Affine Transformations

Transcription

1 Affine Transformations

2 Reading Foley et al., Chapter 5.6 and Chapter 6 Supplemental David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, Second edition

3 2D geometry Pipeline

4 3D Geometry Pipeline

5

6 Affine Geometry Points: location in 3D space Vectors: quantity with a direction and magnitude, but no fixed position Scalar: a real number v s = 5.3 P

7 Affine Spaces Affine space consists of points and vectors related by a set of axioms: Difference of two points is a vector: Head-to-tail rule for vector addition: P P P-Q P-Q P-R Q Q Q-R R

8 Legal affine operations: Affine Operations vector + vector vector scalar vector vector point point vector point + vector point example of an illegal operation: point + point nonsense Useful combination of affine operations: P( α) = P + αv What is it?

9 Affine Combination Affine combination of two points: where α + α = is defined to be the point We can generalize affine combination to multiple points: where Q= αq + α Q Q= Q + a ( Q - Q) 2 2 Q= α Q + α Q + L + α Q 2 2 n n α i =

10 Affine Frame A frame can be defined as a set of vectors and a point: Where v,, L vn ( L O ) v,,, v n form a basis and O is a point in space. Any point P can be written as P= p v + L + p v + O n n And any vector as: u= u v + L + u v n n

11 Matrix representation of points and vectors Coordinate axiom: So every point in the frame can be written as And every vector as P = P= P F = v, v2, K, v n, O P= p v + p v + L + p v + O 2 2 ( ) p p 2 = [ v v2 L vn O ] L pn u= u v + u v + L + u v + O 2 2 u u 2 = [ v v2 L vn O ] L un n n n n

12 Changing frames Given a point P in frame, what are the coordinates of P in frame F ʹ = vʹ, vʹ, K, vʹ, Oʹ ( ) 2 n p pʹ p pʹ 2 2 [ v v L v O] L [ vʹ vʹ L vʹ Oʹ ] P = 2 n = 2 n L pn pʹ n Since each element of can be written in coordinates relative to v = f vʹ + L + f vʹ O i i, i, n n = f vʹ + L + f vʹ + Oʹ n+, n+, n n

13 Written in a matrix form Changing frames cont d pʹ p f, fn, fn+, p L ʹ 2 p 2 M O M M [ vʹ vʹ L vʹ Oʹ ] M = [ vʹ vʹ L vʹ Oʹ ] 2 n 2 n f, n fn, n fn+, n pn pn ʹ pʹ p p f, fn, fn, p L + ʹ 2 p 2 p 2 M O M M = M = F M f, n fn, n fn+, n pʹ n pn pn

14 Euclidean and Cartesian spaces A Euclidean space is an affine space with an inner product: T uv, = u v= uv A Cartesian space is a Euclidean space with a standard orthonormal frame. In 3D: (e, e2, e2, ) e e i j if i= j = otherwise

15 Length: Useful properties and operations in Cartesian spaces v = v v Distance between points: P Q Angle between vectors: Perpendicular (orthogonal): Parallel: u v u v =± Cross product (in 3D): cos u v u v u v= u v= w

16 F: A B Affine Transformations is an affine transformation if it preserves affine combinations: Where α i = Affine coordiantes are preserved: ( α ) i i = αi ( i) F Q F Q. The same applies to vectors. ( O + ) ivi = ( O) + i ( vi) F p F pf Lines map to lines: Paralelism is preserved: FP ( + αv) = FP ( ) + αf( v) FQ ( + βv) = FQ ( ) + βf( v) Ratios are preserved: (,, ) = ( ( ), ( ), ( )) Ratio Q Q Q Ratio F Q F Q F Q 2 2

17 P=[x,y,] P is a column vector P is a row vector 2D Affine Transformations Pʹ = MP xʹ a b c x yʹ d e f y = Pʹ = PM a d xʹ yʹ = x y b e c f [ ] [ ]

18 Identity Doesn t move points at all

19 Translation xʹ c x yʹ f y = xʹ = x+ c yʹ = y+ f

20 Scaling Changing the diagonal elements performs scaling a f If a=f scaling is uniform x' = ax y ' = fy What if a,f<

21 Shearing What about the off-diagonal elements? The matrix d Gives x' = x y' = dx+ y

22 Effect on unit square a b a a+ b b d e d d e e = + M can be determined just by knowing how corners [,,] and [,,] are mapped a and e give x- and y-scaling b and d give x- and y-shearing

23 Rotation Rotation of points [,,] and [,,] by angle around the origin: cos( α) sin( α) sin( α) cos( α)

24 The Matrices Identity (do nothing): Scale by s x in the x and s y in the y direction (s x < or s y < is reflection): Rotate by angle θ (in radians): Shear by amount a in the x direction: s x cos θ sin θ s y a -sin θ cos θ Shear by amount b in the y direction: b Translate by the vector (t x, t y ): t x t y

25 Transformation Composition Applying transformations F to point P and transformation G to the result Pʹ = Pʹ ʹ = FP GPʹ Combining two transformations Pʹ ʹ = = GF ( P) ( GF) P

26 Problems 2,3,4,4,7,8 Let s play a game

27 Rotation around arbitrary point y p θ x

28 Reflection around arbitrary axis y p θ x

29 Reflection around arbitrary axis y v p x

30 Compact representation Fast implementation Easy to invert Easy to compose Properties of Transforms

31 3D Scaling xʹ sx x yʹ sy y = zʹ sz z

32 3D Translation xʹ tx x yʹ t y y = zʹ tz z

33 Rotation in 3D Rotation now has more possibilities in 3D: R R R x y z cosθ sinθ ( θ ) = sinθ cosθ cosθ sinθ ( θ ) = sinθ cosθ cosθ sinθ sinθ cosθ ( θ ) = R z R y R x Use right hand rule

34 Rotation in 3D What about the inverses of 3D rotations? R R R x y z cosθ sinθ ( θ ) = sinθ cosθ cosθ sinθ ( θ ) = sinθ cosθ cosθ sinθ sinθ cosθ ( θ ) = R z R y R x

35 Shearing in 3D Shearing is also more complicated. Here is one example: x' b x y' y = z' z

36 Properties of affine transformations All of the transformations we've looked at so far are examples of affine transformations. Here are some useful properties of affine transformations: Lines map to lines Parallel lines remain parallel Midpoints map to midpoints (in fact, ratios are always preserved) à pq s ratio = = = qr t p'q' q'r'

37 Rotation that aligns 3 orthonormal vectors with the principal axes y v w x z u