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 airplane and autos, in addition to in modeling programs 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 software utilized in this lecture is blossoming, which supplies a sublime labeling of the regulate issues that permits us to research their houses geometrically. Blossoming is used to discover either Bézier and B-spline curves, and specifically to enquire continuity homes, switch of foundation algorithms, ahead differencing, B-spline knot multiplicity, and knot insertion algorithms. We additionally examine triangle diagrams (which are heavily with regards to blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.
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Extra resources for A blossoming development of splines
Ti−1 , ti+k , . . , ti+n−1 have multiplicity no greater than n − k, these control points completely define f and g when k of their arguments are ti , and thus, F and G meet C n−k at ti . Now we will show that the B-spline is not C n−k+1 . We will work with the segment dek−1 ) and fined over [ti−1 , ti ] whose control points are given above. From f (ti−n+k+1 , . . , ti−1 , ti k k−1 , ti+k ). Likewise, a proof by inf (ti−n+k , . . , ti−1 , ti ) we can compute f (ti−n+k , . . , ti−1 , ti duction shows that we can compute the points f ti−n+k+1+ j , .
As a side note, realize that the Lane–Riesenfeld algorithm is a particularly simple way to draw uniform B-splines: at each step, you simply average points together. As you repeatedly apply the Lane–Riesenfeld algorithm, the refined polygon converges to the curve. While exact convergence only occurs after infinite number of refinements, we can achieve a good approximation after four or five levels of refinement, as illustrated in Fig. 9 for a cubic B-spline refined three times. 1 Implementations 1.
If so, support your statement. If not, give a counterexample. (b) Now suppose that f (1, 2, 3) = f (2, 3, 4) = f (3, 4, 5). Is the curve geometrically C 2 at F(3)? If so, support your statement. If not, give a counterexample. 2 Implementations 1. Extend your interactive B-spline editor of the previous assignment in the following ways: • The right mouse button displays the blossom value of the displayed control point closest to the current mouse position. • There are three display modes: – Just the curve.
A blossoming development of splines by Stephen Mann