25- 场景图和变换

– Geometry & attributes – Transformations – Bounding volumes
Transformations
– – – – Basic 2D transformations Matrix representation Matrix composition 3D transformations
5
3D Object Representations
What object representation is best for this?
6
Scene Graphs
Hierarchy (DAG) of nodes, where each may have:
– – – – Geometry representation Modeling transformation Parents and/or children Bounding volume
49
Rotate around a general axis
绕任意轴旋转的五步：
1）平移物体使旋转轴通过坐标原点； 2）旋转物体使旋转轴与某一坐标轴重合； 3）绕坐标轴完成指定的旋转； 4）用逆旋转使旋转轴回到其原始方向； 5）用逆平移使旋转轴回到原来的位置。
y
y
y
u


x x x
z z
u
u
50
y
18
2D Modeling Transformations
19
2D Modeling Transformations
20
2D Modeling Transformations
21
2D Modeling Transformations
22
2D Modeling Transformations
Multiply matrix by column vector
– apply transformation to point
27
Matrix representation
Transformations combined by multiplication
Matrices are a convenient and efficient way to represent a sequence of transformations
Haverkort
10
Scene Graphs
Advantages
– – – – Allows definitions of objects in own coordinate systems Allows use of object definition multiple times in a scene Allows hierarchical processing (e.g., intersections) Allows articulated animation
Transformations
– – – – Basic 2D transformations Matrix representation Matrix composition 3D transformations
41
Matrix Composition
Transformations can be combined by matrix multiplication
54
Transformations in Scene Graphs
典型地，变换节点至少包含一个矩阵来处 理变换，另外节点也可能包含一些自己的 变换参数。为了得到一个对象节点的复合 变 换 矩 阵 （ Composite Transformation Matrix，CTM），需要上溯所有父节点。
55
Scene Graphs & Modeling Transformations
教师：吴壮志 北京航空航天大学计算机学院 2015年4月23日
3D Object Representations
Points
– Range image – Point cloud
Solids
– – – – Voxels BSP tree CSG Sweep
46
3D Transformations
Same idea as 2D transformations
– Homogeneous coordinates: (x,y,z,w) – 4x4 transformation matrices
47
Basic 3D Transformations
48
Basic 3D Transformations
13
2D Modeling Transformations
14
2D Modeling Transformations
15
2D Modeling Transformations
16
2D Modeling Transformations
17
2D Modeling Transformations
39
Cross-Ratio
Definition: for 4 collinear points A, B, C, D
Projective Invariant: (A,B;C,D) = (A’,B’;C’,D’)
40
Overview
Scene graphs
– Geometry & attributes – Transformations – Bounding volumes
4
Overview
Scene graphs
– Geometry & attributes – Transformations – Bounding volumes
Transformations
– – – – Basic 2D transformations Matrix representation Matrix composition 3D transformations
Scene Graphs
Advantages
– Allows definitions of objects in own coordinate systems – Allows use of object definition multiple times in a scene –
42
Matrix Composition
Matrices are a convenient and efficient way to represent a sequence of transformations
– General purpose representation – Hardware matrix multiply – Efficiency with premultiplication(预乘) • Matrix multiplication is associative(结合律)
23
2D Modeling Transformations
24
2D Modeling Transformations
25
Overview
Scene graphs
– Geometry & attributes – Transformations – Bounding volumes
Transformations
11
Scene Graph Example
12
Overview
Scene graphs
– Geometry & attributes – Transformations – Bounding volumes
Transformations
– – – – Basic 2D transformations Matrix representation Matrix composition 3D transformations
y
y
u


x x x
z z
u
u
51
Rotate around a general axis
另外一种方法完成第2)步
Ry ( ) Rx ( )
y
u
P2 v P1 w x
z
52
y
u P2 v P1
w x
z
53
Transformations in Scene Graphs
– 在下面的Scene Graph中，变换t0影响所有的对象，但是变换t2仅影响 obj2和group3的实例（包括对象obj3和obj4的实例） – Note that if you want to use multiple instances of a sub-tree, such as group3 above, you must define it before it’s used • this is so that it’s easier to implement
36
Affine Transformations
37
Projective Transformations
38
Projective Transformations
Will be useful to model (pinhole) cameras: can represent camera projection in same framework as modeling transformations
Surfaces
– – – – Polygonal mesh Subdivision Parametric Implicit
High-level structures
– Scene graph – Application specific