Curves. Hakan Bilen University of Edinburgh. Computer Graphics Fall Some slides are courtesy of Steve Marschner and Taku Komura

Transcription

1 Curves Hakan Bilen University of Edinburgh Computer Graphics Fall 2017 Some slides are courtesy of Steve Marschner and Taku Komura

2 How to create a virtual world? To compose scenes We need to define objects - Characters - Terrains - Objects (trees, furniture, buildings etc)

3 Geometric representations Meshes Triangle, quadrilateral, polygon Implicit surfaces Blobs, metaballs Parametric surfaces / curves Polynomials Bezier curves, B-splines

4 [scene360.com] Motivation Smoothness Many applications require smooth surfaces Can produce smooth surfaces with less parameters Easier to design Can efficiently preserve complex structures

5 Original Spline

6 From draftsmanship to CG Control user specified control points analogy: ducks Smoothness smooth functions usually low order polynomials analogy: physical constraints, optimization

7 What is a curve? A set of points that the pen traces over an interval of time What is the dimensionality? Implicit form: f x, y = x 2 + y 2 1 = 0 Find the points that satisfy the equation Parametric form: x, y = f t = cost, sint, t [0,2π) Easier to draw

8 f t is a parametric curve What is a spline curve? in this context piecewise polynomial function that switches between different functions for different t intervals Example: t 3, if 0 t < 1 f t = 1 t 1 3, if 1 t < 2 0, otherwise

9 Defining spline curves Discontinuities at the integers [t=k] Each spline piece is defined over [k,k+1] (e.g. a cubic spline) f t = at 3 + bt 2 + ct + d Different coefficients for every interval Control of spline curves Interpolate Approximate

10 Today Spline segments Linear Quadratic Hermite Bezier Chaining splines Notation vectors bold and lowercase v points as column vector p = p x p y matrices bold and uppercase M

11 Spline segments Linear Segment A line segment connecting point p o to p 1 p 1 Such that f 0 = p 0 and f 1 = p 1 f x t = 1 t x o + tx 1 f y t = 1 t y o + ty 1 Vector formulation f t = 1 t p o + tp 1 Matrix formulation f t = t 1???? f t = t p 0 p 1 p 0 p 1 p 0 1 t t 1 t p o + tp 1

12 Matrix form of spline f t = t p 0 p 1 Blending functions b t specify how to blend the values of the control point vector f t = b 0 t p 0 + b 1 t p 1 b 0 t = 1 t b 1 t = t b 0 t b 1 t

13 Beyond line segment Quadratic A quadratic (f t = a 0 + a 1 t + a 2 t 2 ) passes through p 0, p 1, p 2 s.t. p 0 = f 0 = a a a 2 p 1 = f 0.5 = a a a 2 p 2 = f 1 = a a a 2 Points can be written in terms of constraint matrix C p 0 p a 0 a 0 p 1 = Ca p 1 = a 1 a 1 p 2 p a 2 a 2 = C 1 f t can be written in terms of basis matrix B = C 1 and points p p 0 f t = tbp = tc 1 p = t 2 t p p 2 p 0 p 1 p 2

14 Matrix form of spline Blending functions f t = t 2 t p 0 p 1 p 2 Blending functions b t specify how to blend the values of the control point vector f t = b 0 t p 0 + b 1 t p 1 + b 2 t p 2 b 0 t = 2t 2 3t + 1 b 1 t = 4t 2 4t b 2 t = 2t 2 1

15 Hermite spline Piecewise cubic (f t = a 0 + a 1 t + a 2 t 2 + a 3 t 3 ) Additional constraint on tangents (derivatives) f t = a 0 + a 1 t + a 2 t 2 + a 3 t 3 f t = a 1 + 2a 2 t + 3a 3 t 2 p 0 = f 0 = a 0 p 1 = f 1 = a 0 + a 1 + a 2 + a 3 v 1 = f (0) = a 1 p 0 v 0 p 1 v 2 = f 1 = a 1 + 2a 2 + 3a 3 Simpler matrix form f t = t 3 t 2 t p 0 p 1 v 1 v 2 v 1

16 Hermite to Bézier q 1 q 2 Specify tangents as points p 0 = q 0, p 1 = q 3, v 0 = 3 q 1 q 0, v 1 = 3 q 3 q 2 p q 0 p q 1 v = q 2 v q 3 Update Hermite eq. (from previous slide) f t = t 3 t 2 t v 0 q 0 p q 0 q 1 q 2 q 3 v 1 p 1 q 3 v 1

17 Bézier matrix f t = t 3 t 2 t q 0 q 1 q 2 q 3 f t = σd n=0 b n,3 q n Blending functions b t has a special name in this case: Bernstein polynomials b n,k = n k tk 1 t n k and that defines Bézier curves for any degree

18 Bézier blending functions The functions sum to 1 at any point along the curve. Endpoints have full weight q 0 q 3 q 1 q 2 10/10/2008

19 Another view to Bézier segments de Casteljau algorithm Blend each linear spline with α and β = 1 α p 1 p 1 α β αp 0 + βp 1 β α p 0 α α 2 p 0 + 2αβp 1 + β 2 p 1 β p 0 p 2

20 Review

21 Today Spline segments Linear Quadratic Hermite Bezier Chaining splines Continuity and local control Hermite curves Bezier curves Catmull-Rom curves B-splines

22 Putting segments together Limited degree of freedom with a single polynomial Will it be smooth enough?

23 Measuring smoothness Continuity Smoothness as degree of continuity zero-order (C 0 ): position match first-order (C 1 ): tangent match second-order (C 2 ): curvature match C N vs G N

24 Putting segments together Limited degree of freedom with a single polynomial Will it be smooth enough? Local control

25 Local control How many knots for m segments with 2 control points each?

26 Local control?

27 Local control changing control point only affects a limited part of spline without this, splines are very difficult to use overshooting fixed computation many likely formulations lack this natural spline polynomial fits (matlab demo)

28 Putting segments together Piecewise linear Blending functions for a linear segment (0<t<1) p 1 p 0 p 1 f t p 0 f t = 1 t p 0 + tp 1

29 Putting segments together Piecewise linear p 1 p 0 p 1 p 1 p 2 f 1 t f 0 t p 2 p 0 f 0 t = 1 t p 0 + tp 1 (0 < t < 1) f 1 t =? p 1 +? p 2 (1 < t < 2) f 1 t = 2 t p 1 + t 1 p 2 (1 < t < 2)

30 Putting segments together How can we chain these segments to a longer curve? Use first segment between t=0 to t=1 Use second segment between t=1 to t=2 f t = f i t i for i t i + 1 Shift blending functions f 0 t = b 0 t p 0 + b 1 t p 1 f 1 t = b 0 t 1 p 1 + b 1 t 1 p 2 Match derivatives at end points to avoid discontinuity

31 Hermite basis How many segments for n control points? (n-2)/2 Continuity? C 1 Slide credit: S. Marschner

32 Chaining Bézier Splines Can we add blending functions as in Hermite? If p 3 p 2 is collinear with p 4 p 3, it is G 1 continuous p 3 p 2 = k(p 4 p 3 ), k > 0 If tangents match, it is C 1 continuous p 3 p 2 = k(p 4 p 3 ), k = 1 p 0 p 1 p 2 p 3

33 Chaining segments Hermite curves are convenient because they can be made long easily Bézier curves are convenient because their controls are all points but it is fussy to maintain continuity constraints and they interpolate every 3rd point, which is a little odd We derived Bézier from Hermite by defining tangents from control points a similar construction leads to the interpolating Catmull-Rom spline

34 Catmull-Rom Would like to define tangents automatically use adjacent control points q 1 q 2 q 0 q 3

35 Hermite to Catmull-Rom p 0 = f 0 p 0 = q 1 p 1 = f 1 p 1 = q 2 v 1 = f (0) v 2 = f 1 v 0 = 1 (q 2 2 q 0 ) v 1 = 1 (q 2 3 q 1 ) f t = t 3 t 2 t p 0 p 1 v 0 v 1 f t = t 3 t 2 t q 0 q 1 q 2 q 3

36 Catmull-Rom Interpolating curve Like Bézier, equivalent to Hermite Continuity? Local control? Add tension p 0 = q 1 p 1 = q 2 v 0 = 1 (1 t)(q 2 2 q 0 ) v 1 = 1 (1 t)(q 2 3 q 1 )

37 B-splines We may want more continuity than C 1 f t = σ n i=1 p i b i (t) parameterized by k control points made of polynomials of degree k-1 is C (k 2) B-splines are a clean, flexible way of making long splines with arbitrary order of continuity Various ways to think of construction a simple one is convolution relationship to sampling and reconstruction

38 Quadratic B-spline b i,3 t = 1 2 u2, if i t < i + 1, u = t i u 2 + u + 1, if i + 1 t < i + 1, u = t (i + 1) (1 u)2, if i + 2 t < i + 3, u = t (i + 2) 1 2 u2, otherwise.

39 B-Spline Smoothing effect

40 Summary Spline segments Linear Quadratic Hermite Bezier Chaining splines Catmull Rom curve and B-splines Suggested reading: B1 Chapter 15