09 - Designing Surfaces. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo

Transcription

1 9 - Designing Surfaces

2 Triangular surfaces A surface can be discretized by a collection of points and triangles Each triangle is a subset of a plane Every point on the surface can be expressed as an affine combination of three vertices The surface is C

3 Barycentric coordinates The points inside a triangle, are usually parametrized using barycentric coordinates w: v 3 v x v2 x v3 x w px v y v2 y v3 y w2 = py p w3 v v 2 The barycentric coordinates naturally defines a parametrization

4 Tensor Surfaces

5 Bilinear Interpolation Linear interpolation Simplest curve between two points Isoparametric curve Bilinear interpolation Simplest surface between four points X X x(u, v) = b i,j B i (u)b j (v) i= = u u j= b b v b b v Domain

6 Bilinear Interpolation Linear interpolation Simplest curve between two points Bilinear interpolation Simplest surface between four points b =( v)b + vb b =( v)b + vb b =( u)b + ub

7 Bézier Patches Build on Bézier curves mx b b b 2 b 2 b 23 b m (u) = b i B m i (u) i= Control points as curves b 33 nx b 3 b i = b i (v) = j= b ij B n j (v) b mx nx b 3 b mn (u, v) = b ij B m i (u)b n j (v) i= j= Keep one parameter fixed: iso-parameter curves

8 Properties Affine invariance

9 Properties Affine invariance Repeated (bi-)linear combinations Convex hull property

10 Properties Affine invariance Repeated (bi-)linear combinations Convex hull property Partition of unity and non-negativity Polynomial boundary curves and corner interpolation

11 Properties Affine invariance Repeated (bi-)linear combinations Convex hull property Partition of unity and non-negativity Polynomial boundary curves and corner interpolation Variation diminishing

12 Properties Affine invariance Repeated (bi-)linear combinations Convex hull property Partition of unity and non-negativity Polynomial boundary curves and corner interpolation Variation diminishing

13 De Casteljau Algorithm Bézier curves created by repeated linear interpolation Surfaces: repeated bilinear interpolation

14 Demo

15 NURBS Surfaces x(u, v) = i j w i,jd i,j N m (u)n n i j (v) (u)n n j (v) i j w i,jn m i Standard in most advanced modeling systems projection of tensor product patches tensor product surface! (basis is not separable)

16 NURBS Surfaces Standard in most advanced modeling systems

17 NURBS Surfaces Influence of weights

18 NURBS Surfaces Influence of weights

19 NURBS Surfaces Influence of weights

20 NURBS Surfaces Influence of weights

21 Subdivision Surfaces

22 Why? Tensor product surfaces are defined on regular grids They cannot be defined on arbitrary meshes

23 A different way of constructing curves!

24 Interpolating: 4 point scheme

25 Interpolating: 4 point scheme

26 Interpolating: 4 point scheme

27 Interpolating: 4 point scheme

28 Catmull-Clark scheme odd even Converges to cubic B-spline

29 Subdivision Methods Principal characteristics: triangular or quadrangular meshes approximating or interpolating smoothness of the limit surface We will study 2 of them: Loop subdivision for triangle meshes (Approximating, C 2 ) Catmull-Clark subdivision for quadrilateral meshes (Approximating, C 2 ) Other famous schemes (see the references for details) Butterfly, Kobbelt, Doo-Sabin, Midedge, Biquartic

30 Loop subdivision w(n) = 5 8 ( cos2 n )2

31 Special stencils Special stencils exist that allow to evaluate: The limit position of a vertex: that is, in a single step you can compute the position of one vertex after an infinite number of subdivisions The tangent plane at a vertex: in this case, two stencils are used and each stencil generates a vector that lies in the tangent plane It is also possible to evaluate the surface analytically, in other words it is possible to find the mapping between every point of the control mesh and the limit surface. For details see

32 Catmull Clark subdivision = 3 2N = 4N On a regular grid, the Catmull clark subdivision is a collection of bicubic Bézier patches

33 Catmull-Clark It is the standard in the animation industry All the major 3D modeling softwares support it Similarly to Loop: stencils for the limit surface stencils for tangent plane it can be evaluated analytically

34 How are they used in practice?

35 Efficient implementation At every subdivision step, we add new vertices and move the existing ones in new positions Every position is computed as a weighted average of existing vertices This means that the process is linear! For a fixed number of levels of subdivision, the vertices of subdivided surface can be computed as: Subdivided vertices p = Sq Control vertices where S is a sparse and fixed matrix

36 T-spline Extension of splines for non-rectangular grids

37 References Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley Chapter 5 Curves and Surfaces for CAGD - Gerald Farin Subdivision Zoo - Denis Zorin

38 Projective Transformations

39 Viewing Transformation object space camera space screen space model camera projection viewport world space canonical view volume

40 Orthographic Projection camera space y (r,t,f) z (l,b,n) x projection M orth = r+l r l r l 2 t+b t b t b 2 n f n+f n f canonical view volume

41 Perspective Projection In Orthographic projection, the size of the objects does not change with distance In Perspective projection, the objects that are far away look smaller Image Plane Image Plane y s y y s = dy z d z

42 Divisions in Matrix Form We would like to reuse the matrix machinery that we built in the previous lectures How do we encode divisions? y s = dy z Image Plane We extend homogeneous coordinates y s y d z

43 Until now What do we have a b c a 2 b 2 c 2 A ya a x + b y + c a 2 x + b 2 y + c 2 A We can use the last row of the a b c a 2 b 2 c 2 e f g A ya a x + b y + c a 2 x + b 2 y + c 2 ex + fy + g A a x+b y+c ex+fy+g a 2 x+b 2 y+c 2 ex+fy+g C A

44 Intuition Purely x y A y/wa w Or as a projection, where each line is identified by a point on the plane z= Note that in this case, you can think of it as a transformation in a space with one more dimension

45 Projective Transformation A transformation of this form is called a projective transformation (or a homography) The points are represented in homogeneous a b c a 2 b 2 c 2 e f g A ya a x + b y + c a 2 x + b 2 y + c 2 ex + fy + g A a x+b y+c ex+fy+g a 2 x+b 2 y+c 2 ex+fy+g C A

46 Example, M 2 3 A 2/3 /3,,3 It transforms a square into a quadrilateral note that straight lines are preserved, but parallel lines are not! Note that you can use homogeneous coordinates for as many transformations as you want, only when you need the cartesian representation you have to normalize, 3,

47 Perspective Projection Perspective projection is easily implementable using this machinery y s = d z Image Plane y d y s ys A y z z

48 ` We will use the same conventions that we used for orthographic: Camera at the origin, pointing negative z We scale x, y and bring along the z z y (l,b,n) (r,t,f) P = n B n n + f fna x

49 Effect on the points z y (l,b,n) (r,t,f) P = n B n n + f fna x P x By za = nx B ny + f)z fna nx z ny B n + f fn z C A z

50 Effect on the points z y (l,b,n) (r,t,f) P = n B n n + f fna x P x By za = nx B ny + f)z fna nx z ny B n + f fn z C A z

51 Orthographic Projection camera space y (r,t,f) z (l,b,n) x projection M orth = r+l r l r l 2 t+b t b t b 2 n f n+f n f canonical view volume

52 Complete Perspective Transformation P = n B n n + f fna M orth = r+l r l r l 2 t+b t b t b n+f n f 2 n f canonical camera space view volume P M orth

53 Parameters? How to set the parameters of the transformation? If we look at the center of the center of the window then the barycenter of the front back should be at (,,f) If we want no distortion on the image we need to keep a fixed aspect ratio: z y x (l,b,n) (r,t,f) width/height = r/t (width and height are the size in pixels of the final image) There is only one degree of freedom left, the field of view angle : tan 2 = t n The parameters can thus by found by fixing n and. You can then compute t an consequently all the other parameters needed to construct the transformation

54 References Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley Chapter 7