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## Bezier - Surfaces

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### #1Oesi  Members

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Posted 03 March 2000 - 04:17 AM

Hi ! I read an article about Bezier surfaces and an algorithm about dynamic LOD of the Bezier surfaces. But for this algorithm I need the second derivate of the Bernstein basis, but I don''t know how to get it. Bernstein: B (u) = (n!/(i!*(n-i)!)*u^i*(1-u)^(n-i) If someone has a way, to get the second derive of that function, I''ll need it, no matter how slow it is. Sorry, for my bad english... cya Oesi

### #2Spellbound  Members

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Posted 03 March 2000 - 05:01 AM

What is the grade of your Bezier surface? Bi-quadratic (Nine control points)? Bi-cubic (16 control points)? More?

For the bi-cubic surface you have the following Berstein polynomes:

B0 = (1-u)3
B1 = 3u(1-u)2
B2 = 3u2(1-u)
B3 = u3

These can be derived separately:

dB0 = -3(1-u)2
dB1 = 3(1-u)2 - 6u(1-u)
dB2 = 6u(1-u) - 3u2
dB3 = 3u2

One more derivation yields the second derivate:

ddB0 = 6(1-u)
ddB1 = -12(1-u) + 6u
ddB2 = 6(1-u) - 12u
ddB3 = 6u

(I believe these are correct but my derivation is a bit rusty so make sure before you use them.)

For a bi-quadratic surface the formulae are even easier:

B0 = (1-u)2
B1 = 2u(1-u)
B2 = u2

(You''ll have to do the derivations yourself )

### #3Oesi  Members

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Posted 03 March 2000 - 08:49 PM

THX, I think this is just what I need.

I tried to derivate the Bernstein - basis in one thing and failed at n!/(i!*(n-1)!)

Ok, so I have to derive it for each grade I need.

cya Oesi

### #4Spellbound  Members

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Posted 04 March 2000 - 01:31 AM

That is what I would do.

Tip: Rational B-Spline surfaces are just as easy to implement as bi-quadratic Bezier surfaces, but you have a lot more control over the appearance.

### #5Oesi  Members

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Posted 04 March 2000 - 07:30 PM

Sorry, but what are B-Splines exactly ?

I read a little about them but I still don''t know really what they are and how to implement.

### #6Spellbound  Members

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Posted 05 March 2000 - 02:28 AM

With a Bezier surface each control point in the characteristic polyhedron affects the whole surface according to the Bernstein polynomes. With B-Spline surfaces each control point only affects the immediate surrounding area of the surface. How large this area is is determined by the grade of the B-Spline. A grade of three means that each point on the surface is affected by at most 9 control points.

A B-Spline surface of grade 3 can when rendered be divided into smaller patches, each controlled by 9 control points, and rendered just as the bi-quadratic Bezier surface. With slightly different weight functions of course.

Here is an article on NURB curves. It explains the NURB curve (Duh!). The Non-Uniform part can be a little tricky, but fortunately you'll most likely not need it, at least not anytime soon. The B-Spline is just a specialization of the NURB where the knot-vector is uniform, instead of non-uniform.

Edited by - Spellbound on 3/5/00 8:32:44 AM

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