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Computer Graphics 2016 11. Complex 3D Hongxin Zhang State Key Lab of CAD&CG, Zhejiang University 2016-12-12

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 Graphics @ ZJU Hongxin Zhang, 2016

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

Bézier curve

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.

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.

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. 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.

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].

B-spline basis Computer Graphics @ ZJU Hongxin Zhang, 2016

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)

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

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 Graphics @ ZJU Hongxin Zhang, 2016

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

NURBS surface NURBS surface ω ij : weights

Sweeping

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

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 Graphics @ ZJU Hongxin Zhang, 2016

Polygonal Mesh

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 Graphics @ ZJU Hongxin Zhang, 2016

polygonal mesh representations Computer Graphics @ ZJU Hongxin Zhang, 2016

Computer Graphics @ ZJU Hongxin Zhang, 2016

Wavefront.obj file # List of Vertices, with (x,y,z[,w]) coordinates, w is optional and defaults to 1.0. v 0.123 0.234 0.345 1.0 v...... # Texture coordinates, in (u,v [,w]) coordinates, these will vary between 0 and 1, w is optional and default to 0. vt 0.500 1 [0] vt...... # Normals in (x,y,z) form; normals might not be unit.. vn 0.707 0.000 0.707 vn...... # Parameter space vertices in ( u [,v] [,w] ) form; free form geometry statement ( see below ) vp 0.310000 3.210000 2.100000 vp...... # Face Definitions (see below) f 1 2 3 f 3/1 4/2 5/3 f 6/4/1 3/5/3 7/6/5 f...... Computer Graphics @ ZJU Hongxin Zhang, 2016

Wavefront.obj file - Vertex positions - v, vt, vn - Face definitions - f v1 v2 v3 v4... - f v1/vt1 v2/vt2 v3/vt3... - f v1/vt1/vn1 v2/vt2/vn2 v3/vt3/vn3... - f v1//vn1 v2//vn2 v3//vn3... - Referencing materials - mtllib [external.mtl file name] - usemtl [material name] Computer Graphics @ ZJU Hongxin Zhang, 2016

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 1 -... - # Smooth shading can be disabled as well. - s off -... Computer Graphics @ ZJU Hongxin Zhang, 2016

Subdivision surfaces

Chaiken (1974)

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.

subdivision surface

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 Graphics @ ZJU Hongxin Zhang, 2016

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 Graphics @ ZJU Hongxin Zhang, 2016

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

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 Graphics @ ZJU Hongxin Zhang, 2016

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 Graphics @ ZJU Hongxin Zhang, 2016

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 Graphics @ ZJU Hongxin Zhang, 2016

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 Graphics @ ZJU Hongxin Zhang, 2016

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 Graphics @ ZJU Hongxin Zhang, 2016

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

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 Graphics @ ZJU Hongxin Zhang, 2016

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

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

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

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

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

subdivision surface Catmull-Clark subdivision surface:

subdivision surface Doo-Sabin surface

不不同细分曲面的比较

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

Geri s Game Computer Graphics @ ZJU Hongxin Zhang, 2016

subdivision + random terrain generation

terrain generation Computer Graphics @ ZJU Hongxin Zhang, 2016

More complex 3D

Boolean operation Boolean union Boolean difference Boolean intersection Constructive Solid Geometry

Constructive Solid Geometry

Self-similar Self-affine Invariant fractal set Fractal

Fractal Julia Set

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

L-system for tree

Sketch Tree Demo

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

粒子系统 Computer Graphics @ ZJU Hongxin Zhang, 2016