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Computer Graphics

Computer Graphics. Lecture Notes #11 Curves and Surfaces II. Knot. Control point. m=9 (10 control points) m-1 knots m-2 knot intervals. Uniform Non-rational B-Splines. For each i  4 , there is a knot between Q i-1 and Q i at t = t i .

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Computer Graphics

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  1. Computer Graphics Lecture Notes #11 Curves and Surfaces II

  2. Knot. Control point. m=9 (10 control points) m-1 knots m-2 knot intervals. Uniform Non-rational B-Splines. • For each i 4 , there is a knot between Qi-1 and Qi at t = ti. • Initial points at t3 and tm+1 are also knots. The following illustrates an example with control points set P0 … P9: Lecture Notes #11

  3. Knot. Control point. P1 P3 Q3 P0 P2 Uniform Non-rational B-Splines. • First segment Q3 is defined by point P0 through P3 over the range t3 = 0 to t4 = 1. So m at least 3 for cubic spline. m=9 (10 control points) m-1 knots m-2 knot intervals. Lecture Notes #11

  4. Knot. Control point. P1 P3 P4 Q4 P2 Uniform Non-rational B-Splines. • Second segment Q4 is defined by point P1 through P4 over the range t4 = 1 to t5= 2. m=9 (10 control points) m-1 knots m-2 knot intervals. Lecture Notes #11

  5. 0 6 8 m t m+1 Uniform Non-Rational B-Splines • The unweighted spline set will look as follows, 10 control points, 10 splines, but only 8 knots and 7 knot intervals. • You can see why t3 to t4 is the first interval with a curve since it is the first with all four B-Spline functions. • t9 to t10 is the last interval, t3 to tm+1 are the knots. 3 4 Lecture Notes #11

  6. t 4 8 12 Cubic B-Splines • We have talked about a bi-infinite set of splines. What does this look like? • The unweighted B-Splines are shown for clarity. • How does a weighted set affect the shape of the curve? Lecture Notes #11

  7. X(t) t t Generating a curve Opposite we see an example of a shape to be generated. Here we see the curve again with the weighted B-Splines which generated the required shape. Lecture Notes #11

  8. B-Spline example • A closed curve is rather like a bi-infinite set – a periodic set of N knots in which: for all i, ti+N = ti and ti = i for i = 0..N-1 • Consider a closed curve with 5 control points with 5 knots. N=5, ie. x–1 x4, etc. • Control points are: x0 = (2,0); x1 = (1,1); x2 = (-1,1); x3 = (-1,-1) and x4 = (1,-1). Lecture Notes #11

  9. x2 x1 x0 x3 x4 B-Spline example Draw the interval: t = [4..5] , the others are handled the same way. x1 x4 Lecture Notes #11

  10. How smooth is a B-Spine? • Smoothness increases with order k in Bi,k • Quadratic, k = 3, gives up to C1 order continuity. • Cubic, k = 4 gives up to C2 order continuity. • However, we can lower continuity order too with Multiple Knots, ie. ti = ti+1= ti+2 = … Knots are coincident and so now we have non-uniform knot intervals. • A knot with multiplicity m is continuous to the (k-1-m)th derivative. • A knot with multiplicity k has no continuity at all, ie. the curve is broken at that knot. Lecture Notes #11

  11. Non-Uniform Non-Rational B-Splines • Note: as m increases by 1, an extra B-Spline function is attached to that knot: qiBi,k • B-Splines have special forms at multiple knots. • Obtained from a recursive formula defining how B-Splines are built, by setting m successive knots to be equal: tk+1 = tk+2 = … = tk+m • No point in having m>k • Thus for endpoints to be on curve, they must have a multiplicity of k, multiplicity of 4 in cubic case. Lecture Notes #11

  12. B-Splines at multiple knots • The multiple-knot B-Splines are as follows: B0,4(t) B1,4(t) B2,4(t) B3,4(t) t i=0,1,2,3 4 5 6 7 B0,4 is totally discontinuous at t = t0 , B1,4 is position continuous, B2,4 is gradient continuous, B3,4 is curvature continuous. Lecture Notes #11

  13. P0 P2 P3 P1 B-Spline continuity example. First knot shown with 4 control points, and their convex hull. Lecture Notes #11

  14. P0 P2 P4 P3 P1 B-Spline continuity example. First two curve segments shown with their respective convex hulls. Centre Knot must lie in the intersection of the 2 convex hulls. Lecture Notes #11

  15. P0 P3 P4 P1=P2 Repeated control point. First two curve segments shown with their respective convex hulls. Knot is forced to lie on the line that joins the 2 convex hulls. Curve is only C1 continuous Lecture Notes #11

  16. P0 P4 P1=P2=P3 Triple control point. First two curve segments shown with their respective convex hulls. Both convex hulls collapse to straight lines – all the curve must lie on these lines. Curve is only C0 continuous (Curiously it is actually C2 continuous because tangent vector magnitude falls to zero at join. ) Lecture Notes #11

  17. Non-uniform non-rational B-splines. • Parametric interval between knots does not have to be equal. • Blending functions no longer the same for each interval. • Multiple knots may have to be spaced apart. • Advantages • Continuity at selected control points can be reduced to C1 or lower – allows us to interpolate a control point without side-effects. • Can interpolate start and end points. • Easy to add extra knots and control points. Lecture Notes #11

  18. Summary of B-Splines. • Functions that can interpolate a series of control points with C2 continuity and local control. • Don’t pass through their control points, although can be forced. • note that if an order k B-Spline has k points locally colinear then a straight-line section will result midway in the set, and will touch (tangentially) the convex hull for (k – 1) colinear control points. • Uniform • Knots are equally spaced in t. • Non-Uniform • Knots are unequally spaced • Allows addition of extra control points anywhere in the set. Lecture Notes #11

  19. Summary cont. • For interactive curve modelling • B-Splines are very good. • To interpolate a number of positions • Catmull-Rom splines are best. • To interpolate with tangent control • Hemite or Bézier forms are useful and most often used. • To draw a spline. • Brute force method : Evaluate matrix to get parametric expressions for coordinates. Then for small increments of t, join with line segments. • Better : use method of forward differences to express spline in t form. Lecture Notes #11

  20. Surfaces – a simple extension • Easy to generalise from cubic curves to bicubic surfaces. • Surfaces defined by parametric equations of two variables, s and t. • ie. a surface is approximated by a series of crossing parametric cubic curves • Result is a polygon mesh and decreasing step size in s and t will give a mesh of small near-planar quadrilateral patches and more accuracy. Lecture Notes #11

  21. Example Bézier surface Lecture Notes #11

  22. Control of surface shape • Control is now a 2D array of control points. • The two parameter surface function, forming the tensor product with the blending functions is: • Use appropriate blending functions for Bézier and B-Spline surface functions. • Convex Hull property is preserved since bicubic is still a weighted sum (1). Lecture Notes #11

  23. Bézier example • Matrix formulation is as follows: Substitute suitable values for s and t (20 in the above ex.) Lecture Notes #11

  24. B-Spline surfaces • Break surface into 4-sided patches choosing suitable values for s and t. • Points on any external edges must be multiple knots of multiplicity k. • Lot more work than Bézier. • There are other types of spline systems and NURBS modelling packages are available to make the work much easier. • Use polygon packages for display, hidden-surface removal and rendering. (Bézier too) Lecture Notes #11

  25. Continuity of Bicubic patches. • Hermite and Bézier patches • C0 continuity by sharing 4 control points between patches. • C1 continuity when both sets of control points either side of the edge are collinear with the edge. • B-Spline patch. • C2 continuity between patches. Lecture Notes #11

  26. Displaying Bicubic patches. • Can calculate surface normals to bicubic surfaces by vector cross product of the 2 tangent vectors. • Normal is expensive to compute • Formulation of normal is a biquintic (two-variable,fifth-degree) polynomial. • Display. • Can use brute-force method – very expensive ! • Forward differencing method very attractive. Lecture Notes #11

  27. Reading for this lecture • Foley at al., Chapter 11, sections 11.2.3, 11.2.4, 11.2.9, 11.2.10, 11.3 and 11.5. • Introductory text, Chapter 9, sections 9.2.4, 9.2.5, 9.2.7, 9.2.8 and 9.3. • Other texts: look for B-Splines UNRBS, NUNRBS, something on forward difference method, comparison of systems, and Bézier and B-Spline surface modelling. Lecture Notes #11

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