This pdf includes the following topics:- Translation Scale Reflection Rotation Classes of Transformations Matrix Notation Reflection Shear Rotation and many more.
1. 2D Geometrical Transformations Foley & Van Dam, Chapter 5
3. 2D Geometrical Transformations Assumption: Objects consist of points and lines. A point is represented by its Cartesian coordinates: P = (x, y) Geometrical Transformation: Let (A, B) be a straight line segment between the points A and B. Let T be a general 2D transformation. T transforms (A, B) into another straight line segment (A’, B’), where: A’=TA and B’=TB
11. Classes of Transformations • Rigid transformation (distance preserving): Translation + Rotation • Similarity transformation (angle preserving): Translation + Rotation + Uniform Scale • Affine transformation (parallelism preserving): Translation + Rotation + Scale + Shear All above transformations Affine are groups where Similarity Rigid ⊂ Similarity ⊂ Affine Rigid
12. Matrix Notation • Let’s treat a point (x, y) as a 2x1 matrix (column vector): x y • What happens when this vector is multiplied by a 2x2 matrix? a b x ax + by c = d y cx + dy
13. 2D Transformations • 2D object is represented by points and lines that join them • Transformations can be applied only to the the points defining the lines • A point (x, y) is represented by a 2x1 column vector, so we can represent 2D transformations by using 2x2 matrices: x ' a b x y ' = c d y
14. Scale • Scale (a, b): (x, y) → (ax, by) a 0 x ax 0 = b y by •If a or b are negative, we get reflection • Inverse: S-1(a,b) = S(1/a, 1/b)
15. Reflection • Reflection through the y axis: − 1 0 0 1 • Reflection through the x axis: 1 0 0 − 1 • Reflection through y = x: 0 1 1 0 • Reflection through y = -x: 0 − 1 − 1 0
16. Shear • Shear (a, b): (x, y) → (x+ay, y+bx) 1 a x x + ay b = 1 y y + bx
17. Rotation • Rotate(θ): (x, y) → (x cos(θ)+y sin(θ), -x sin(θ)+y cos(θ)) cos θ sin θ x x cos θ + y sin θ − sin θ = cos θ y − x sin θ + y cos θ • Inverse: R-1(q) = RT(q) = R(-q)
18. Composition of Transformations • A sequence of transformations can be collapsed into a single matrix: x x [ A ][B ][C ] = [D ] y y •Note: Order of transformations is important! translate rotate rotate translate
19. Translation • Translation (a, b): x x + a y → y + b Problem: Cannot represent translation using 2x2 matrices Homogeneous Coordinates
20. Homogeneous Coordinates Is a mapping from Rn to Rn+1: ( x , y ) → ( X , Y , W ) = ( tx , ty , t ) Note: All triples (tx, ty, t) correspond to the same non-homogeneous point (x, y) Example (2, 3, 1) ≡ (6, 9, 3). Inverse mapping: X Y ( X ,Y ,W ) → , W W
21. Translation • Translate(a, b): 1 0 a x x + a 0 1 b y = y + b 0 0 1 1 1 Inverse: T-1(a, b) = T(-a, -b) Affine transformations now have the following form: a b e c d f 0 0 1
22. Geometric Interpretation A 2D point is mapped to a line (ray) in 3D The non-homogeneous points are obtained by projecting the rays onto the plane Z=1 W Y (X,Y,W) 1 y x (X,Y,1) X
23. Example Rotation about an arbitrary point (x0,y0) θ 1. Translate the coordinates so that the origin is at (x0,y0) 2. Rotate by θ 3. Translate back 1 0 x 0 cos θ sin θ 0 1 0 − x 0 x 0 1 y − sin θ cos θ 0 0 1 − y y = 0 0 0 0 1 0 0 1 0 0 1 1 cos θ − sin θ x ( 1 − cos θ ) + y 0 0 sin θ x = sin θ cos θ y 0 ( 1 − cos θ ) − x 0 sin θ y 0 0 1 1
24. Example Reflection about an arbitrary line p2 L = p1 + t (p2-p1) = t p2 + (1-t) p1 p1 1. Translate the coordinates so that P1 is at the origin 2. Rotate so that L aligns with the x-axis 3. Reflect about the x-axis 4. Rotate back 5. Translate back
25. Change of Coordinates It is often required to transform the description of an object from one coordinate system to another Rule: Transform one coordinate frame towards the other in the opposite direction of the representation change ion entat res y’ R ep x’ y at ion s f orm x Tra n
26. Example • Change of coordinates: Represent P = (xp, yp, 1) in the (x’, y’) coordinate system P ' = MP cos θ sin θ 0 1 0 − x 0 M = − sin θ cos θ 0 0 1 − y 0 0 0 1 0 0 1 y’ x’ y θ (x0, y0) (xp, yp) x
27. Example • Change of coordinates: Alternative method: assume x’ = (ux, uy) and y’ = (vx, vy) in the (x, y) coordinate system P ' = MP u x u 0 1 0 − x y 0 M = v x v y 0 0 1 − y 0 0 0 1 0 0 1 y’ x’ y (vx, vy) (ux, uy) (x0, y0) x
28. Example Reflection about an arbitrary line p2 L = p1 + t (p2-p1) = t p2 + (1-t) p1 p1 Define a coordinate systems (u, v) parallel to P1P2: p2 − p1 u x u = ≡ p 2 − p1 u y − u v v = y = v x u x y 1 0 p 1x u x vx 0 1 0 0 u x uy 0 1 0 − p 1x M = 0 1 p 1y u y vy 0 0 −1 0 v x vy 0 0 1 − p 1y 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1
29. 3D Viewing Transformation Pipeline Viewing coordinates World Coordinates Object in World p ing m ap :2 D 3 D 2D:2D mapping Viewport Device Coordinates
30. World to Viewing Coordinates In order to define the viewing window we have to •Windowing-coordinate origin P0 = (x0,y0) •View vector up v = (vx,vy) •Using v, we can find u: u = v x (0,0,1) yv ie w (vx,vy). y world (x0,y0)x v ie w (ux,uy). x world Transformation from world to viewing coordinates : ux uy 0 1 0 − x0 M wc − vc = vx vy 0 0 1 − y 0 0 0 1 0 0 1
31. Window to Viewport Coordinates (xmax,ymax) (xmin,ymin) Window is Viewing Coordinates Window translated to origin (umaxn,vmaxn) (umin,vmin) Window scaled and translated to Window scaled to Normalized Viewport location in device coordinates Viewport size 1 0 0 1 0 u min u max − u min 0 0 x max − x min 1 0 − x min 1 M vc − dc= 0 1 v min 0 v max − v min 0 0 0 0 1 − y min 0 0 y max − y min 1 0 0 1 0 0 1 0 0 1 Normalized Device Coordinates
32. Efficiency Considerations A 2D point transformation requires 9 multiplies and 6 a b c x ax + by + cz d e f y = d x + e y + fz g h i z g x + h y + i z But since affine transformations have always the form: a b c x ax + by + c d e f y = dx + ey + f 0 0 1 1 1 The number of operations can be reduced to 4 multiplies and 4 adds
33. Efficiency Considerations The rotation matrix is: cos θ sin θ x x cos θ + y sin θ − sin θ = cos θ y − x sin θ + y cos θ When rotating of small angles θ, we can use the fact that cos(θ) ≅ 1 and simplify 1 sin θ x x + y sin θ − sin θ = 1 y − x sin θ + y
34. Determinant of a Matrix a b = ad − bc c d a b c d e f = aei + bfg + cdh − ceg − afh − bdi g h i e f d f d e = a −b +c h i g i g h If P is a polygon of area AP, transformed by a matrix M, the area of the transformed polygon is AP∗|M|
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