"Fl" == Fred lunnon <fred.lunnon@gmail.com> writes:
Fl> I didn't understand this question at all, I'm afraid. Fl> Where you wrote "loci", I think you probably meant "foci". Yes. An uncaught typo. Fl> The "radii" presumably refers to major and minor radius of an ellipse. Of course. And the logical equivs for parabolae and hyperbolae. Fl> The coefficients are presumably those of the parameters in the Fl> coordinate functions. As I explain below I was thinking of Bernstein basis polys, but I Fl> A circle is rationally parameterised by [w,x,y] = [1+t^2, 1-t^2, 2t], Fl> the Cartesian coordinates of the conic then being [x/w, y/w]. Fl> Why should this not disprove your assertion? Fl> A reference to the relevant background might be of some assistance! So perhaps then it only holds for bezier and b-splines? Or, since cubics are the norm for graphics, I may have just forgotten to consider quadradics.... :-/ The standard cubic NURB-Spline for a circle with radius=1 centered at [0,0] has four arcs with (normalized) knot vector: [ 0 0 0 1/4 1/4 1/2 1/2 3/4 3/4 1 1 1 ] and coefficient matrix: x = [ 1 sqrt(1/2) 0 -sqrt(1/2) -1 -sqrt(1/2) -0 sqrt(1/2) 1 ] y = [ 0 sqrt(1/2) 1 sqrt(1/2) 0 -sqrt(1/2) -1 -sqrt(1/2) 0 ] w = [ 1 sqrt(1/2) 1 sqrt(1/2) 1 sqrt(1/2) 1 sqrt(1/2) 1 ] for a circle with radius=1 centered at [0,0]. Whereas that same circle can be drawn with a single arc NURB-Spline using quintics with knot vector [ 0 0 0 0 0 0 1 1 1 1 1 1 ] and coefficient matrix: x = [ 0 4 2 -2 -4 0 ] y = [ -5 -1 3 3 -1 -5 ] w = [ 5 1 1 1 1 5 ] I don't beleive that one can reduce that to a quartic and retain the lack of radicals. -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6