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- 1. 1 B-Spline Surfaces B-Spline Surfaces and their construction AML710 CAD LECTURE 25 – K,l=degree of polynomial in respective parameters • Can be 2 to the number of control points – If k,l set to 1, then only a plot of the control points • Bi,j is the input set of (n+1)x(m+1) control points (polygon net vertices) • Parameters u,w now depend on how we choose the other parameters (no longer locked to 0-1) • N i,k , M j,l , blending functions Polynomials of degree k-1, l-n in each parameter and at each interval xi≤ u≤xi+1, yj≤ w≤yj+1 12,12;; )()(),( maxminmaxmin 1 1 1 1 ,,, +≤≤+≤≤≤≤≤≤ = + = + = mlnkwwwuuu wMuNBwuQ n i m j ljkiji Definition of B-Spline Surface Extending the Idea of B-Spline curve, we obtain a Cartesian product B-splie surface
- 2. 2 B-Spline blending functions defined by Cox-deBoor recursion formula 1 1,1 1 1, , )()()()( )( ++ −++ −+ − − − + − − = iki kiki iki kii ki xx uNux xx uNxu uN and ≤≤ = ≤≤ = + + otherwise ywyif wM otherwise xuxif uN ii j ii i 0 1 )( 0 1 )( 1 1, 1 1, 1 1,1 1 1, , )()()()( )( ++ −++ −+ − − − + − − = jlj ljlj jlj ljj lj yy wMwy yy wMyw wM Properties of B-spline surfaces • The highest order in each parametric direction is limited to the number of defining polygon vertices in that direction • The continuity of the surface in each parametric direction is k-2,l-2 respectively • The surface is invariant to an affine transformation • The variation diminishing property of B-spline surface is not well known • The influence of any polygon net vertex is limited to ±±±±k/2, ±±±±l/2 spans in the respective parametric direction. • If the number of polygon net vertices is equal to the order of basis in that direction and if there are no interior knot values, then the B-spline surface reduces to a Bezier surface
- 3. 3 B-spline surface Subdivision • A B-spline surface is subdivided by separately subdividing polygon grid lines in one or both parametric direction • The flexibility of B-spline curves and surfaces is increased by raising the order of the basis function and hence the defining polygon/grid segments. • An alternative to degree raising is increasing the knot values in the knot vector used. • The basic idea of degree raising or knot insertion is to achieve the flexibility without changing the shape of the curve or surface. • The nature of the knot vector is preserved (uniform, open) even after insertion of new knot values. • Rational b-spline surface is a single precise mathematical representation capable of representing common analytical surfaces – planes, conic surfaces including sphere, free- form surfaces, quadric and sculptured surfaces used in the CAD applications • Non-uniform rational b-splines (NURBS) forms the basis of initial graphics exchange specification (IGES) Rational B-Spline Surface
- 4. 4 • Definition: A rational b-spline surface is the projection of a non rational (polynomial) b-spline defined in 4-dimensional homogenous coordinate space back into 3D physical space 1,1,2,,,;0 ),( )()( )()( ),( maxmaxminmin 1 1 1 1 ,,1 1 1 1 ,,, 1 1 1 1 ,,,, ++≤≤≤≤≥ == + = + = + = + = + = + = mnlkwuwuwuh wuSB wMuNh wMuNBh wuQ n i m j jijin i m j ljkiji n i m j ljkijiji Rational B-Spline Surface + = + = = 1 1 1 1 ,,, ,,, , )()( )()( ),( n i m j ljkiji ljkiji ji wMuNh wMuNh wuS RationalB-spline Curve
- 5. 5 RationalB-spline Curve -NURBS •The legs of the control triangle are of equal length (i.e. it is isosceles). •The chord connecting the first and the last control points meets each leg at an angle φ equal to half the angular extent of the arc. •The weight of the inner control point is cos(φ ). •The open uniform knot vector is [0,0,0,1,1,1]. Rational B-spline Curve -NURBS We see below how to get a whole circle with only one NURBS. The knot vectors are [0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1] and [0, 0, 0, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 1, 1, 1].

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