1. Math for CSTutorial 11 Course Outline Homogeneous Coordinates

2. Math for CSTutorial 12 Matrix Multiplication Rule Matrices make linear transformations of vectors

3. Math for CSTutorial 13 Translation, Scaling, Rotation of Vectors

4. Math for CSTutorial 14 Translation Now, we can write the translation as the multiplication by specially designed matrix: Translation in Homogeneous coordinates

5. Math for CSTutorial 15 Two translations We can check that the matrix representing two sequential translations can be written as the multiplication of their matrices. Sequential Translations in Homogeneous Coordinates

6. Math for CSTutorial 16 Scaling in Homogeneous coordinates Scaling Scaling matrix looks similar to what it was for ordinary coordinates: What is the Matrix for Scaling 0.1x 1 and 10x 2 ?

7. Math for CSTutorial 17 Two scalings The matrix of two successful scalings is the multiplication of two scaling matrices: Several Scalings in Homogeneous coordinates What is the Matrix for Scaling 0.1x 1 and 10x 2 and then 20x 1 and 0.1x 2 ?

8. Math for CSTutorial 18 Rotation Two Rotations Easy to check, that clock-wise rotation on angle θ is given by: Two successful rotations can be represented by multiplication of their matrices: Rotation in Homogeneous coordinates

9. Math for CSTutorial 19 Homogeneous coordinates scaled by a constant, represent the same point. x2x2 x1x1 W2W3 W1 Scaling of Homogeneous coordinates

10. Math for CSTutorial 110 Scaling of Homogeneous coordinates

11. Math for CSTutorial 111 Rotation How to write the rotation around a point ? Bring p back Bring p to the origin Scaling Bring p back Bring p to the origin Bring the point p to the origin; make a rotation, bring it back: … the same procedure for scaling: Rotation around arbitrary point

12. Math for CSTutorial 112 1). Write down the matrix for: Rotation θ=90° around p=(2,5), translation (-2,2), scaling (x2) around p=(-1,1). Translation Rotation Scaling ( T(-1,1)·S(2,2)·T(1,-1) ) ·T(-2,2) ·( T(2,5) ·R(90) ·T(-2,-5) ) Example 1. Series of transformations

13. Math for CSTutorial 113 PointTranslation Scaling The translation and scaling are very similar in 3D: Homogeneous coordinates in 3D

14. Math for CSTutorial 114 The Rotation in 3D can be done around arbitrary axis. Euler angles representation. Any rotation is the composition of three basic rotation, a rotation around the axis x of an angle , a rotation around the axis y of an angle  and a rotation around the angle z of an angle  are called Euler angles In right hand coordinated these rotations are defined as follows Simple representation Order-dependent:  Not suitable for animation, because the interpolation between the angles of rotation leads to false locations Rotation in 3D: Axis needed

15. Math for CSTutorial 115 2). Rotation θ=90° around x followed by rotation θ=90° around y. find the axis of rotation. R=R1·R2; If c – rotation axis, then: Rc=c; Solve v.r.t c; c=(a,a,-a,1) 3). Prove that rotation is not commutative: Rx(θ 1 )·Ry(θ 2 )≠ Ry(θ 2 ) ·Rx(θ 1 ) Examples 2,3. Rotations in 3D

16. Math for CSTutorial 116 The line equation in Euclidian Coordinates is: The Euclidian point (x,y) in Homogeneous Coordinates can be written as p=(x,y,1) or p=(αx, αy, α) Therefore, denoting Euclidian line (1) by the homogeneous triple u=(a,b,c) we obtain, that the point p lies on the line u iff: For example, point p=(1,2,1) lies on the line y=x+1 which can be written by u=(1,-1,1) Lines in Homogeneous Coordinates

17. Math for CSTutorial 117 Now, let us look for the interception point p=(x, y, w) of two lines u 1 =(a 1,b 1,c 1 ) and u 2 =(a 2,b 2,c 2 ): Form the first equation: Substituting this into the second: Taking convenient choice of scaling, in which Intersection of two lines 1/2

18. Math for CSTutorial 118 We obtain On the other hand, writing formally We have showed, that the point p of the intersection of two lines u 1 and u 2 is described by Intersection of two lines 2/2

19. Math for CSTutorial 119 Now, let us look for the line u=(a, b, c) passing through two points p 1 =(x 1,y 1,w 1 ) and p 2 =(x 2,y 2,w 2 ): Similarly to the solution of the lines intersection, we obtain The third point p 3 =(x 3,y 3,w 3 ) lies on the line u if (p 3,u)=0, which via definitions of determinant can be written as Due to duality of representation of lines and points in Homogeneous Coordinates, if the three lines u 1, u 2 and u 3 intersect in a single point, they satisfy Three points on the line