Re: [Algorithms] Spline Surface Extremal Points ?

[Dave E. <eb...@ma...>]

From: "Steve Wood" <Ste...@im...>
> I don't know anything about spline patches...but extrema is defined as the
> point where the the slope of the tangent=0 AND the slope of the tangent to
> the left of that point is the opposite sign from the slope of the tangent
to
> the right of that point...ie...it's not an inflection point.

Your definition is for the graph of a function y = f(x).  A local maximum
occurs at x0 if f'(x0) = 0 and f"(x0) < 0.  Unfortunately these extrema
are tied into the coordinate system.  Consider f(x) = 1-x^2 for |x| <= 1.
The local maximum occurs at x = 0.  Now rotate the curve (x,f(x)) about
the origin a small amount (so that no tangent line becomes vertical).
The new curve is a graph of another function, but the local maximum
for it is not at the same point as in the original coordinate system.

For a graph z = (x,y,f(x,y)), a local maximum occurs at (x0,y0) if
Gradient(f)(x0,y0) = (0,0) and Hessian(f)(x0,y0) is a negative
definite matrix.  Just as in the one dimensional case, if the graph
has no vertical tangents, and if you rotate the graph a small amount
so that you still have no vertical tangents, the new surface is a
graph of another function, but the local maxima are not at the
same location as in the original coordinate system.

If the original poster has a coordinate system in mind for the
surface, then the definition above will locate the extremal
points.  But if the idea is to find some "intrinsic" extrema,
for example, the vertices of an ellipsoid that do not change
with orientation of the ellipsoid, you need a different concept
for extremum.  The usual ones involve measuring principal
curvatures and deriving some scalar function from them, then
applying the definition mentioned above for finding extrema
of that function.  This approach is what is used for defining
"ridges"  (curves of extreme points) on a surface.

--
Dave Eberly
eb...@ma...
http://www.magic-software.com