Computergrafik. Matthias Zwicker. Herbst 2010

Transcription

1 Computergrafik Matthias Zwicker Universität Bern Herbst 2010

2 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling

3 Piecewise Bézier curves Each segment spans four control points Each segment contains four Bernstein polynomials Each control point belongs to one Bernstein polynomial x 2 (t) u=0 x 0 (t) x 1 (t) x 3 (t) u=12 Bernstein polynomials 0 4 Segment u

4 B-splines Same idea, but different polynomial blending functions UniformB-splines have only one type of blending function: B-spline (basis) function B-spline function of degree n is C n-1 continuous Local support, at each point u exactly n+1 functions are non-zero Uniform B-spline (basis) functions of degree 3 u

5 B-splines Weighted average of control points p i using B- spline functions b i i( (u) (no details here) Positive, partition of unity => convex hull property Matrix form (note different basis matrix; caution: last lecture, matrices were transposed) p i p i p i p p i 3 T 1 x(u) t 3 t 2 t 1 6 B B spline G B spline where t u i and i u

6 B-splines Widely used for curve and surface modeling Advantages over Bézier curves Built-in continuity Local support: curve only affected by nearby control points Intuitive behavior Techniques for inserting new points, joining curves, etc. Doesn t interpolate endpoints Not a problem for closed curves p Can be generalized to interpolate any point along the curve (non-uniform curves, later)

7 Further generalizations Rational B-splines Non-uniform B-splines Non-uniform rational B-splines (NURBS) Interactive explanation Interactive explanation

8 Rational curves Big drawback of all polynomial curves Can t make circles, ellipses, nor arcs, nor conic sections Rational B-spline Add a weight to each control point Homogeneous point: use w Polynomial curve (b-spline, Bézier) Rational curve Not polynomial l any more!

9 Rational curves Weight causes point to pull more (or less) With proper points & weights, can do circles Polynomial curve Rational curve pull less pull more

10 Rational curves Can generate curves for circles etc. with appropriate weights Need extra user interface to adjust the weights Often, hand-drawn curves are unweighted

11 NURBS Non-Uniform Rational B-splines Nonuniform: Introduce knot vector Knot vector provides additional degrees of freedom to define B-spline basis (blending) functions

12 NURBS Knot vector is a vector of locations on the u axis, usually called t i B-spline function of degree n uses n+2 knots (Uniform) B-splines use a uniform knot vector Nonuniform B-splines use an arbitrary knot vector Knot vector determines shape of basis functions knots Uniform knot vector u Nonuniform knot vector u

13 Nonuniform B-spline bases Construction Recursive Generate higher order bases step by step from lower order bases Can prove Partition of unity (i.e., convex hull property) Built-in continuity

14 Nonuniform B-spline bases Nonuniform knot vector Nonuniform, linear B-spline bases Linear weighting function u u Multiply & add Quadratic B-spline basis u

15 For your reference... Recursive definition of non-uniform B-spline basis functions b basis functions b j,n Function b j,n has degree n Knot vector {t j } Note: notation here uses t instead of u

16 Knot vectors Uniform B-splines have knot vector t j =j Cubic Bézier curves {t j }= [ ] [0,0,0,0,1,1,1,1] Can make corners (C 1 discontinuity) Allows mixing interpolating (e.g. at endpoints) and approximating 4 coinciding knots Bézier curve as B-spline with nonuniform knot vector u

17 NURBS Math is more complicated Very widely used for curve and surface modeling Supported by virtually all 3D modeling tools Open source modeling tool: Techniques for cutting, inserting, merging, g, revolving, etc Applets en html

18 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling

19 Curved surfaces Curves Described by a 1D series of control points A function x(t) Segments joined together to form a longer curve Surfaces Described by a 2D mesh of control points Parameters have two dimensions (two dimensional parameter domain) A function x(u,v) Patches joined together to form a bigger surface

20 Parametric surface patch x(u,v) describes a point in space for any given (u,v) pair u,v each range from 0 to 1 v x(0.8,0.7) z y u v 1 x 0 1 u 2D parameter domain

21 Parametric surface patch x(u,v) describes a point in space for any given (u,v) pair u,v each range from 0 to 1 v x(0.8,0.7) x(0.4,v) 4 z y u x(u,0.25) v 1 Parametric curves x 0 1 u 2D parameter domain For fixed u 0, have a v curve x(u 0,v) For fixed v 0, have a u curve x(u,v 0 ) For any point on the surface, there is one pair of parametric curves that go through point

22 Tangents The tangent to a parametric curve is also tangent to the surface For any point on the surface, there are a pair of (parametric) tangent vectors Note: not necessarily perpendicular to each other v x v x u u

23 Tangents Notation Tangent along u direction x (u, v) u u x(u, v) x u(u, v) or or Tangent along v direction x (u, v) v or or v x(u, v) x v(u, v) Tangents are vector valued functions, i.e., vectors!

24 Surface normal Cross product of the two tangent vectors x u (u, v) x v (u, v) Order matters (determines normal orientation) Usually, want unit normal Need to normalize by dividing through length n x v x u

25 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling

26 Bilinear patch Control mesh with four points p 0, p 1, p 2, p 3 Compute x(u,v) using a two-step construction p 2 p 3 v p 0 p1 u

27 Bilinear patch (step 1) For a given value of u, evaluate the linear curves on the two u-direction edges Use the same value u for both: q 1 =Lerp(u,p 2,p 3 ) q 1 p 2 p 3 v p 0 q 0 p1 u q 0 =Lerp(u,p 0,p 1 )

28 Bilinear patch (step 2) Consider that q 0, q 1 define a line segment Evaluate it using v to get x x Lerp(v,q 0,q 1 ) q 1 p 2 p 3 v x p 0 q 0 p1 u

29 Bilinear patch Combining the steps, we get the full formula x(u,v) Lerp(v, Lerp(u,p 0,p 1 ), Lerp(u,p 2,p 3 )) q 1 p 2 p 3 v x p 0 q 0 p1 u

30 Bilinear patch Try the other order Evaluate first in the v direction r 0 Lerp(v,p 0,p 2 ) r 1 Lerp(v,p 1,p 3 ) p 2 p 3 r 0 r v 1 p 0 p1 u

31 Bilinear patch Consider that r 0, r 1 define a line segment Evaluate it using u to get x x Lerp(u,r( 0,r 1 ) p 2 p 3 r x 0 v r 1 p 0 p1 u

32 Bilinear patch The full formula for the v direction first: x(u,v) ( ) Lerp(u, Lerp(v,p p 0,pp 2 ), Lerp(v,p p 1,pp 3 )) p 2 p 3 r x 0 v r 1 p 0 p1 u

33 Bilinear patch It works out the same either way! x(u,v) Lerp(v, Lerp(u,p 0,p 1 ), Lerp(u,p 2,p 3 )) x(u,v) Lerp(u, Lerp(v,p 0,p 2 ), Lerp(v,p 1,p 3 )) q 1 p 2 p 3 r x 0 v r 1 p 0 q 0 p1 u

34 Bilinear patch Visualization

35 Bilinear patches Weighted sum of control points Bilinear polynomial Matrix form exists, too

36 Properties Interpolates the control points The boundaries are straight line segments If all 4 points of the control mesh are co-planar, the patch is flat If the points are not coplanar, get a curved surface saddle shape, AKA hyperbolic paraboloid The parametric curves are all straight line segments! a (doubly) ruled surface: has (two) straight lines through every point Not terribly useful as a modeling primitive

37 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling

38 Bicubic Bézier patch Grid of 4x4 control points, p 0 through p 15 Four rows of control points define Bézier curves along u p 0,p 1,p 2,p 3 ; p 4,p 5,p 6,p 7 ; p 8,p 9,p 10,p 11 ; p 12,p 13,p 14,p 15 Four columns define Bézier curves along v p 0,p 4,p 8,p 12 ; p 1,p 6,p 9,p 13 ; p 2,p 6,p 10,p 14 ; p 3,p 7,p 11,p 15 p 12 p13 v p 8 p 9 p 4 p 5 p 14 p15 p 0 p 1 p 10 p 11 p 6 p 7 u p 2 p 7 p 3

39 Bicubic Bézier patch (step 1) Evaluate four u-direction Bézier curves at u Get points q 0 q 3 q 0 Bez(u,p 0,p 1,p 2,p 3 ) q 1 Bez(u,p 4,p 5,p 6,p 7 ) q 2 Bez(u,p( 8,p 9,p 10,p 11 ) q 3 Bez(u,p 12,p 13,p 14,p 15 ) p 12 p13 q 3 v p 8 p 9 p 4 p 5 p 0 p 1 q 1 q 2 p 14 p 10 p 11 p15 u q 0 p 2 p 6 p 7 p 3

40 Bicubic Bézier patch (step 2) Points q 0 q 3 define a Bézier curve Evaluate it at v x(u,v) Bez(v,q 0,q 1,q 2,q 3 ) p 12 p13 q 3 v p 8 p 9 p 4 p 5 p 0 p 1 q 1 q 2 x p 14 p 10 p 11 p15 u q 0 p 2 p 6 p 7 p 3

41 Bicubic Bézier patch Same result in either order (evaluate u before v or vice versa) q 0 Bez(u,p 0,p 1,p 2,p 3 ) q 1 Bez(u,p 4,p 5,p 6,p 7 ) r 0 Bez(v,p 0,p 4,p 8,p 12 ) r 1 Bez(v,p 1,p 5,p 9,p 13 ) q 2 Bez(u,pp 8,pp 9,pp 10,pp 11 ) r 2 Bez(v,pp 2,pp 6,pp 10,pp 14 ) q 3 Bez(u,p 12,p 13,p 14,p 15 ) x(u,v) Bez(v,q 0,q 1,q 2,q 3 ) r 3 Bez(v,p 3,p 7,p 11,p 15 ) x(u,v) Bez(u,r 0,r 1,r 2,r 3 ) v p 12 r p13 0 r 3 1 p 8 p 9 xq p 4 p 5 p 0 p 1 q 1 q 2 p 14 p15 r 2 r 3 p 10 p 11 u q 0 p 2 p 6 p 7 p 3

42 Tensor product formulation Corresponds to weighted average formulation Construct two-dimensional weighting g function as product of two one-dimensional functions Bernstein polynomials B i, B j as for curves j Same tensor product construction applies to higher h order Bézier and NURBS surfaces

43 Bicubic Bézier patch: properties Convex hull: any point on the surface will fall within the convex hull of the control points Interpolates 4 corner points Approximates other 12 points, which act as handles The boundaries of the patch are the Bézier curves defined by the points on the mesh edges The parametric curves are all Bézier curves

44 Tangents of Bézier patch Remember parametric curves x(u,v 0 ), x(u 0,v) where v 0, u 0 is fixed Tangents to surface = tangents to parametric curves Tangents are partial derivatives of x(u,v) Normal is cross product of the tangents x p p v v v 0 p 8 p 9 x p 14 p 4 p 5 x p15 p 0 p 1 u u 0 p 2 p 6 p 10 p 7 u p 11 p 3

45 Tangents of Bézier patch x v q 0 Bez(u,p 0,p 1,p 2,p 3 ) q 1 Bez(u,pp 4,pp 5,pp 6,pp 7 ) q 2 Bez(u,p 8,p 9,p 10,p 11 ) q 3 Bez(u,p 12,p 13,p 14,p 15 ) (u,v) Be z (v,q 0,q 1,q 2,q 3 ) x u r 0 Bez(v,p 0,p 4,p 8,p 12 ) r 1 Bez(v,pp 1,pp 5,pp 9,pp 13 ) r 2 Bez(v,p 2,p 6,p 10,p 14 ) r 3 Bez(v,p 3,p 7,p 11,p 15 ) (u,v) Be z (u,r 0,r 1,r 2,r 3 ) v p 13 p 12 r 0 r 1 q 3 p 8 p 9 x p 14 p 4 x p 5 p15 q 2 u q r 2 r p p 1 u q 0 p 2 x v p 6 p 10 p 7 p 11 p 3

46 Tessellating a Bézier patch Uniform tessellation is most straightforward Evaluate points on a grid of u, v coordinates Compute tangents at each point, take cross product to get pervertex normal Draw triangle strips (several choices of direction) Adaptive tessellation/recursive ti i subdivision i i Potential for cracks if patches on opposite sides of an edge divide differently Tricky to get right, but can be done

47 Piecewise Bézier surface Lay out grid of adjacent meshes of control points For C 0 continuity, must share points on the edge Each edge of a Bézier patch is a Bézier curve based only on the edge mesh points So if adjacent meshes share edge points, the patches will line up exactly But we have a crease Grid of control points Piecewise Bézier surface

48 C 1 continuity Want parametric curves that cross each edge to have C 1 continuity Handles must be equal-and-opposite across edge C 0 continuous C 1 continuous [

49 Modeling with Bézier patches Original Utah teapot specified as Bézier Patches

50 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling

51 Advanced surface modeling B-spline/NURBS patches instead of Bézier For the same reason as using B-spline/NURBS curves More flexible (can model spheres) Better mathematical properties, continuity Higher order NURBS patch

52 Modeling headaches Original Teapot is not watertight Spout & handle intersect with body No bottom Hole in spout Gap between lid and body

53 Modeling headaches NURBS surfaces are flexible Conic sections Can blend, merge, trim but Any surface will be made of quadrilateral patches (quadrilateral topology) Because of tensor product formulation p Grid of horizontal and vertical curves

54 Quadrilateral topology Makes it hard to join or abut curved pieces build surfaces with awkward topology or build surfaces with awkward topology or structure

55 Advanced surface modeling Trim curves Cut away part of surface Implement as part of tessellation/rendering Still hard to fit different parts together

56 Subdivision surfaces Goal Create smooth surfaces from small number of control points, like splines More flexibility for the topology of the control points (not restricted to quadrilateral grid) Idea Start with initial coarse polygon mesh Create smooth surface recursively by 1. Splitting (subdividing) mesh into finer polygons 2. Smoothing the vertices of the polygons 3. Repeat from 1.

57 Subdivision surfaces Input mesh Subdivision Subdivision Subdivision & smoothing & smoothing & smoothing Limit surface

58 Subdivision schemes Various schemes available to subdivide and smooth Doo-Sabin Loop All provide certain guarantees for smoothness of limit surface Demo

59 Subdivision surfaces Arbitrary mesh of control points Arbitrary topology or connectivity Not restricted to quadrilateral topology No global u,vparameters Work by recursively subdividing mesh faces Used in particular for character animation One surface rather than collection of patches Can deform geometry without creating cracks Subdivision surfaces

60 Next time More shaders