The Ideal CAGD System

An interactive virtual environment
featuring idealized virtual materials
and means to specify constraints on them.

Choose a material -- defined by its behavior:
Virtual clay -- no constraints at all ?
Sheet metal -- developability criteria
Soap film -- minimal surface; mean curvature = 0
Bernoulli Elastica -- minimizes bending energy
Minimum Variation Surface -- minimizes square of derivative of curvature
Some other desirable properties -- find corresponding functional

Define various local constraints:
Positional constraints: points that the surface must interpolate.
Tangential constraints: a predefined surface normal.
Curvature constraints: predefined Dupin indicatrix (best fitting quadric).
Any combinations of the above.
Note that it is admissible to have tangent and curvature constraints
  at a point that in itself has no positional constraint and can thus
  float around while maintaining the given constraints.

Define some global constraints:
Total arc-length of a curve,
Area of a surface patch,
Volume enclosed by surface,
Area enclosed by curve.

System will continuously solve a constrained optimization problem,
trying to minimize the specified cost functional
while maintaining all the given constraints.
(The global constraints may cause serious trouble).

Implementation ?
Need to find an appropriate representation for the geometry
on which we can play the above games.
That representation must be rich enough to express the above constraints
as well as the desired properties.
E.g., if we want to obtain G2 continuity, 
we may want at least 
biquintic Bezier patches and/or sextic Bernstein triangles,
or a bicubic B-spline system.
We then need enough patches / control points so that we have
many more degrees of freedom to start with
than we have constraints to satisfy.
We need to have several DOF's left over to minimize the cost functional.

If the task demands to model features at many diffeent scales,
then it would be desirable to have hierarchical system
or one based on wavelets.