|
NURBS Curves by Byung-Gook Lee, Applied Mathematics
NURBS
¢Á Non-Uniform Rational B-Splines ¢Á de facto industry standard for the representation, design, and data exchange of geometric information ¢Á Many national and international standards, e.g., IGES, STEP, and PHIGS ¢Á designing with NURBS is intuitive ¢Á NURBS algorithms are fast and numerically stable ¢Á NURBS provide a unified mathematical basis for representaing both analytic shapes, such as conic sections and quadric surfaces, as well as free-form entries, such as car bodies and ship hulls ¢Á NURBS are generations of nonrational B-splines and rational and nonrational Bezier curves and surfaces
Bezier Curves
where the {b_k} are the control points(forming a control polygon), and {B_{k,n}(u)} are the nth-degree Bernstein polynomials given by
A Bezier Curves
Properties of Bezier Curves
¢Á Endpoint interpolation : b^n(0)=b_0 and b^n(1)= b_n ¢Á Symmetry : The curves that correspond to the two different orderings look same ¢Á Affine invariance : Bezier curves are invariant under affine maps. ¢Á Convex hull property : the curves are contained in the convex hulls of their defining contrl points ¢Á Pseudo-local control : The Berstein polynomial B_i,n has only one maximum and attains at u=i/n ¢Á Designing with Bezier curves
Bezier Curve Topics ¢Á Degree Elevation ¢Á Degree Reduction
Rational Bezier Curves
where the {w_i} are the weights,
A Rational Bezier Curves
B-spline Curves
where the {P_i} are the control points(forming a control polygon),and {N_{i,p}(u)} are the pth-degree B-spline basis functions defined on the knot vector
The B-spline basis functions of degree p
Uniform or Nonuniform
uniform if all interior knots are equal spaced i.e., if there exists a real number, d, such that d=u_i+1 - u_i for all p<=i<=m-p-1; otherwise it is nonuniform
A B-spline Curves
Properties of B-spline Curves ¢Á If n=p and U= {0, ldots, 0, 1, ldots, 1}, then C(u) is a Bezier curve, C(u) is a piecewise polynomial curve ¢Á Endpoint interpolation : C(a) = P_0 and C(b) = P_n ¢Á Affine invariance ¢Á Strong convex hull property : the curve is contained in the convex hull of its control polygon ¢Á Local modification scheme
B-spline Curve Topics ¢Á Knot insertion ¢Á Knot refinement ¢Á Knot removal ¢Á Degree elevation ¢Á Degree reduction
NURBS A NURBS curve is a vector-valued piecewise rational polynomial function
where the {w_i} are the weights, the {P_i} are the control points(forming a control polygon), and {N_{i,p}(u)} are the pth-degree B-spline basis functions defined on the nonuniform knot vector
The B-spline basis functions of degree p
|
