Review of Thursday. xa + yb xc + yd

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1 Review of Thursday Vectors and points are two/three-dimensional objects Operations on vectors (dot product, cross product, ) Matrices are linear transformations of vectors a c b x d y = xa + yb xc + yd Objects are transformed by applying transformation matrix to its vertices (coordinate transformation) 29

2 Review of Thursday Objects are transformed by applying transformation matrix to its vertices (coordinate transformation) anisotropic scaling = = = det(m) =6 30

3 A Brief Review of Linear Algebra Rotation cos α sin α sin α cos α counterclockwise (positive direction of rotation) Determinant: Inverse cos α cos α sin α ( sin α) =1 M 1 = M T 31

4 A Brief Review of Linear Algebra Constructing a rotation matrix M(1, 0) T = (cos α, sin α) T M(0, 1) T =( sin α, cos α) T (a, b) T = a(1, 0) T + b(0, 1) T M(a, b) T = am(1, 0) T + bm(0, 1) T (linearity of M) M(a, b) T =(acos α b sin α, a sin α + b cos α) T cos α sin α sin α cos α columns correspond to axes of new coordinate frame 32

5 A Brief Review of Linear Algebra Shear transformation (shearing) displacement along direction proportional to (signed) distance to a line Reflection

6 A Brief Review of Linear Algebra Applying multiple transformations in sequence M scal M rot v = v (M scal M rot ) v = v Order of operations is important! 34

7 Notes on Matrices in OpenGL OpenGL <3.0 includes matrix operations and stack glmatrixmode( ) glloadmatrix( ) glmultmatrix( ) glpushmatrix() //push current matrix to top of stack glpopmatrix() //take matrix from top of stack Assignment 2: You may use glloadmatrix, glmultmatrix or pass the matrices to the vertex shader as uniform variables (do not use gltranslate, glrotate). Matrix operations and stack deprecated in OpenGL 3.0+ Implement own transformation matrices Pass to vertex shader (OpenGL/GLSL: uniform variables) 35

8 Notes on Matrices in OpenGL OpenGL <3.0 vertex shader example void main(void) { gl_position = gl_projectionmatrix * gl_modelviewmatrix * gl_vertex; } OpenGL 3.0+ vertex shader example attribute vec3 in_position; //vertex attribute is part of VBO/vertex array uniform mat4 m_modelview; //custom uniform variable for modelview matrix uniform mat4 m_projection; //custom uniform variable for projection matrix void main(void) { gl_position = m_projection * m_modelview * vec4(in_position,1.0); } 36

9 Homogeneous Coordinates Problems so far: No translation No explicit notion of local and global coordinates No distinction between points and vectors Homogeneous coordinates solve these problems. 37

10 Homogeneous Coordinates Distinguishing points and vectors In world frame vectors are defined by a local coordinate frame. v = α 1 v 1 + α 2 v 2 + α 3 v 3 In world frame points are defined by a local coordinate frame and origin. P = α 1 v 1 + α 2 v 2 + α 3 v 3 + P 0 38

11 Homogeneous Coordinates Distinguishing points and vectors Local coordinate system as rows in matrix v = α 1 v 1 + α 2 v 2 + α 3 v 3 P = α 1 v 1 + α 2 v 2 + α 3 v 3 + P 0 v 1 v 2 v 3 P 0 v =(α 1,α 2,α 3, 0) P =(α 1,α 2,α 3, 1) v 1 v 2 v 3 P 0 v 1 v 2 v 3 P 0 39

12 Transformation in Homogeneous Coordinates Affine transformation now represented by 4x4 matrix A = α 11 α 12 α 13 α 14 α 21 α 22 α 23 α 24 α 31 α 32 α 33 α A transforms points p = x y and vectors z v = 1 x y z 0 40

13 Translation P = x y z v = 1 P = P = P + v v x v y v z 0 P = v x v y v z P translation matrix T x y z 1 = x + v x y + v y z + v z 1 41

14 Transformation in Homogeneous Coordinates Example coordinate transformation (scaling and translation) p 1 v world = T (p)s(0.5, 0.5, 0.5) v local = p p 3 v local

15 Rotation in 3D R z (α) = R x (α) = R y (α) = cos(α) sin(α) 0 0 sin(α) cos(α) cos(α) sin(α) 0 0 sin(α) cos(α) cos(α) 0 sin(α) sin(α) 0 cos(α) upper left sub-matrix identical to 2D matrix inversion: R(α) 1 = R( α) cos( α) = cos(α) sin( α) = sin(α) R(α) 1 = R(α) T 43

16 Rotation: Fixed Point Rotation always performed around origin Changing the fixed point requires translations: v world = T (p)r z (α)t ( p) v local v world = cos(α) sin(α) 0 p 1 p 1 cos(α)+p 2 sin(α) sin(α) cos(α) 0 p 2 p 1 sin(α) p 2 cos(α) v local

17 Arbitrary Rotation Arbitrary rotations can be expressed as successive rotations around coordinate axes R = R x (α)r y (β)r z (γ) Decomposition is not unique R z (90 )R y (90 )=R y (90 )R x ( 90 ) 45

18 Rotation Around Arbitrary Axis Rotation around arbitrary axis v by α : Move fixed point (e.g., center of object) to origin: T ( p) Rotate axis of rotation, such that it aligns with z-axis: R y (β 2 )R x (β 1 ) α Rotate around z-axis by : R z (α) Undo alignment rotation: R x ( β 1 )R y ( β 2 ) Undo translation: T (p) A = T (p)r x ( β 1 )R y ( β 2 )R z (α)r y (β 2 )R x (β 1 )T ( p) Only need to find β 1,β 2 46

19 Rotation Around Arbitrary Axis A sequence of two rotations can align any vector with the z-axis 47

20 Rotation Around Arbitrary Axis β 1,β 2 do not need to be computed explicitly d = v 2 y + v 2 z R x (β 1 )= R y (β 2 )= v z /d v y /d 0 0 v y /d v z /d d 0 v x v x 0 d

21 Homogeneous Coordinates How to interpret a transformation matrix v world = p p p 3 v local Local coordinate frame (source) expressed in global coordinates (target). Inverse matrix reverses transformation. 49

22 Review of Tuesday Transformation matrix in homogeneous coordinates v world = p p p 3 v local Local coordinate frame (source) expressed in global coordinates (target). Transforms points x x p = y z v = y and vectors z

23 Review of Tuesday Origin of current coordinate system is fixed point Identical transformation matrix has different effects based on relative position of the object and coordinate origin. 51

24 Review of Tuesday More complex operations can be performed by concatenating transformations. For example: v world = T (p)r z (α)t ( p) v local Rotation around arbitrary point (2D) A = T (p)r x ( β 1 )R y ( β 2 )R z (α)r y (β 2 )R x (β 1 )T ( p) Rotation around arbitrary axis (3D) 52

25 Sequences of Transformations How do we interpret a sequence of transformation matrices? We can read transformations from right to left or from left to right Reading from right to left: Interpret operations as being performed in (fixed) world coordinate system. Reading from left to right: Interpret operations as being performed in (dynamic) object coordinate system. 53

26 Object Transformation The vertex processor transforms vertices by applying transformation of current transformation matrix settransformationmatrix(matrix1) renderobject(object1) //matrix1 is active settransformationmatrix(matrix2) renderobject(object2) //matrix2 is active Pseudo code Caution when using OpenGL matrix operations (OpenGL <3.0): OpenGL does matrix post-multiplication as opposed to pre-multiplication. 54

27 Coordinate Transformations - Summary Homogeneous coordinates allow us to Incorporate local coordinate system origins (translation) Distinguish points and vectors Do a full transform between coordinate systems Important notions Points are different from vectors (cf. vertices and normals) Order of transformations matters Rotation and translation are rigid-body transformations Programming: be aware of row-major vs. column major matrices 55

28 Chapter 4 - Summary Object geometry (vertices and connectivity) is passed to the graphics pipeline together with transformation operations Transformation operations are represented by matrices Objects are transformed by applying the transformation to each of its vertices (vertex processor does this in parallel) The transformed objects need to be projected into a 2D coordinate system and mapped to the screen ( Where do objects end up on screen? - next chapter) 56