2 d transformations and homogeneous coordinates
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1. 2D transformations like translation, rotation, and scaling can be represented using homogeneous coordinates and matrix multiplication. 2. Homogeneous coordinates add a third coordinate to allow a consistent matrix notation for all transformations. 3. Transformations can be combined by multiplying their matrices in the appropriate order.
2 d transformations and homogeneous coordinates
- 1. 2D transformations and homogeneous coordinates TARUN GEHLOTS
- 2. Map of the lecture • Transformations in 2D: – vector/matrix notation – example: translation, scaling, rotation • Homogeneous coordinates: – consistent notation – several other good points (later) • Composition of transformations • Transformations for the window system
- 3. Transformations in 2D • In the application model: – a 2D description of an object (vertices) – a transformation to apply • Each vertex is modified: • x’ = f(x,y) • y’ = g(x,y) • Express the modification
- 4. Translations • Each vertex is modified: • x’ = x+tx • y’ = y+ty Before After
- 5. Translations: vector notation • Use vector for the notation: – makes things simpler • A point is a vector: x y • A translation is merely a vector sum: P’ = P + T
- 6. Scaling in 2D • Coordinates multiplied by the scaling factor: • x’ = sx x • y’ = sy y Before After
- 7. Scaling in 2D, matrix notation • Scaling is a matrix multiplication: P’ = SP x sx 0 x y 0 sy y
- 8. Rotating in 2D • New coordinates depend on both x and y • x’ = cos x - sin y • y’ = sin x + cos y Before After
- 9. Rotating in 2D, matrix notation • A rotation is a matrix multiplication: P’=RP x cos sin x y sin cos y
- 10. 2D transformations, summary • Vector-matrix notation simplifies writing: – translation is a vector sum – rotation and scaling are matrix-vector multiplication • I would like a consistent notation: – that expresses all three identically – that expresses combination of these also identically • How to do this?
- 11. Homogeneous coordinates • Introduced in mathematics: – for projections and drawings – used in artillery, architecture – used to be classified material (in the 1850s) • Add a third coordinate, w x • A 2D point is a 3 coordinates vector: y w
- 12. Homogeneous coordinates (2) • Two points are equal if and only if: x’/w’ = x/w and y’/w’= y/w • w=0: points at infinity – useful for projections and curve drawing • Homogenize = divide by w. • Homogenized points: x y 1
- 13. Translations with homogeneous x 1 0 tx x x x w w tx y 0 1 ty y y y w 0 0 1 w w w ty x x wt x y y wt y w w
- 14. Scaling with homogeneous x sx 0 0 x x x y 0 sy 0 y w s x w y y w 0 0 1 w w s y w x sx x y sy y w w
- 15. Rotation with homogeneous x cos sin 0 x y sin cos 0 y x x y w cos w sin w w 0 0 1 w y x y w sin w cos w x cos x sin y y sin x cos y w w
- 16. Composition of transformations • To compose transformations, multiply the matrices: – composition of a rotation and a translation: M = RT • all transformations can be expressed as matrices – even transformations that are not translations, rotations and scaling
- 17. Rotation around a point Q • Rotation about a point Q: – translate Q to origin (TQ), – rotate about origin (R ) – translate back to Q (- TQ). P’=(-TQ)R TQ P
- 18. Beware! • Matrix multiplication is not commutative • The order of the transformations is vital – Rotation followed by translation is very different from translation followed by rotation – careful with the order of the matrices! • Small commutativity: – rotation commute with rotation, translation with translation…
- 19. From World to Window • Inside the application: – application model – coordinates related to the model – possibly floating point • On the screen: – pixel coordinates – integer – restricted viewport: umin/umax, vmin/vmax
- 20. From Model to Viewport ymax ymin xmin xmax
- 21. From Model to Viewport • Model is (xmin,ymin)-(xmax,ymax) • Viewport is (umin,vmin)-(umax,vmax) • Translate by (-xmin,-ymin) • Scale by ( umax-umin , vmax-vmin ) xmax-xmin ymax-ymin • Translate by (umin,vmin) M = T’ST
- 22. From Model to Viewport Pixel Coordinates Model Coordinates u x v M y w w
- 23. Mouse position: inverse problem • Mouse click: coordinates in pixels • We want the equivalent in World Coord – because the user has selected an object – to draw something – for interaction • How can we convert from window coordinates to model coordinates?
- 24. Mouse click: inverse problem • Simply inverse the matrix: 1 1 M (T ST) Model Coordinates Pixels coordinates x u 1 y M v w w
- 25. 2D transformations: conclusion • Simple, consistent matrix notation – using homogeneous coordinates – all transformations expressed as matrices • Used by the window system: – for conversion from model to window – for conversion from window to model • Used by the application: – for modeling transformations