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Advanced Graphics Lecture Five

Advanced Graphics Lecture Five. Revenge of the NURBS. Alex Benton, University of Cambridge – A.Benton@damtp.cam.ac.uk Supported in part by Google UK, Ltd. History.

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Advanced Graphics Lecture Five

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  1. Advanced GraphicsLecture Five Revenge of the NURBS Alex Benton, University of Cambridge – A.Benton@damtp.cam.ac.uk Supported in part by Google UK, Ltd

  2. History • The term ‘spline’ comes from the shipbuilding industry: long, thin strips of wood or metal would be bent and held in place by heavy ‘ducks’, lead weights which acted as control points of the curve. • Wooden splines can be described by Cn-continuous Hermite polynomials which interpolate n+1 control points. Top: Fig 3, P.7, Bray and Spectre, Planking and Fastening, WoodenBoat Pub (1996) Bottom: http://www.pranos.com/boatsofwood/lofting%20ducks/lofting_ducks.htm

  3. History • Curves which must have n+1 control points are hard to work with. (n+1)x(m+1) patches are even moreso. • But continuity—smooth curves—is essential to the perception of quality of manufacture. Especially in cars, and some Apple products. • Splines were generalized in the 1960s by Bezier (Renault), de Casteljau (Citroen), de Boor (GM).

  4. Beziers—a quick review • A Bezier cubic is a function P(t) defined by four control points: • P0 and P3 are the endpoints of the curve; P1 and P2 define the other two corners of the bounding trapezoid. • The curve fits entirely within the bounding trapezoid. P1 P2 P0 P3

  5. Joining Bezier splines • To join two Bezier splines with C0 continuity, set P3=Q0. • To join two Bezier splines with C1 continuity, require C0 and make the tangent vectors equal: set P3=Q0 and P3-P2=Q1-Q0. Q1 Q0 P3 P2

  6. Joining Bezier patches • This works for Bezier patches as well: to join two patches with C1 continuity at an edge, ensure that their control points and tangent vectors on that edge are equal. • NURBS patches arejoined in exactly thesame way.

  7. Beziers • Cubics, of course, are just one example of Bezier splines: Linear: P(t) = (1-t)P0 + tP1 Quadratic: P(t) = (1-t)2P0 + 2t(1-t)P1 + t2P2 Cubic: P(t) = (1-t)3P0 + 3t(1-t)2P1 + 3t2(1-t)P2 + t3P3 ... General: “n choose v” = n! / i!(n-i)!

  8. Beziers P1 (1-t)P1+tP2 • Thinking about Beziers • Linear interpolations • The linear Bezier is a linear interpolation between two points. • The quadratic Bezier can be seen as a linear interpolation between two lines: P(t) = (1-t) ((1-t)P0+tP1) + (t) ((1-t)P1+tP2) • The cubic is a linear interpolation between linear interpolations between linear interpolations… etc. • Another way to see Beziers is as a weighted average between the control points. (1-t)P0+tP1 P2 P(t) P0

  9. Bernstein polynomials P(t) = (1-t)3P0 + 3t(1-t)2P1 + 3t2(1-t)P2 + t3P3 • The four control functions are the four Bernstein polynomials for n=3. • General form: • Bernstein polynomials in 0 ≤ t ≤ 1 always sum to 1:

  10. We can parameterize this chain over t by saying that instead of going from 0 to 1, t moves smoothly through the intervals [0,1,2,3] The curve C(t) would be: C(t) = P(t) • ((0 ≤ t <1) ? 1 : 0) + Q(t-1) • ((1 ≤ t <2) ? 1 : 0) + R(t-2) • ((2 ≤ t <3) ? 1 : 0) [0,1,2,3] is, in a way, a type of knot vector. 0, 1, 2, and 3 are the knots. This is not strictly true, but the idea is right: knots define intervals. Consider a chain of splines with many control points… P = {P0, P1, P2, P3} Q = {Q0, Q1, Q2, Q3} R = {R0, R1, R2, R3} …with C1 continuity… P3=Q0, P2-P3=Q0-Q1 Q3=R0, Q2-Q3=R0-R1 Parameterizing a chain of Beziers Q1 Q2 Q0 Q3 P3 R0 P2 R1

  11. NURBS • NURBS (“Non-Uniform Rational B-Splines”) are a generalization of Beziers. • NU: Non-Uniform. The control points don’t have to be weighted equally. • R: Rational. The spline may be defined by rational polynomials (homogeneous coordinates.) • BS: B-Spline. A chained series of Bezier splines with controllable degree.

  12. B-Splines • A Bezier cubic is a polynomial of degree three: it must have four control points per spline, it must begin at the first and end at the fourth, and it assumes that all four control points are equally important. • Beziers are actually a type of B-spline. B-splines are a piecewise parameterization of Beziers that supports an arbitrary number of control points and lets you specify the degree of the polynomial which interpolates them.

  13. B-Splines • We’ll build our definition of a B-spline from: • n+1, the number of control functions • k, the degree of the curve • {P0…Pn}, a list of n+1control points • [t0,t1,…,tk+n+1], a knot vector of parameter values • n intervals → n+1 control points • k is the degree of the curve, so k+1 is the number of control points which influence a single interval. k must be at least 1 (two control points linked by each linear line segments) and no more than n (the number of intervals.) • There are k+n+2 knots, and ti ≤ ti+1 for all ti.

  14. B-Splines • The equation for a B-spline is • Ni,k(t) is the basis function of control point Pi. Ni,k(t) is defined recursively:

  15. B-Splines t0 t1 t2 t3 t4 … N1,0(t) N2,0(t) N3,0(t) … N0,0(t) N1,1(t) N2,1(t) … N0,1(t) N1,2(t) N0,2(t) … N0,3(t) …

  16. B-Splines 0 1 1 2 2 3 3 4 4 5 N0,0(t) N1,0(t) N2,0(t) N3,0(t) N3,0(t) Knot vector = {0,1,2,3,4,5}, k = 0

  17. B-Splines N0,1(t) N1,1(t) N2,1(t) N3,1(t) Knot vector = {0,1,2,3,4,5}, k = 1

  18. B-Splines N0,2(t) N1,2(t) N2,2(t) Knot vector = {0,1,2,3,4,5}, k = 2

  19. B-Splines At k=1 the function is piecewiselinear, depends on P0,P1,P2,P3, and is fully defined on [t1,t4). At k=2 the function is piecewisequadratic, depends on P0,P1,P2, and is fully defined on [t2,t3). Each degree-k control function depends on k+1 knot values, so Ni,k depends on ti through ti+k, inclusive. So six knots → five discontinuous functions → four piecewise linear interpolations → three quadratics, interpolating three control points. n+1=3 control points, k=2 degree, n+1+k+1=6 knots. Knot vector = {0,1,2,3,4,5}

  20. Non-Uniform B-Splines • The knot vector {0,1,2,3,4,5} is uniform: ti+1-ti = ti+2-ti+1 ti. • Varying the size of an interval changes the parametric-space distribution of the weights assigned to the control functions. • Repeating a knot value reduces the continuity of the curve in the affected span by one degree. • Repeating a knot k+1 times will lead to a control function being influenced only by that knot value; the spline will pass through the corresponding control point with C0 continuity.

  21. Open vs Closed • A knot vector which repeats its first and last knot values k+1 times is called closed, otherwise open. • Repeating the knots k+1 times is the only way to force the curve to pass through the first or last control point. • Without this, the functions N0,kand Nn,kwhich weight P0and Pn would still be ‘ramping up’ and not yet equal to one at the first and last ti.

  22. Open vs Closed • Two examples you may recognize: • k=2, n+1=3 control points, knots={0,0,0,1,1,1} • k=3, n+1=4 control points, knots={0,0,0,0,1,1,1,1}

  23. Non-Uniform Rational B-Splines • Repeating knot values is a clumsy way to control the curve’s proximity to the control point. • We want to be able to slide the curve nearer or farther without losing continuity or introducing new control points. • The solution: homogeneous coordinates. • Associate a ‘weight’ with each control point: ωi.

  24. Non-Uniform Rational B-Splines • Recall: [x, y, z, ω]H→ [x / ω, y / ω, z / ω] • Or: [x, y, z,1] → [xω, yω, zω, ω]H • The control point Pi=(xi, yi, zi) becomes the homogeneous control point PiH =(xiωi, yiωi, ziωi) • A NURBS in homogeneous coordinates is:

  25. Non-Uniform Rational B-Splines • To convert from homogeneous coords to normal coordinates:

  26. Non-Uniform Rational B-Splines • A piecewise rational curve is thus defined by: with supporting rational basis functions: • This is essentially an average re-weighted by the ω’s. • Such a curve can be made to pass arbitrarily far or near to a control point by changing the corresponding weight.

  27. Non-Uniform Rational B-Splines in action

  28. Tensor products for surfaces • The tensor product of two vectors is a matrix. • Can also take the tensor of two polynomials.

  29. NURBS surfaces • The tensor product of the polynomial coefficients of two NURBS splines is a matrix of polynomial coefficients. • If curve A has degree k and n+1 control points and curve B has degree j and m+1 control points then A⊗B is an (n+1)x(m+1) matrix of polynomials of degree max(j,k). • So all you need is (n+1)(m+1) control points and you’ve got a rectangular surface patch! • This approach generalizes to triangles and arbitrary n-gons.

  30. References • Les Piegl and Wayne Tiller, The NURBS Book, Springer (1997) • Alan Watt, 3D Computer Graphics, Addison Wesley (2000) • G. Farin, J. Hoschek, M.-S. Kim, Handbook of Computer Aided Geometric Design, North-Holland (2002) Really good book!

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