

y^{2} = a_{2} sin x^{0} sin x^{1} cos x^{2},

y^{3} = a_{3} sin x^{0} sin x^{1} sin x^{2}.

EleNTorus software uses this formula recursion to compute a n-ellipsoid. It also computes tangent and curvature at every point as described below.

If 1-ellipsoid (ellipse) is rotated about an axis not passing through the center, a torus results.
The offsetting distance is the torus radius *b _{i}*.
If the torus radius is greater than the ellipse radius, then the torus has a hole.
The 1-torus is the ellipse not centered on the origin

```
y
```^{0} = a_{0} cos x^{0},

y^{1} = a_{1} sin x^{0} + b_{1}.

The 2-torus is

```
y
```^{0} = a_{0} cos x^{0},

y^{1} = (a_{1} sin x^{0} + b_{1}) cos x^{1},

y^{2} = (a_{2} sin x^{0} + b_{1}) sin x^{1} + b_{2}.

The 3-torus is

```
y
```^{0} = a_{0} cos x^{0},

y^{1} = (a_{1} sin x^{0} + b_{1}) cos x^{1},

y^{2} = ((a_{2} sin x^{0} + b_{1}) sin x^{1} + b_{2}) cos x^{2},

y^{3} = ((a_{3} sin x^{0} + b_{1}) sin x^{1} + b_{2}) sin x^{2} + b_{3}.

EleNTorus software uses this formula recursion to compute a n-torus.
The n-torus formula includes other important manifolds as special cases.
The n-ellipsoid has *b _{i}*=0.
The unit n-sphere has

Like all parametric representations, these formulas imply directional tangents and curvatures at every point.
These are derived from first and second partial derivatives *y ^{r}_{,i}* and

```
y
```^{0}_{,0} = -sin x^{0},

y^{0}_{,1} = 0,

y^{0}_{,2} = 0,

y^{1}_{,0} = cos x^{0} cos x^{1},

y^{1}_{,1} = -sin x^{0} sin x^{1},

y^{1}_{,2} = 0,

y^{2}_{,0} = cos x^{0} sin x^{1} cos x^{2},

y^{2}_{,1} = sin x^{0} cos x^{1} cos x^{2},

y^{2}_{,2} = -sin x^{0} sin x^{1} sin x^{2},

y^{3}_{,0} = cos x^{0} sin x^{1} sin x^{2},

y^{3}_{,1} = sin x^{0} cos x^{1} sin x^{2},

y^{3}_{,2} = sin x^{0} sin x^{1} cos x^{2}.

The unit 3-sphere Hessian matrix is

```
y
```^{0}_{,00} = -cos x^{0},

y^{0}_{,01} = 0,

y^{0}_{,02} = 0,

y^{0}_{,10} = y^{0}_{,01},

y^{0}_{,11} = 0,

y^{0}_{,12} = 0,

y^{0}_{,20} = y^{0}_{,02},

y^{0}_{,21} = y^{0}_{,12},

y^{0}_{,22} = 0,

y^{1}_{,00} = -sin x^{0} cos x^{1},

y^{1}_{,01} = -cos x^{0} sin x^{1},

y^{1}_{,02} = 0,

y^{1}_{,10} = y^{1}_{,01},

y^{1}_{,11} = -sin x^{0} cos x^{1},

y^{1}_{,12} = 0,

y^{1}_{,20} = y^{1}_{,02},

y^{1}_{,21} = y^{1}_{,12},

y^{1}_{,22} = 0,

y^{2}_{,00} = -sin x^{0} sin x^{1} cos x^{2},

y^{2}_{,01} = cos x^{0} cos x^{1} cos x^{2},

y^{2}_{,02} = -cos x^{0} sin x^{1} sin x^{2},

y^{2}_{,10} = y^{2}_{,01},

y^{2}_{,11} = -sin x^{0} sin x^{1} cos x^{2},

y^{2}_{,12} = -sin x^{0} cos x^{1} sin x^{2},

y^{2}_{,20} = y^{2}_{,02},

y^{2}_{,21} = y^{2}_{,12},

y^{3}_{,22} = -sin x^{0} sin x^{1} cos x^{2},

y^{3}_{,00} = -sin x^{0} sin x^{1} sin x^{2},

y^{3}_{,01} = cos x^{0} cos x^{1} sin x^{2},

y^{3}_{,02} = cos x^{0} sin x^{1} cos x^{2},

y^{3}_{,10} = y^{3}_{,01},

y^{3}_{,11} = -sin x^{0} sin x^{1} sin x^{2},

y^{3}_{,12} = sin x^{0} cos x^{1} cos x^{2},

y^{3}_{,20} = y^{3}_{,02},

y^{3}_{,21} = y^{3}_{,12},

y^{3}_{,22} = -sin x^{0} sin x^{1} sin x^{2}.

For the 3-torus, the Jacobian matrix is

```
y
```^{0}_{,0} = -a_{0} sin x^{0},

y^{0}_{,1} = 0,

y^{0}_{,2} = 0,

y^{1}_{,0} = a_{1} cos x^{0} cos x^{1},

y^{1}_{,1} = -(a_{1} sin x^{0} + b_{1}) sin x^{1},

y^{1}_{,2} = 0,

y^{2}_{,0} = a_{2} cos x^{0} sin x^{1} cos x^{2},

y^{2}_{,1} = (a_{2} sin x^{0} + b_{1}) cos x^{1} cos x^{2},

y^{2}_{,2} = -((a_{2} sin x^{0} + b_{1}) sin x^{1} + b_{2}) sin x^{2},

y^{3}_{,0} = a_{3} cos x^{0} sin x^{1} sin x^{2},

y^{3}_{,1} = (a_{3} sin x^{0} + b_{1}) cos x^{1} sin x^{2},

y^{3}_{,2} = ((a_{3} sin x^{0} + b_{1}) sin x^{1} + b_{2}) cos x^{2}.

The 3-torus Hessian matrix is

```
y
```^{0}_{,00} = -a_{0} cos x^{0},

y^{0}_{,01} = 0,

y^{0}_{,02} = 0,

y^{0}_{,10} = y^{0}_{,01},

y^{0}_{,11} = 0,

y^{0}_{,12} = 0,

y^{0}_{,20} = y^{0}_{,02},

y^{0}_{,21} = y^{0}_{,12},

y^{0}_{,22} = 0,

y^{1}_{,00} = -a_{1} sin x^{0} cos x^{1},

y^{1}_{,01} = -a_{1} cos x^{0} sin x^{1},

y^{1}_{,02} = 0,

y^{1}_{,10} = y^{1}_{,01},

y^{1}_{,11} = -(a_{1} sin x^{0} + b_{1}) cos x^{1},

y^{1}_{,12} = 0,

y^{1}_{,20} = y^{1}_{,02},

y^{1}_{,21} = y^{1}_{,12},

y^{1}_{,22} = 0,

y^{2}_{,00} = -a_{2} sin x^{0} sin x^{1} cos x^{2},

y^{2}_{,01} = a_{2} cos x^{0} cos x^{1} cos x^{2},

y^{2}_{,02} = -a_{2} cos x^{0} sin x^{1} sin x^{2},

y^{2}_{,10} = y^{2}_{,01},

y^{2}_{,11} = -(a_{2} sin x^{0} + b_{1}) sin x^{1} cos x^{2},

y^{2}_{,12} = -(a_{2} sin x^{0} + b_{1}) cos x^{1} sin x^{2},

y^{2}_{,20} = y^{2}_{,02},

y^{2}_{,21} = y^{2}_{,12},

y^{2}_{,22} = -((a_{2} sin x^{0} + b_{1}) sin x^{1} + b_{2}) cos x^{2},

y^{3}_{,00} = -a_{3} sin x^{0} sin x^{1} sin x^{2},

y^{3}_{,01} = a_{3} cos x^{0} cos x^{1} sin x^{2},

y^{3}_{,02} = a_{3} cos x^{0} sin x^{1} cos x^{2},

y^{3}_{,10} = y^{3}_{,01},

y^{3}_{,11} = -(a_{3} sin x^{0} + b_{1}) sin x^{1} sin x^{2},

y^{3}_{,12} = (a_{3} sin x^{0} + b_{1}) cos x^{1} cos x^{2},

y^{3}_{,20} = y^{3}_{,02},

y^{3}_{,21} = y^{3}_{,12},

y^{3}_{,22} = -((a_{3} sin x^{0} + b_{1}) sin x^{1} + b_{2}) sin x^{2}.

The Jacobian matrix has (*n*+1)X*n* elements.
The Hessian matrix has (*n*+1)X*n*^{2} elements.
That is 1100 elements when *n*=10.
Although some of these are equal due to symmetry, the complexity is daunting.
Fortunately, they can be evaluated in a smart way.
Formula recursion described above is inherited by partial derivative formulas.

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EleNTorus software performs several optimizations:

- Sine and cosines need only
*n*evaluations each. - Common factors appear repeatedly in both the representation and partial derivative formulas.

EleNTorus software is implemented in the `manifold.c`

routine in the Geodes geodesic computing package
stored in the Netlib repository of scientific software.

EleGeodesic software computes geodesics on n-dimensional manifolds, including all EleNTorus manifolds.

EleSalient software computes salient assemblages (recursive bumps) on n-dimensional manifolds, including all EleNTorus manifolds.