Reading. 4. Affine transformations. Required: Watt, Section 1.1. Further reading:

Transcription

1 Reading Required: Watt, Section.. Further reading: 4. Affine transformations Fole, et al, Chapter David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2 nd Ed., McGraw-Hill, New York, 990, Chapter 2.

2 Geometric transformations Geometric transformations will map points in one space to points in another: (',',z') = f(,,z). Representation We can represent a point, p = (,), in the plane These tranformations can be ver simple, such as scaling each coordinate, or comple, such as nonlinear twists and bends. We'll focus on transformations that can be represented easil with matri operations. We'll start in 2D... as a column vector as a row vector [ ] 2

3 Representation, cont. We can represent a 2-D transformation M b a matri a c b d If p is a column vector, M goes on the left: p' = Mp ' a b = ' c d If p is a row vector, M T goes on the right: Two-dimensional transformations Here's all ou get with a 2 2 transformation matri M: So: ' a b ' = c d ' = a + b ' = c + d We will develop some intimac with the elements a, b, c, d p' = pm a c = b d [ ' '] [ ] We will use column vectors. T 3

4 Identit Suppose we choose a=d=, b=c=0: Gives the identit matri: 0 0 Doesn't move the points at all Scaling Suppose we set b=c=0, but let a and d take on an positive value: Gives a scaling matri: a 0 0 d Provides differential (non-uniform) scaling in and : ' = a ' = d

5 Suppose we keep b=c=0, but let either a or d go negative. Eamples: Now let's leave a=d= and eperiment b.... The matri b 0 gives: ' = + b ' = 0 5

6 Effect on unit square Let's see how a general 2 2 transformation M affects the unit square: a c b d [ p q r s] = [ p' q' r' s' ] a b a a+ b b c d 0 0 = 0 c c+ d d Effect on unit square, cont. Observe: Origin invariant under M M can be determined just b knowing how the corners (,0) and (0,) are mapped a and d give - and -scaling b and c give - and -shearing s r p q 6

7 Rotation From our observations of the effect on the unit square, it should be eas to write down a matri for rotation about the origin : Linear transformations The unit square observations also tell us the 22 matri transformation implies that we are representing a point in a new coordinate sstem: θ p' = Mp a b = c d = [ u v] = u+ v 0 where u=[a c] T and v=[b d] T are vectors that define a new basis for a linear space. 0 The transformation to this new basis (a.k.a., change of basis) is a linear transformation. Thus, M= R( θ) = 7

8 Limitations of the 2 2 matri A 2 2 linear transformation matri allows Scaling Rotation Reflection Shearing Q: What important operation does that leave out? Affine transformations In order to incorporate the idea that both the basis and the origin can change, we augment the linear space u, v with an origin t. We call u, v, and t (basis and origin) a frame for an affine space. Then, we can represent a change of frame as: p' = u + v + t This change of frame is also known as an affine transformation. How do we write an affine transformation with matrices? 8

9 Homogeneous coordinates Idea is to loft the problem up into 3-space, adding a third component to ever point: Rotation about arbitrar points Until now, we have onl considered rotation about the origin. With homogeneous coordinates, ou can specif a rotation, θ, about an point q = [q q ] T with a matri: And then transform with a 3 3 matri: ' 0 t ' = T() t = 0 t w ' 0 0 q θθθτ θ Translate q to origin 2. Rotate 3. Translate back... gives translation! Note: Transformation order is important!! 9

10 Points and vectors From now on, we can represent points as have an additional coordinate of w=. Barcentric coordinates A set of points can be used to create an affine frame. Consider a triangle ABC and a point P: Vectors have an additional coordinate of w=0. Thus, a change of origin has no effect on vectors. Q: What happens if we multipl a matri b a vector? These representations reflect some of the rules of affine operations on points and vectors: vector + vector scalar vector point - point point + vector point + point We can form a frame with an origin C and the vectors from C to the other vertices: u= A C v= B C t= C We can then write P in this coordinate frame: P= αu+ β v+ t = One useful combination of affine operations is: p() t = po + tu Q: What does this describe? The coordinates (α, β, γ) are called the barcentric coordinates of P relative to A, B, and C. 0

11 Computing barcentric coordinates In the triangle eample: Cross products Consider the cross-product of two vectors, u and v. What is the geometric interpretation of this cross-product? we can compute the barcentric coordinates of P: A B C α P αa+ βb+ γc= A B C β = P γ Simple matri analsis gives the solution: P B C A P C A B P P B C A P C A B P α = β = γ = A B C A B C A B C A B C A B C A B C Computing the determinant of the denominator gives: BC BC + AC AC + AB AB A cross-product can be computed as: i j k u v= u u uz v v vz = ( uv z uv z ) i + ( uv z uv z) j + ( uv uv ) k uv z uv z = uv z uv z uv uv What happens when u and v lie in the - plane? What is the area of the triangle the span?

12 Barcentric coords from area ratios Now, let s rearrange the equation from two slides ago: BC BC + AC AC + AB AB = ( B A )( C A ) ( B A )( C A ) The determinant is then just the z-component of (B-A) (C-A), which is two times the area of triangle ABC! Thus, we find: SArea( PBC ) SArea( APC ) SArea( ABP ) α = β = γ = SArea( ABC) SArea( ABC) SArea( ABC) Where SArea(RST) is the signed area of a triangle, which can be computed with crossproducts. Affine and conve combinations Note that we seem to have added points together, which we said was illegal, but as long as the have coefficients that sum to one, it s ok. We call this an affine combination. More generall: P= α A + L+ α A n n is a proper affine combination if: n i= α = Note that if the α i s are all positive, the result is more specificall called a conve combination. Q: Wh is it called a conve combination? i 2

13 Basic 3-D transformations: scaling Some of the 3-D transformations are just like the 2-D ones. For eample, scaling: ' s ' 0 s 0 0 = z' 0 0 sz 0z Translation in 3D ' 0 0 t ' 0 0 t = z' 0 0 tz z z z z z 3

14 Rotation in 3D Rotation now has more possibilities in 3D: Shearing in 3D Shearing is also more complicated. Here is one eample: cosθ sinθ 0 R ( θ) = 0 sinθ cosθ cosθ 0 sin θ R ( θ) = sinθ 0 cosθ cosθ sin θ 0 0 sinθ cosθ 0 0 Rz ( θ) = z R R R z Use right hand rule z ' b 0 0 ' = z' 0 0 0z z We call this a shear with respect to the -z plane. 4

15 Preservation of affine combinations A transformation F is an affine transformation if it preserves affine combinations: F( α A + L+ α A ) = α F( A) + L+ α F( A ) n n n n where the A i are points, and: n i= α = Clearl, the matri form of F has this propert. i Properties 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 alwas preserved) One special eample is a matri that drops a dimension. For eample: = z This transformation, known as an orthographic projection is an affine transformation. s p : t r q pq s ratio = = = qr t p' q' s : t p'q' q'r' r' We ll use this fact later 5

16 Summar What to take awa from this lecture: All the names in boldface. How points and transformations are represented. What all the elements of a 2 2 transformation matri do and how these generalize to 3 3 transformations. What homogeneous coordinates are and how the work for affine transformations. How to concatenate transformations. The rules for combining points and vectors The mathematical properties of affine transformations. 6