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Piecewise Polynomial Parametric Curves

Piecewise Polynomial Parametric Curves. 2001. 1. 31. Sun-Jeong Kim. Overview. Introduction to mathematical splines Bezier curves Continuity conditions (C 0 , C 1 , C 2 , G 1 , G 2 ) Creating continuous splines C 2 -interpolating splines B-splines Catmull-Rom splines. Introduction.

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Piecewise Polynomial Parametric Curves

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  1. Piecewise Polynomial Parametric Curves 2001. 1. 31. Sun-Jeong Kim

  2. Overview • Introduction to mathematical splines • Bezier curves • Continuity conditions (C0, C1, C2, G1, G2) • Creating continuous splines • C2-interpolating splines • B-splines • Catmull-Rom splines - 2 -

  3. Introduction • Mathematical splines are motivated by the “loftsman’s spline”: • Long, narrow strip of wood or plastic • Used to fit curves through specified data points • Shaped by lead weights called “ducks” • Gives curves that are “smooth” or “fair” • Such splines have been used for designing: • Automobiles • Ship hulls • Aircraft fuselages and wings - 3 -

  4. Requirements • Here are some requirements we might like to have in our mathematical splines: • Predictable control • Multiple values • Local control • Versatility • Continuity - 4 -

  5. Mathematical Splines • The mathematical splines we’ll use are: • Piecewise • Parametric • Polynomials • Let’s look at each of these terms… - 5 -

  6. Parametric Curves • In general, a “parametric” curve in the plane is expressed as: • Example: A circle with radius r centered at the origin is given by: • By contrast, an “implicit” representation of the circle is: - 6 -

  7. Parametric Polynomial Curves • A parametric “polynomial” curve is a parametric curve where each function x(t), y(t) is described by a polynomial: • Polynomial curves have certain advantages • Easy to compute • Infinitely differentiable - 7 -

  8. Piecewise Parametric Polynomial Curves • A “piecewise” parametric polynomial curve uses different polynomial functions for different parts of the curve • Advantage: Provides flexibility • Problem: How do you guarantee smoothness at the joints? (Problem known as “continuity”) • In the rest parts of this lecture, we’ll look at: • Bezier curves – general class of polynomial curves • Splines – ways of putting these curves together - 8 -

  9. Bezier Curves • Developed simultaneously by Bezier (at Renault) and by deCasteljau (at Citroen), circa 1960 • The Bezier Curve Q(u) is defined by nested interpolation: • Vis are “control points” • {V0, …, Vn} is the “control polygon” - 9 -

  10. Example - 10 -

  11. Bezier Curves: Basic Properties • Bezier Curves enjoy some nice properties: • Endpoint interpolation: • Convex hull: The curve is contained in the convex hull of its control polygon • Symmetry: • Q(u) is defined by {V0, …, Vn} Q(1-u) is defined by {Vn, …, V0} - 11 -

  12. Example: 2D Bezier Curves - 12 -

  13. Bezier Curves: Explicit Formulation • Let’s give Vi a superscript Vij to indicate the level of nesting • An explicit formulation Q(u) is given by the recurrence: - 13 -

  14. is the i ‘th Bernstein polynomial of degree n Explicit Formulation, cont. • For n=2, we have: • In general: - 14 -

  15. Example - 15 -

  16. Bezier Curves: More Properties • Here are some more properties of Bezier curves • Degree: Q(u) is a polynomial of degree n • Control points: How many conditions must we specify to uniquely determine a Bezier curve of degree n - 16 -

  17. More Properties, cont. • Tangent: • k ‘th derivatives: In general, • Q(k)(0) depends only on V0, …, Vk • Q(k)(1) depends only on Vn, …, Vn-k • At intermediate points , all control points are involved every derivative - 17 -

  18. Explanation - 18 -

  19. Cubic Curves • For the rest of this discussion, we’ll restrict ourselves to piecewise cubic curves • In CAGD, higher-order curves are often used • Gives more freedom in design • Can provide higher degree of continuity between pieces • For Graphics, piecewise cubic let’s you do about anything • Lowest degree for specifying points to interpolate and tangents • Lowest degree for specifying curve in space • All the ideas here generalize to higher-order curves - 19 -

  20. Matrix Form of Bezier Curves • Bezier curves can also be described in matrix form: - 20 -

  21. Display: Recursive Subdivision • Q: Suppose you wanted to draw one of these Bezier curves – how would you do it? • A: Recursive subdivision - 21 -

  22. Display, cont. • Here’s pseudo code for the recursive subdivision display algorithm ProcedureDisplay({V0, …, Vn}) if {V0, …, Vn} flat within e then Output line segment V0Vn else Subdivide to produce {L0, …, Ln} and {R0, …, Rn} Display({L0, …, Ln}) Display({R0, …, Rn}) end if end procedure - 22 -

  23. Splines • To build up more complex curves, we can piece together different Bezier curves to make “splines” • For example, we can get: • Positional (C0) Continuity: • Derivative (C1) Continuity: • Q: How would you build an interactive system to satisfy these conditions? - 23 -

  24. Advantages of Splines • Advantages of splines over higher-order Bezier curves: • Numerically more stable • Easier to compute • Fewer bumps and wiggles - 24 -

  25. Tangent (G1) Continuity • Q: Suppose the tangents were in opposite directions but not of same magnitude – how does the curves appear? • This construction gives “tangent (G1) continuity” • Q: How is G1 continuity different from C1? - 25 -

  26. Qn+1(u) Qn+1(u) Qn+1(u) Explanation • Positional (C0) continuity Qn(u) • Derivative (C1) continuity Qn(u) • Tangent (G1) continuity Qn(u) - 26 -

  27. Curvature (C2) Continuity • Q: Suppose you want even higher degrees of continuity – e.g., not just slopes but curvatures – what additional geometric constraints are imposed? • We’ll begin by developing some more mathematics..... - 27 -

  28. Operator Calculus • Let’s use a tool known as “operator calculus” • Define the operator D by: • Rewriting our explicit formulation in this notation gives: • Applying the binomial theorem gives: - 28 -

  29. Taking the Derivative • One advantage of this form is that now we can take the derivative: • What’s (D-1)V0? • Plugging in and expanding: • This gives us a general expression for the derivative Q’(u) - 29 -

  30. W0 Q’(1) Q’(0) W1 W2 W3 Specializing to n=3 • What’s the derivative Q’(u) for a cubic Bezier curve? • Note that: • When u=0: Q’(u) = 3(V1-V0) • When u=1: Q’(u) = 3(V3-V2) • Geometric interpretation: • So for C1 continuity, we need to set: V0 V1 V2 V3 - 30 -

  31. Taking the Second Derivative • Taking the derivative once again yields: • What does (D-1)2 do? - 31 -

  32. Q’’(0) Q’’(1) Q’(0) Q’(1) Second-Order Continuity • So the conditions for second-order continuity are: • Putting these together gives: • Geometric interpretation V0 V1 V3 V2 - 32 -

  33. W2 W2 W3 W1 W2 W1 W1 W1 W2 W3 W0 W0 W0 W3 W0 W3 C3 Continuity • Summary of continuity conditions • C0 straightforward, but generally not enough • C3 is too constrained (with cubics) V0 V3 V1 V2 - 33 -

  34. Creating Continuous Splines • We’ll look at three ways to specify splines with C1 and C2 continuity • C2 interpolating splines • B-splines • Catmull-Rom splines - 34 -

  35. joint C2 Interpolating Splines • The control points specified by the user, called “joints”, are interpolated by the spline • For each of x and y, we needed to specify 3 conditions for each cubic Bezier segment. • So if there are m segments, we’ll need 3m-1 conditions • Q: How many these constraints are determined by each joint? - 35 -

  36. In-Depth Analysis, cont. • At each interior joint j, we have: • Last curve ends at j • Next curve begins at j • Tangents of two curves at j are equal • Curvature of two curves at j are equal • The m segments give: • m-1 interior joints • 3 conditions • The 2 end joints give 2 further constraints: • First curve begins at first joint • Last curve ends at last joint • Gives 3m-1 constraints altogether - 36 -

  37. End Conditions • The analysis shows that specifying m+1 joints for m segments leaves 2 extra degree of freedom • These 2 extra constraints can be specified in a variety of ways: • An interactive system • Constraints specified as user inputs • “Natural” cubic splines • Second derivatives at endpoints defined to be 0 • Maximal continuity • Require C3 continuity between first and last pairs of curves - 37 -

  38. C2 Interpolating Splines • Problem: Describe an interactive system for specifying C2 interpolating splines • Solution: • 1. Let user specify first four Bezier control points • 2. This constraints next 2 control points – draw these in. • 3. User then picks 1 more • 4. Repeat steps 2-3. - 38 -

  39. Global vs. Local Control • These C2 interpolating splines yield only “global control” – moving any one joint (or control point) changes the entire curve! • Global control is problematic: • Makes splines difficult to design • Makes incremental display inefficient • There’s a fix, but nothing comes for free. Two choices: • B-splines • Keep C2 continuity • Give up interpolation • Catmull-Rom Splines • Keep interpolation • Give up C2 continuity – provides C1 only - 39 -

  40. B-Splines • Previous construction (C2 interpolating splines): • Choose joints, constrained by the “A-frames” • New construction (B-splines): • Choose points on A-frames • Let these determine the rest of Bezier control points and joints • The B-splines I’ll describe are known more precisely as “uniform B-splines” - 40 -

  41. V2 V2 V1 V3 V0 Explanation • C2 interpolating spline V1 V0 • B-splines B2 B1 B0 B3 - 41 -

  42. n+1 control points polynomials of degree d-1 B-Spline Construction • The points specified by the user in this construction are called “deBoor points” - 42 -

  43. B-Spline Properties • Here are some properties of B-splines: • C2 continuity • Approximating • Does not interpolate deBoor points • Locality • Each segment determined by 4 deBoor points • Each deBoor point determine 4 segments • Convex hull • Curves lies inside convex hull of deBoor points - 43 -

  44. Algebraic Construction of B-Splines • V1 = (1 - 1/3)B1 + 1/3 B2 • V2 = (1- 2/3)B1 + 2/3 B2 • V0 = ½ [(1- 2/3)B0 + 2/3 B1] + ½ [(1 - 1/3)B1 + 1/3 B2] = 1/6 B0 + 2/3 B1 + 1/6 B2 • V3 = 1/6 B1 + 2/3 B2 + 1/6 B3 - 44 -

  45. Algebraic Construction of B-Splines, cont. • Once again, this construction can be expressed in terms of a matrix: - 45 -

  46. Drawing B-Splines • Drawing B-splines is therefore quite simple procedureDraw-B-Spline ({B0, …, Bn}) for i=0 to n-3 do Convert Bi, …, Bi+3 into a Bezier control polygon V0, …, V3 Display ({V0, …, Vn}) end for end procedure - 46 -

  47. Multiple Vertices • Q: What happens if you put more than one control point in the same place? • Some possibilities: • Triple vertex • Double vertex • Collinear vertices - 47 -

  48. End Conditions • You can also use multiple vertices at the endpoints • Double endpoints • Curve tangent to line between first distinct points • Triple endpoints • Curve interpolates endpoint • Start out with a line segment • Phantom vertices • Gives interpolation without line segment at ends - 48 -

  49. Catmull-Rom Splines • The Catmull-Rom splines • Give up C2 continuity • Keep interpolation • For the derivation, let’s go back to the interpolation algorithm. We had a conditions at each joint j: • 1. Last curve end at j • 2. Next curve begins at j • 3. Tangents of two curves at j are equal • 4. Curvature of two curves at j are equal • If we… • Eliminate condition 4 • Make condition 3 depend only on local control points • … then we can have local control! - 49 -

  50. =V3 =V0 Derivation of Catmull-Rom Splines • Idea: (same as B-splines) • Start with joints to interpolate • Build a cubic Bezier curve between successive points • The endpoints of the cubic Bezier are obvious: • Q: What should we do for the other two points? B0 B3 B2 B1 - 50 -

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