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Elementary 3D Transformations - a "Graphics Engine"

Elementary 3D Transformations - a "Graphics Engine". Transformation procedures Transformations of coordinate systems Translation Scaling Rotation. Coordinate systems. left handed right handed. Transformation procedures. A scene is made up of objects

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Elementary 3D Transformations - a "Graphics Engine"

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  1. Elementary 3D Transformations - a "Graphics Engine" Transformation procedures Transformations of coordinate systems Translation Scaling Rotation

  2. Coordinate systems left handed right handed

  3. Transformation procedures • A scene is made up of objects • Objects can be made of separately defined parts • Each object / part defined by a list of points (vertices) • Any part of the object can be moved or distorted by applying a transformation to the list of points which define it

  4. Basic transformations • Translation (shift) • Scaling • Rotation

  5. Y D D C A C A B Z B X Translation x’ = x + Tx y’ = y + Ty z’ = z + Tz T = (Tx,Ty,Tz)

  6. D C A D B C A B Scalingabout the origin x’ = x * Sx y’ = y * Sy z’ = z * Sz S > 1 - enlarge 0 < S < 1 - reduce S < 0 - mirror Y Z X

  7. Scalingabout an arbitrary point Scaling about a fixed point ( xc, yc, zc ) x' = xc + ( x – xc ) * Sx y' = xc + ( y – yc ) * Sy z' = zc + ( z – zc ) * Sz Can also be achieved by a composite transformation.

  8. Rotation The direction of rotation in the left-handed system Positive angle of rotation when looking from a positive axis toward the origin a 90oclockwise rotation transforms one positive axis into the other.

  9. Y Clockwise X Positive angle of rotation for Z axis Looking from the positive end of Z axis towards the origin

  10. Rotation Axis of rotation is Direction of positive rotation is X from Y to Z Y from Z to X Z from X to Y

  11. C B D D A C A B Rotation about Z axis x' = x·cos  - y·sin  y' = x·sin  + y·cos  z' = z Y Z X

  12. D A B D C C A B Rotation about X axis • y' = y·cos  - z·sin  • z' = y·sin  + z·cos  • x' = x Y Z X

  13. D D A C B A B C Rotation about Y axis • z' = z·cos - x·sin • x' = z·sin + x·cos • y' = y Y Z X

  14. Matrix representationHomogeneous coordinates • common notation for ALL transformations • common computational mechanism for ALL transformations • simple mechanism for combining a number of transformations => computational efficiency

  15. Common matrix operation for all transformations??? • Translate (shift) point P • Scale point P • Rotate point P Point (vector) P = [ xp yp zp ] Matrix ???

  16. Homogeneous coordinates • Point P = (x, y, z ) represented by a vector P = • Transformations All represented by a 4 x 4 matrix T T =

  17. Point transformation in homogeneous coordinates • Implemented by matrix multiplication P’ = T · P

  18. Transformation matrices for elementary transformations • 4 x 4 matrix • Homogeneous coordinates • Translation, scaling, rotation and perspective projection, all defined through matrices

  19. Translation x’ = x + Tx y’ = y + Ty z’ = z + Tz T =

  20. Scaling x’ = x * Sx y’ = y * Sy z’ = z * Sz S =

  21. Rotation about Z axis x' = x·cos  - y·sin  y' = x·sin  + y·cos  z' = z Rz =

  22. Rotation about X axis y' = y·cos  - z·sin  z' = y·sin  + z·cos  x' = x Rx =

  23. Rotation about Y axis z' = z·cos - x·sin x' = z·sin + x·cos y' = y Ry =

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