Equidistance surface

EQUIDISTANCE SURFACE, SYMMETRY SET, SKELETTON

Other names: mediatrix surface.

Obtaining the equation of the equidistance curve: eliminate from .

Particular case of a surface and a plane: ; alors avec .

The equidistance surface between two surfaces and is the locus of the points M located on a normal at M₁ to and on a normal at M₂ to with MM₁ = MM₂ . This is the generalisation to 3 dimensions of the equidistance curve of 2 curves.
Therefore, it is also:
- the locus of points located on two parallel surfaces respectively to the two surfaces, at the same distance.
- the locus of the centres of the spheres tangent to both surfaces.

Examples:
- the equidistance surface of two parallel curves is yet another parallel surface.
- the equidistance surface of two surfaces symmetric about a plane is a portion of that plane (for example, the equidistance surface of two secant planes is composed of the two bisector planes).
- the equidistance surface of a sphere and a plane is composed of one or two paraboloids; writing F the centre of the circle, R its radius, P the line, then the equidistance curve is defined by where H is the projection of M on P. The focus of the paraboloid is F and their directrix plane is parallel to P.

Various cases: disjoint, tangent, or secant sphere and plane.
The second paraboloid is not traced.

- the equidistance surface of two circles is composed of two bifocal quadrics; Writing (centre,radius) of the circles as (F,R) and (F',R') respectively, the equidistance surface is defined by .

These two quadrics are...

...two ellipsoids when one of the surfaces is inside the other: ...an ellipsoid and a plane when one of the sphere is inside the other and tangent to it:

...an ellipsoid and a hyperboloid when the spheres are secant: ...a hyperboloid and a plane when the spheres are tangent and outside one another:

...two hyperboloids when the spheres are outside one another Opposite, the equidistance surface of a plane and a paraboloid.
Algebraic curve of degree 6.

We can consider cases where the two surfaces are one. The equidistance surface, which then takes the name of the symmetry set, is the locus of the centers of the bitangent (or multitangent) spheres to the surface, and therefore also the locus of the points where the parallel surfaces intersect themselves.

Examples

The symmetry set of an ellipsoid is

The symmetry set of the envelope of a family of spheres is the locus of the centres (but, be careful, some spheres may have an imaginary contact with the surface: their centre will not be counted in the real symmetry set); for example, the symmetry set of an anallagmatic surface is its deferent.

The notion of mediatrix set of two subsets of the plane is similar to that of equidistance surface. It is the set of points at equal
distance from the two subsets (the distance to a set being the infimum of the distances to a point in said set); when the two subsets are curves, the mediatrix is in
general included in the equidistance surface.

Here is, for example, the mediatrix set of two secant spheres, always composed of an ellipsoid, but the hyperboloid of which has only one sheet:

The notion similar to that of symmetry set is therefore the skeleton of a compact subset X of the space, which is the closure of the set of centres of the maximal disks contained in the subset (a maximal ball is a ball that is not included in any other disk).

For example, the skeletton of a full tetrahedron is composed of the six full triangles that join the center of the insphere to the edges.

For a full convex polyhedron, the skeleton is composed of portions of bisector planes of couples of faces, delimiting the "attraction zones" of each face
(locus of the points of the polyhedron that are closer to this face than to the others).

The notion can be extended to the case where X is any subset; for example, if X is the complement of a finite subset Y, the skeleton is composed of the boundaries of the Voronoi cells associated to the points of Y.

surface suivante surface précédente courbes 2D courbes 3D surfaces fractals polyèdres

...two ellipsoids when one of the surfaces is inside the other:		...an ellipsoid and a plane when one of the sphere is inside the other and tangent to it:
...an ellipsoid and a hyperboloid when the spheres are secant:		...a hyperboloid and a plane when the spheres are tangent and outside one another:
...two hyperboloids when the spheres are outside one another		Opposite, the equidistance surface of a plane and a paraboloid. Algebraic curve of degree 6.