By Stephen Mann
During this lecture, we learn Bézier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces which are universal in CAD platforms and are used to layout plane and cars, in addition to in modeling applications utilized by the pc animation undefined. Bézier/B-splines characterize polynomials and piecewise polynomials in a geometrical demeanour utilizing units of keep an eye on issues that outline the form of the outside. the first research device utilized in this lecture is blossoming, which provides a sublime labeling of the regulate issues that enables us to research their homes geometrically. Blossoming is used to discover either Bézier and B-spline curves, and specifically to enquire continuity houses, switch of foundation algorithms, ahead differencing, B-spline knot multiplicity, and knot insertion algorithms. We additionally examine triangle diagrams (which are heavily concerning blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.
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Extra info for A blossoming development of splines
Draw a triangle diagram for evaluating a cubic scaled monomial F(u) with blossom ¯ 0, ¯ 0), ¯ f ∗ (0, ¯ 0, ¯ δ), f ∗ (0, ¯ δ, δ), f ∗ (δ, δ, δ) at t. 9 FAST EVALUATION How do we evaluate a curve quickly? For univariate polynomials, if we evaluate a degree n polynomial in monomial form by evaluating x i for each i and multiplying by ci , it takes n additions and n(n + 1)/2 multiplications for each dimension of our range. cls September 25, 2006 16:36 POLYNOMIAL CURVES 33 When speed is a concern, Horner’s rule is the common technique for fast evaluation: p(x) = a + b x + c x 2 + d x 3 = a + x(b + x(c + d x)) Thus, each evaluation requires only n additions and n multiplications for each dimension of the range.
F ∗ (u, (n − j )! (n − j )! n− j n! (n − j )! n− j n! (n − j )! n− j n− j j = = n− j = k=0 ( j) F k=0 k n− j −k j j +k (n − j )! (n − j − k)! (n − j − k)! n! (0) k! n− j −k n− j ¯ . . , 0¯ , δ, . . , δ )u k f ∗ (0, k k=0 k=0 ( j +k) = F (u) n− j ¯ . . , 0¯ , δ, . . , δ )u k f ∗ (δ, . . 4. ) = Note that this is multilinear: each term has either u i or wi as a linear term. Thus, α f ∗ (u, ¯ . . ). f ∗ (α u, Now let us evaluate our multilinear blossom at f ∗ (u¯ 1 , u¯ 2 , δ). Then F(u) = 3u 3 + 2u 2 + 6u + 1 f ∗ (u¯ 1 , u¯ 2 , u¯ 3 ) = 3u 1 u 2 u 3 + 2(u 1 u 2 w3 + u 2 u 3 w1 + u 3 u 1 w2 )/3 +2(u 1 w2 w3 + u 2 w3 w1 + u 3 w1 w2 ) + w1 w2 w3 f ∗ (u¯ 1 , u¯ 2 , δ) = 3u 1 u 2 + 2(0 + u 2 w1 + u 1 w2 )/3 + 2(0 + 0 + w1 w2 ) + 0 = 3u 1 u 2 + 2(u 2 w1 + u 1 w2 )/3 + 2w1 w2 ¯ u, ¯ δ) = 3u 2 + 4u/3 + 2 f ∗ (u, = F (1) (u)/3 By computing the derivative of F in the usual fashion, we see that the last step is true.
N− j j where u¯ = (u, 1) and δ = (1, 0). Here, 1 is a unit vector in P, and represents a direction for a directional derivative. A nonunit vector may also be used, but then the resulting evaluation of the blossom will need to be rescaled by the length of this vector raised to the j th power. The theorem can be generalized to allow for different δs for each of the j arguments, leading to mixed directional derivatives. cls September 25, 2006 16:36 POLYNOMIAL CURVES 21 Proof. Proof of (1): This is a generalization of the blossoming principle proof.