By Stephen Mann
During this lecture, we examine Bézier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces which are universal in CAD structures and are used to layout plane and vehicles, in addition to in modeling applications utilized by the pc animation undefined. Bézier/B-splines symbolize polynomials and piecewise polynomials in a geometrical demeanour utilizing units of keep watch over issues that outline the form of the skin. the first research software utilized in this lecture is blossoming, which provides a sublime labeling of the keep an eye on issues that permits us to investigate their homes geometrically. Blossoming is used to discover either Bézier and B-spline curves, and particularly to enquire continuity homes, swap of foundation algorithms, ahead differencing, B-spline knot multiplicity, and knot insertion algorithms. We additionally examine triangle diagrams (which are heavily regarding 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
However, there are also several other methods. We will look at the one based on repeated knot insertion. Suppose we have a B-spline curve as defined above, and we wish to insert a new knot to get a new knot vector and a new set of control points for the same curve. The new knot vector we know (it is just the old one with the new knot inserted). cls 40 QC: IML/FFX T1: IML September 25, 2006 16:37 A BLOSSOMING DEVELOPMENT OF SPLINES Thus, starting with the original set of control points, we remove f (t1 , t2 , t3 ) and f (t2 , t3 , t4 ) and add f (t1 , t2 , t), f (t2 , t, t3 ), f (t, t3 , t4 ).
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.
U¯ , δ, . . , δ ) = 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).
A blossoming development of splines by Stephen Mann
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