Re: [Closed REQ 6002]: quads with curved boundaries

[daemon]

> How can I define quadrilaterals with curved (parabolic function) 
> boundaries?

You can get that effect in Geomview by using Bezier patches.

For parabolic boundary curves in particular, you can use bi-quadratic patches,
for which the header key word will be

BEZ223 	  # degree-2 in u and in v, 3-D (as opposed to 4-D rational) points

The matrix of control points for each patch will then be (2+1) x (2+1); I'll
write them as

b00  b01  b02
b10  b11  b12
b20  b21  b22

where each of the bij is a 3-component (x y z) vector.

The point at parameter t (0<=t<=1) on a quadratic Bezier *curve*,
with control points p0 p1 p2, lies at
   (p0*(1-t) + p1*t)*(1-t) + (p1*(1-t) + p2*t)*t

 = p0*(1-t)^2 + 2*p1*t*(1-t) + p2*t^2

This passes through p0 at t=0, p2 at t=1, and has tangents pointing toward p1
at each endpoint.

Each of the patch's boundary curves have this form, with each triple of
control points on the matrix's edge playing the role of p0 p1 p2 in the above.

Hope this makes sense; if not, many computer-graphics texts have practical
sections on using spline curves and surfaces.