For those who know the basics of Beziers... A...Z = vertices a, c, e = 0.0 to 1.0, inclusive b = (1 - a), d = (1 - c), f = (1 - e) (Note: I'll use "aa" to represent "a squared") Point: (a+b)^0 = 1 vO = A Curve: Two-vertex (a+b)^1 = 1 vO = Aa + Bb Three-vertex (a+b)^2 = 1 vO = Aaa + B2ab + Cbb Surface: Four-Vertex (a+b)(c+d) = 1 vO = Aac + Bad + Cbc + Dbd Nine-Vertex (a+b)^2 * (c+d)^2 = 1 vO = Aaacc + B2aacd + Caadd + D2abcc + E4abcd + F2abdd + Gbbcc + H2bbcd + Ibbdd Now, my question. Would the below equation produce a solid? Solid: Eight-Vertex (a+b)(c+d)(e+f) = 1 vO = Aace + Bacf + Cade + Dadf + Ebce + Fbcf + Gbde + Hbdf No, this is not homework. This is obsession. EDIT: Silly mistake. [edited by - Wyrframe on September 17, 2002 2:55:36 AM]

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Original post by Wyrframe
No, this is not homework. This is obsession.

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OK, cool. It is a fairly interesting theoretical question.

To answer your question, the answer is yes , your last equation does represent a solid. Which you''ve implicitly explained in your definitions of surfaces and curves. A curve is essentially a "locus of points defined by a function of one parameter," that parameter being a in your case. A surface is a "locus of points defined by a function of two parameters," a and c in your case. Finally, a solid is a "locus of points defined by a function of three parameters," a , c and e in your case.

If you want to continue your obsession, consider this extension. You have not said that your vertices A....Z are points in 3-space (xyz space). Suppose that A...Z are actually points in 2-space (xy only). A curve is still a curve. And a surface is still a surface. Both are subspaces of 2-space. The curve is just a curve in a plane. And the surface is a fragment of the plane. But in this case, what is the solid? Think about it. Then go the other way. Suppose your vertices A....Z are points in 4-space (or even n-space). What are the meanings of a curve, surface, and solid in n-space? That one''s a bit more difficult to think about. For purposes of thinking, you might want to treat the higher dimensions as spatial dimensions and not the time dimension. Just for kicks, .

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Okay, thanks. I already knew the theory, I just wanted some confirmation from someone else before I go and kill myself with the math...

As soon as I''ve got one or two usable forms of the "solid" equation, I have been planning to write a decent tutorial on these things... it''s taken me a year and a quarter to find and comprehend even the simple quadratic bezier curve, (a+b)^2 = 1.

Thanks for your confirmation!