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1 Computer Graphics Complex 3D Hongxin Zhang State Key Lab of CAD&CG, Zhejiang University

2 General spline curves parametric curve basis functions P(t) = X i P i B i (t) t 2 [t 0,t 1 ) control pints Computer ZJU Hongxin Zhang, 2016

3 Bézier curve Bézier curve where, P i (i=0,1,,n) are control points. Bernstein basis

4 Bézier curve

5 Bézier curve Rational Bézier Curve where B i,n (t) is Bernstein basis, ω i is the weight at p i. It s a generalization of Bézier curve, which can express more curves, such as circle.

6 Bézier surface Bézier surface Bézier surface: where B i,n (u) 和 B j,m (v) Bernstein basis with n degree and m degree, respectively, (n+1) (m+1) P i,j (i=0,1,,n; j=0,1,,m) construct the control meshes.

7 Bézier surface normal vector of Bézier surface partial derivation of Bézier surface S(u,v): normal N(u,v) :

NURBS curve B-spline curve disadvantages of Bézier curve: 1. control points determine the degree of the curve. many control points means high degree.

8 NURBS curve B-spline curve disadvantages of Bézier curve: 1. control points determine the degree of the curve. many control points means high degree. 2. It s global. A control point influences the whole curve. de Boor et al. replaced Bernstein basis with B-spline basis to generate B-spline curve.

9 NURBS curve B-spline curve: Where P 0, P 1,,P n are control points, u=[u 0 =a, u 1,, u i,, u n+k+1 =b].

10 B-spline basis Computer ZJU Hongxin Zhang, 2016

11 NURBS curve B-spline---de Boor algorithm to calculate the point of B-spline curve C(u) at u: 1. find the interval where u lies in : u [u j,u j+1 ); 2. curve in u [u j,u j+1 ) is only determined by P j-p, P j-p+1,, P j ; 3. calculate 4. = C(u)

12 NURBS curve Catmull-Clark and Doo-Sabin subdivision Start from Catmull-Clark rules Doo-Sabin rules:

13 B-spline surface (n+1) (m+1) control points: P i,j (Degrees of u, v: p, q); nodes: U=[u 0,u 1,,u n+p+1 ], V = [v 0,v 1,, v m+q+1 ], Then a tensor B-spline surface with degree p q : Computer ZJU Hongxin Zhang, 2016

14 NURBS surface NURBS (Non-uniform Rational B-spline) NURBS curves: 重节点

15 NURBS surface NURBS surface ω ij : weights

16 Sweeping

17 More complex objects? Challenge: surface continuity? Computer ZJU Hongxin Zhang, 2016

18 Homework 4 - Write a report on car shape design - In A4 paper, at least 2 pages - Major solution, processing pipeline - Key techniques and standard - bonus: how to render the shape Computer ZJU Hongxin Zhang, 2016

19 Polygonal Mesh

20 What is polygonal mesh? - A polygon mesh is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling Computer ZJU Hongxin Zhang, 2016

21 What is polygonal mesh? - A polygon mesh is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling Computer ZJU Hongxin Zhang, 2016

22 polygonal mesh representations Computer ZJU Hongxin Zhang, 2016

23 polygonal mesh representations Computer ZJU Hongxin Zhang, 2016

24 Computer ZJU Hongxin Zhang, 2016

25 Wavefront.obj file # List of Vertices, with (x,y,z[,w]) coordinates, w is optional and defaults to 1.0. v v # Texture coordinates, in (u,v [,w]) coordinates, these will vary between 0 and 1, w is optional and default to 0. vt [0] vt # Normals in (x,y,z) form; normals might not be unit.. vn vn # Parameter space vertices in ( u [,v] [,w] ) form; free form geometry statement ( see below ) vp vp # Face Definitions (see below) f f 3/1 4/2 5/3 f 6/4/1 3/5/3 7/6/5 f Computer ZJU Hongxin Zhang, 2016

26 Wavefront.obj file - Vertex positions - v, vt, vn - Face definitions - f v1 v2 v3 v f v1/vt1 v2/vt2 v3/vt f v1/vt1/vn1 v2/vt2/vn2 v3/vt3/vn f v1//vn1 v2//vn2 v3//vn Referencing materials - mtllib [external.mtl file name] - usemtl [material name] Computer ZJU Hongxin Zhang, 2016

27 Wavefront.obj file - Named objects and polygon groups are specified via the following tags. - o [object name] - g [group name] - Smooth shading across polygons is enabled by smoothing groups. - s # Smooth shading can be disabled as well. - s off -... Computer ZJU Hongxin Zhang, 2016

28 Subdivision surfaces

29 Chaiken (1974)

31 subdivision surface subdivision curves: starting from a set of points, generate new points in every step under some rules, when such step goes on infinitely, the points will be convergent to a smooth curve.

32 subdivision surface

33 细分曲面的基本概念是一种过程式 ( 迭代 ) 的曲面构造方法输入 : 多边形网格 => 控制网格输出 :a. 加密网格 => 有限次迭代 [ 用于绘制 ] b. 连续曲面 => 极限曲面 [ 用于分析 ] 离散连续 Computer ZJU Hongxin Zhang, 2016

34 Classical subdivision schemes Catmull-Clark surface. [Catmull 1978] Doo-Sabin surface. [Doo 1978] Loop s subdivision schemes. [Loop 1987] Butterfly subdivision schemes. [Dyn 1990] 离散连续 Computer ZJU Hongxin Zhang, 2016

35 advantage of subdivision surface - topologically complex shape - stable; - easy to implement; - no need to merge between the surfaces - LOD Computer ZJU Hongxin Zhang, 2016

36 细分曲面的基本步骤细分操作 : 拓拓扑规则 : 加密采样, 重新构造网格几何规则 : 光顺网格 Computer ZJU Hongxin Zhang, 2016

37 Loop subdivision surface 拓拓扑规则 : 怎样加密三角网格? 对分每条边, 并连接新顶点将每个三角形剖分成四个更更小的三角形 Computer ZJU Hongxin Zhang, 2016

38 Loop subdivision surface 几何规则 : 怎样放置新顶点的位置? 利利用原始网格中与新顶点相邻的顶点来做加权平均如果顶点的连接度不不是 6 该怎么办? Computer ZJU Hongxin Zhang, 2016

39 Loop subdivision surface 几何规则 : 怎样放置新顶点的位置? 奇异顶点处的规则 Computer ZJU Hongxin Zhang, 2016

40 Loop subdivision surface 怎样选择系数 Beta? 分析极限曲面性质与曲面的连续性和光滑性相关需要计算相关 ( 细分系数 ) 矩阵的特征结构 Computer ZJU Hongxin Zhang, 2016

41 Loop subdivision surface 细分结果可达到较好的连续性光顺性! Computer ZJU Hongxin Zhang, 2016

42 Subdivision schemes 各种不不同的细分方法不不同的处理理拓拓扑加密方法不不同的布置顶点位置方法逼近 v.s. 插值 Computer ZJU Hongxin Zhang, 2016

43 细分曲面的基本思想向上采样割角平均回插 - 补偿 Computer ZJU Hongxin Zhang, 2016

44 四边网格上的拓拓扑规则 Catmull-Clark 型 Primal Dual Doo-Sabin 型 10/09/11 Math Dept. ZJU. 13

45 Doo-Sabin 型细分曲面 10/09/11 Math Dept. ZJU. 14

46 subdivision surface Catmull-Clark and Doo-Sabin subdivision start from Catmull-Clark rules Doo-Sabin rules:

47 Catmull-Clark subdivision 面点模板边点模板顶点模板 10/09/11 Math Dept. ZJU. 15

48 subdivision surface Catmull-Clark subdivision surface:

49 subdivision surface Doo-Sabin surface

50 不不同细分曲面的比较

51 不不同细分曲面的比较

52 参考资料料普林林斯顿大学的课件半静态回插细分 From Pixar, Geri s Game

53 Geri s Game Computer ZJU Hongxin Zhang, 2016

54 subdivision + random terrain generation

55 terrain generation Computer ZJU Hongxin Zhang, 2016

56 More complex 3D

57 Boolean operation Boolean union Boolean difference Boolean intersection Constructive Solid Geometry

58 Constructive Solid Geometry

59 Self-similar Self-affine Invariant fractal set Fractal

60 Fractal Julia Set

61 字符 : F 常数 : +, 公理 : F++F++F 规则 : F F F++F F F : 向前 - : 左转 60 + : 右转 60 L-system

62 L-system for tree

63 Sketch Tree Demo

64 procedure (script) modeling A lot of research Natural scene modeling City modeling (shape grammar) Amazing software nodebox v.s. processing

65 粒子系统 Computer ZJU Hongxin Zhang, 2016

17 Prelight Apply Color Paint Vertex Color Tool Prelight Apply Color Paint Vertex Color Tool 242 Apply Color, Prelight Maya Shading Smooth

17 Prelight Apply Color Paint Vertex Color Tool Prelight Apply Color Paint Vertex Color Tool 242 Apply Color, Prelight Maya Shading Smooth 17 Prelight 233 234 242 Apply Color Paint Vertex Color Tool Prelight Apply Color Paint Vertex Color Tool 242 Apply Color, Prelight Maya Shading Smooth Shade All Custom Polygon DisplayOptions Color in Shaded