Reading. Affine transformations. Vector representation. Geometric transformations. x y z. x y. Required: Angel 4.1, Further reading:

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1 Reading Required: Angel 4.1, Further reading: Affine transformations Angel, the rest of Chapter 4 Fole, et al, Chapter David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2 nd Ed., McGraw-Hill, New York, 1990, Chapter Geometric transformations Vector representation Geometric transformations will map points in one space to points in another: (x',, z ) = f (x,, z). These transformations can be ver simple, such as scaling each coordinate, or complex, such as nonlinear twists and bends. We'll focus on transformations that can be represented easil with matrix operations. First, a review of vectors and some basic operations on them We can represent a point, p = (x,), in the plane or p=(x,,z) in 3D space as column vectors as row vectors [ x ] [ x z] z 3 4

2 Canonical axes Vector length and dot products 5 6 Vector cross products 2D matrices We can represent a 2-D transformation M b a matrix a b M = c d If p is a column vector, M goes on the left: p' = Mp x' a b = ' c d If p is a row vector, M T goes on the right: p' = pm a c x = x b d [ ' '] [ ] We will use column vectors. T 7 8

3 Two-dimensional transformations Here's all ou get with a 2 x 2 transformation matrix: So: x ' a b ' = c d x' = ax+ b ' = cx + d Identit Suppose we choose a=d=1, b=c=0: Gives the identit matrix: Doesn't move the points at all We will develop some intimac with the elements a, b, c, d 9 10 Scaling Suppose we set b=c=0, but let a and d take on an positive value: Gives a scaling matrix: a 0 0 d Provides differential (non-uniform) scaling in x and : x ' = ax ' = d Suppose we keep b=c=0, but let either a or d go negative. Examples:

4 Now let's leave a=d=1 and experiment with b.... The matrix 1 b 0 1 gives: x ' = x+ b ' = Effect on unit square Let's see how a general 2 x 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 c c+ d d Effect on unit square, cont. Observe: Origin invariant under M M can be determined just b knowing how the corners (1,0) and (0,1) are mapped a and d give x- and -scaling b and c give x- and -shearing Rotation From our observations of the effect on the unit square, it should be eas to write down a matrix for rotation about the origin : Thus, M= Rθ ( ) = 15 16

5 Degrees of freedom For an transformation, we can count its degrees of freedom the number of independent (though not necessaril unique) parameters needed to specif the transformation. One wa to count them is to add up all the apparentl free variables and subtract the number of equations that constrain them. How man degrees of freedom does an arbitrar 2X2 transformation have? Limitation of the 2 x 2 matrix A 2 x 2 linear transformation matrix allows Scaling Rotation Reflection Shearing Q: What important operation does that leave out? How man degrees of freedom does a 2D rotation have? Homogeneous coordinates We can loft the problem up into 3-space, adding a third component to ever point: 1 Adding the third w component puts us in homogenous coordinates. Then, transform with a 3 x 3 matrix: x' 1 0 t x ' = T( t) = 0 1 t w ' Affine transformations The addition of translation to linear transformations gives us affine transformations. In matrix form, 2D affine transformations alwas look like this: a b tx A t M= c d t = D affine transformations alwas have a bottom row of [0 0 1]. An affine point is a linear point with an added w- coordinate which is alwas 1:... gives translation! p plin = = 1 aff 1 Appling an affine transformation gives another affine point: Aplin + t Mpaff = 1 20

6 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 x q 1] T with a matrix: Points and vectors Vectors have an additional coordinate of w=0. Thus, translation has no effect on vectors. Q: What happens if we multipl a vector b an affine matrix? These representations reflect some of the rules of affine operations on points and vectors: vector + vector scalar vector point - point point + vector point + point 1. Translate q to origin 2. Rotate 3. Translate back One useful combination of affine operations is: p() t = po + tu Q: What does this describe? Note: Transformation order is important!! Basic 3-D transformations: scaling Translation in 3D Some of the 3-D affine transformations are just like the 2-D ones. In this case, the bottom row is alwas [ ]. For example, scaling: x' tx ' t = z ' tz z x' sx ' 0 s 0 0 = z ' 0 0 sz 0z

7 Rotation in 3D Rotation now has more possibilities in 3D: Rotation in 3D (cont d) How man degrees of freedom are there in an arbitrar rotation? cosα sinα 0 R α x ( ) = 0 sinα cosα cosβ 0 sinβ R β = ( ) sinβ 0 cosβ cosγ sinγ 0 0 sinγ cosγ 0 0 R γ = z ( ) R R x R z Use right hand rule How else might ou specif a 3D rotation? Shearing in 3D Shearing is also more complicated. Here is one example: x' 1 b 0 0 ' = z ' z 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) pq s ratio = = = qr t p'q' q'r' We call this a shear with respect to the x-z plane

8 Affine transformations in OpenGL OpenGL maintains a modelview matrix that holds the current transformation M. The modelview matrix is applied to points (usuall vertices of polgons) before drawing. It is modified b commands including: glloadidentit() M I set M to identit gltranslatef(t x, t, t z ) M MT translate b (t x, t, t z ) glrotatef(θ, x,, z) M MR rotate b angle θ about axis (x,, z) glscalef(s x, s, s z ) M MS scale b (s x, s, s z ) Summar What to take awa from this lecture: All the names in boldface. How points and transformations are represented. How to compute lengths, dot products, and cross products of vectors, and what their geometrical meanings are. What all the elements of a 2 x 2 transformation matrix do and how these generalize to 3 x 3 transformations. What homogeneous coordinates are and how the work for affine transformations. How to concatenate transformations. The mathematical properties of affine transformations. Note that OpenGL adds transformations b postmultiplication of the modelview matrix