Envelope of a family of surfaces

ENVELOPE SURFACE OF A FAMILY OF SURFACES

1) Case of a family of surfaces with one parameter.

If is defined by the Cartesian equation (1): f(x, y, z, t) = 0, then the equation of the envelope is obtained by eliminating t between (1) and the equation (2): .
If is parametrically defined by (M(u,v,t))_u,v, then solving gives, by elimination of one of the parameters u,v,t, the parametrization of the envelope.
If is a plane, passing by and with normal vector , then the envelope is the ruled surface union of the lines passing by and with direction vector .

The envelope of a family of surfaces with one parameter is the surface union of the characteristic curves of the surfaces , limit curves when goes to t of the intersection curves between and ; the surface is tangent at any of its points to a surface and "in general", any surface is tangent along a curve to ; the restrictive cases are the following ones:
- on an interval, the surfaces pass by a fixed curve, in which case, this curve belongs to the envelope.
- the surfaces do not intersect with one another (for example, concentric spheres, or surfaces for which the intersection points are imaginary).

The family of characteristic curves then has, in general, an envelope, which is the cuspidal edge of the surface .
With the above notations in the parametric case, since the condition is symmetrical in u,v and t, the two envelopes of the surfaces loci of the points (M(u,v,t))_u,t and the surfaces loci of the points (M(u,v,t))_v,t are the same as those of the surfaces ; the common envelope is in fact the locus of the points where the surfaces of the three families are tangent along a line.

When the surfaces are planes, the characteristic curve is a line that remains tangent to the cuspidal edge of the envelope (which then is a ruled developable surface).

Examples:
- the polar developable of a curve is the envelope of its normal planes.
- the Dupin cyclides and the tubes are envelopes of spheres.

2) Case of a family with two parameters.

If is defined by the Cartesian equation (1): f(x, y, z, t, t') = 0, then the equation of the envelope can be obtained by eliminating t and t' between (1) , (2): and (3): .
If is defined parametrically by (M(u,v,t, t'))_u,v, then solving ??? (condition for these 4 vectors to be coplanar) gives, by elimination of two of the parameters u,v,t,t', the parametrization of the envelope.

The envelope of a family of surfaces with two parameters is the surface generated by the characteristic points of the surfaces , limit points when (t₁ ,t'₁) goes to (t ,t') of the intersections between and ???; the surface is tangent at any of its points to a surface and "in general", every surface is tangent at at least one point to .

With the above notations in the parametric case, since the condition is symmetrical in u,v,t,t' , then envelope of the is also the envelope of 3 other families with two parameters: the common envelope is in fact the locus of the points where the surfaces of the 4 families are tangent.

Examples:
- Every surface is the envelope of its tangent planes.
- The parallel surfaces of a surface are the envelopes of spheres with constant radius centered on this surface.
- The pedal of a surface with respect to a point O is the envelope of the spheres with diameter [OM] when M describes .
- The (reciprocal) polar of a surface with respect to a sphere (S) is the envelope of the polar planes with respect to (S) of the points of .
- The envelope of the plane of a triangle [ABC] the vertices of which move on the axes Ox, Oy et Oz is
- an astroidal surface: when the distance from the center of gravity to O is constant
- the cubic surface: xyz = a³ when the tetrahedron OABC keeps a constant volume

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