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 structures and are used to layout plane and cars, in addition to in modeling programs utilized by the pc animation undefined. Bézier/B-splines signify 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 chic labeling of the regulate issues that enables us to research their houses geometrically. Blossoming is used to discover either Bézier and B-spline curves, and specifically 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 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
That control point is redundant, since we can compute it from the control points on either side. That is, if f i is parameterized over [r, s ] and f i+1 is parameterized over [s , t], then (t − s ) (s − r ) (r t − r s ) + (ts − tr ) f i (r, s , s ) + f i+1 (s , s , t) = f ,s,s t −r t −r t −r = f i (s , s , s ) Thus, we would only need to store 2L + 2 control points (Fig. 1). Likewise, if we have a C 2 piecewise cubic, then we would only need to store L + 3 control points (Fig. 1: Reducing data used to represent C 1 joined cubic B´ezier segments shown for two cubic with uniform knots in the figure, the missing B´ezier control points between two gray points are located at 1/3 and 2/3 away along the segment between the gray points (the ends of the curve are special cases, where the missing B´ezier point lies half-way between the black and the gray points).
Instead, we have hidden this index in the definition of a B-spline. While we can evaluate the blossom from one of these sets of control points on argument bags that are not all contained within the defining interval, the value may be ill-defined: if we evaluate a different set of control points on the same argument bag, we may get a different value. In fact, except for the cases given by the continuity conditions, we will expect to get different results. 4, having a uniform knot vector allows us to use a very simple rendering method.
I n−i i F (i) (0)u i /i! by Taylor expansion, and since the monomials form a basis, we have F (i) (0) = n! ¯ . . , 0¯ , δ, . . , δ ) f ∗ (0, (n − i)! n−i i Now, we have the following: n! n! ¯ . . , 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, . .
A blossoming development of splines by Stephen Mann