VIST 375

1. VIST 375 Foundations of Visualization Week of 10/17/16

2. Coordinates • A way to talk about spatial positions and relationships • The mathematical and computational basis for representing spatial relationships

3. Coordinate Systems • 3D • cartesian • cylindrical • spherical

4. Matrix Notation X’ = aX + bY as [P’] = [A][P] Y’ = cX + dY where [P’] = X’ [A] = a b [P] = X Y’ c d Y

5. 3D Coordinates • positions • vectors • transformations • projections

6. Right handed or left handed? • Relationship of the three axes • Applies to 3D cartesian coordinates • Similar axes relationships apply in spherical and cylindrical coordinates y y z right left x x z

7. 3D Transformations • Scaling, Rotation, Translation • in Matrix Notation [ X’ ] = [ X ] [ T ] or [ X’ ] = [ T ] [ X ] (preferred)

8. 2D Scaling [ P’ ] = [ S ] [ P ] where [ S ] = Sx 0 0 Sy X’ = Sx*X Y’ = Sy*Y

9. 3D Scaling X’ = Sx * X Y’ = Sy * Y or Z’ = Sz * Z x’ y’ z’ Sx 0 0 0 Sy 0 0 0 Sz x y z =

10. 2D Rotation Positive rotation about origin x’ = x cos  - y sin  y’ = x sin  + y cos 

11. 3D Rotation • About X axis • About Y axis • About Z axis • About “arbitrary” axis

12. 3D Rotation about the X axis X’ = X Y’ = cos Y - sin Z Z’ = sin Y + cos Z or x y z 1 0 0 0 cos -sin 0 sin cos x’ y’ z’ =

13. 3D Rotation about the Y axis X’ = cosX + sin Z Y’ = Y Z’ = -sin X + cos Z

14. 3D Rotation about the Z axis X’ = cos X - sin Y Y’ = sin X + cos Y Z’ = Z

15. 2D Translation X’ = X + Tx Y’ = Y + Ty • Remenber doesn’t fit the pattern • What to do?

16. 3D Translations X’ = X + Tx Y’ = Y + Ty Z’ = Z + Tz Requires Homogeneous coordinate to put into matrix form

17. 2D Homogeneous Translation [ P’ ] = [ T ][ P ] where [ T ] = 1 0 Tx [ P ] = X [ P’] = X’ 0 1 Ty Y Y’ 0 0 1 1 w x’ = X’/w y’ = Y’/w

18. 3D Homogeneous Translation [ P’ ] = [ T ] [ P ] where x’ = X’/ W y’ = Y’/ W z’ = Z’/ W X’ Y’ Z’ W 1 0 0 Tx 0 1 0 Ty 0 0 1 Tz 0 0 0 1 X Y Z 1 =

19. Transformation Duality Transforming the point or Transforming the coordinate system?

20. Transformation Ordering • Order of transformations determines results • For example, translation then rotation is generally not the same as rotation then translation

21. Projections • Orthogonal • ignores distance from viewer • Perspective • takes viewer distance into account

22. Orthogonal Projection • 3D world into 2D plane • 3D environment onto 2D screen • Essentially just ignores depth or the Z coordinate

23. Perspective Projection • Again 3D world into 2D plane, but • Modifies projected x,y position based on distance from eye or z coordinate X x,y,z image plane x’,y’ line of sight eye Z f

24. Projection Transformation [ P’ ] = [ Mp ][ P ] where [ P’] = [ P ] = [ Mp ] = x’ = X’ / w y’ = Y’/ w z’ = Z’/ w x’ = x / (z / f) y’ = y / (z / f) z’ = f x’ y’ z’ w x y z 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1/f 0

25. Alternative Form X eye Z d image plane

26. Alternative Form [P’ ] = [Mp ][P] where [P’] = [P] = [Mp ]= x’ = X’/ w y’ = Y’/ w z’ = 0/w x’ = x/((z/d)+1) y’ = y/((z/d)+1) x’ y’ z’ w x y z 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1/d 1

27. Projection Matrices Orthogonal Perspective 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1/d 1

28. 3D Vector Cross Products C = A x B where A, B, and C are all vectors C is normal to the plane of A and B Cx = AyBz-ByAz Cy = AzBx-AxBz Cz = AxBy-BxAy and |C| = |A||B| sin 

29. Vector Cross Product Can be define as the ‘determinant’ i j k C = Ax Ay Az Bx By Bz where i, j, k are unit vectors in the x, y, z directions

30. Transformation Concatenation Transformations can be successively applied in matrix form as follows X’ = [ T3 ] [ T2 ] [ T1 ] [ X ] which means X’ = ( [ T3 ] ( [ T2 ] ( [ T1 ] [ X ] ) ) ) or X’ = ( ( ( [ T3 ] [ T2 ] ) [ T1 ] ) [ X ] )