A Blossoming Development of Splines
In this lecture, author Stephen Mann presents Bezier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces that are common in CAD systems. They are used to design aircraft and automobiles, as well as having uses in modeling packages used by the computer animation industry. Bezier/B-splines represent polynomials and piecewise polynomials in a geometric manner using sets of control points that define the shape of the surface.The primary analysis tool used in this lecture is blossoming, which gives an elegant labeling of the control points that allow us to analyze their properties geometrically. Blossoming is used to explore both Bezier and B-spline curves, and in particular to investigate continuity properties, change of basis algorithms, forward differencing, B-spline knot multiplicity, and knot insertion algorithms. We also look at triangle diagrams (which are closely related to blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.
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2-to-l subdivision afﬁne affine combinations affine transformation B-spline basis functions B-spline curve B-spline segments B´ezier control points B´ezier curve barycentric coordinates basis functions Bernstein basis Bernstein polynomials bilinear interpolation blending functions blossom arguments blossom f blossom of F blossom values blossoming theorem C1 continuity Casteljau’s algorithm coefﬁcients compute construction control polygon cubic B-spline cubic B´ezier curve cubic curve dataflow diagram deﬁned deﬁnition degree n polynomial derivative of F domain triangle edge example Exercise F and G FIGURE ﬁnd ﬁrst derivative forward differencing full multiplicity geometric modeling give intersect interval 0,1 knot insertion labeled Lagrange polynomials Lane—Riesenfeld algorithm last control point linear meet with C2 monomial monomial form mouse button multiafﬁne multilinear blossom nodes Note NURBS panels parametric continuity proof quadratic repeated curve evaluation Section simplex splines Suppose surface patch tensor-product surface triangle diagram triangular Bezier patch uniform knot vector vector space weight zero