Geometric Transformations
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1 Geometric Transformations CS 4620 Lecture Steve Marschner 1
2 A little quick math background Notation for sets, functions, mappings Linear and affine transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication Implicit vs. explicit geometry 2017 Steve Marschner 2
3 Implicit representations Equation to tell whether we are on the curve Example: line (orthogonal to u, distance k from 0) (u is a unit vector) Example: circle (center p, radius r) {v (v p) (v p) r 2 =0} Always define boundary of region (if f is continuous) 2017 Steve Marschner 3
4 Explicit representations Also called parametric Equation to map domain into plane Example: line (containing p, parallel to u) Example: circle (center b, radius r) Like tracing out the path of a particle over time Variable t is the parameter 2017 Steve Marschner 4
5 Transforming geometry Move a subset of the plane using a mapping from the plane to itself Parametric representation: Implicit representation: 2017 Steve Marschner 5
6 Translation Simplest transformation: Inverse: Example of transforming circle 2017 Steve Marschner 6
7 Translation Simplest transformation: Inverse: Example of transforming circle 2017 Steve Marschner 6
8 Linear transformations One way to define a transformation is by matrix multiplication: Such transformations are linear, which is to say: (and in fact all linear transformations can be written this way) 2017 Steve Marschner 7
9 Geometry of 2D linear trans. 2x2 matrices have simple geometric interpretations uniform scale non-uniform scale rotation shear reflection Reading off the matrix 2017 Steve Marschner 8
10 Linear transformation gallery Uniform scale 2017 Steve Marschner 9
11 Linear transformation gallery Nonuniform scale 2017 Steve Marschner 10
12 Linear transformation gallery Rotation apple Steve Marschner 11
13 Linear transformation gallery Reflection can consider it a special case of nonuniform scale 2017 Steve Marschner 12
14 Linear transformation gallery Shear 2017 Steve Marschner 13
15 Composing transformations Want to move an object, then move it some more We need to represent S o T ( S compose T ) and would like to use the same representation as for S and T Translation easy Translation by ut then by us is translation by ut + us commutative! 2017 Steve Marschner 14
16 Composing transformations Linear transformations also straightforward Transforming first by M T then by M S is the same as transforming by M S M T only sometimes commutative e.g. rotations & uniform scales e.g. non-uniform scales w/o rotation Note M S M T, or S o T, is T first, then S 2017 Steve Marschner 15
17 Combining linear with translation Need to use both in single framework Can represent arbitrary seq. as e. g. Transforming by M T and u T, then by M S and u S, is the same as transforming by M S M T and u S + M S u T This will work but is a little awkward 2017 Steve Marschner 16
18 Homogeneous coordinates A trick for representing the foregoing more elegantly Extra component w for vectors, extra row/column for matrices for affine, can always keep w = 1 Represent linear transformations with dummy extra row and column 2017 Steve Marschner 17
19 Homogeneous coordinates Represent translation using the extra column 2017 Steve Marschner 18
20 Homogeneous coordinates Composition just works, by 3x3 matrix multiplication This is exactly the same as carrying around M and u but cleaner and generalizes in useful ways as we ll see later 2017 Steve Marschner 19
21 Affine transformations The set of transformations we have been looking at is known as the affine transformations straight lines preserved; parallel lines preserved ratios of lengths along lines preserved (midpoints preserved) 2017 Steve Marschner 20
22 Affine transformation gallery Translation 2017 Steve Marschner 21
23 Translation 2017 Steve Marschner 22
24 Translation 2017 Steve Marschner 22
25 Translation 2017 Steve Marschner 22
26 Translation 2017 Steve Marschner 22
27 Affine transformation gallery Uniform scale 2017 Steve Marschner 23
28 Affine transformation gallery Nonuniform scale 2017 Steve Marschner 24
29 Scaling 2017 Steve Marschner 25
30 Scaling 2017 Steve Marschner 25
31 Scaling 2017 Steve Marschner 25
32 Scaling 2017 Steve Marschner 25
33 Affine transformation gallery Rotation 2017 Steve Marschner 26
34 Rotation about z axis 2017 Steve Marschner 27
35 Rotation about z axis 2017 Steve Marschner 27
36 Rotation about x axis 2017 Steve Marschner 28
37 Rotation about x axis 2017 Steve Marschner 28
38 Rotation about y axis 2017 Steve Marschner 29
39 Rotation about y axis 2017 Steve Marschner 29
40 General Rotation Matrices A rotation in 2D is around a point A rotation in 3D is around an axis so 3D rotation is w.r.t a line, not just a point there are many more 3D rotations than 2D a 3D space around a given point, not just 1D convention: positive rotation is CCW convention: positive rotation is CCW when axis vector is pointing at you 2D 3D 2017 Steve Marschner 30
41 Euler angles An object can be oriented arbitrarily Euler angles: simply compose three coord. axis rotations e.g. x, then y, then z: heading, attitude, bank (common for airplanes) roll, pitch, yaw (common for vehicles) pan, tilt, roll (common for cameras) R( x, y, z )=R z ( z )R y ( y )R x ( x ) 2017 Steve Marschner 31
42 Euler angles An object can be oriented arbitrarily Euler angles: simply compose three coord. axis rotations e.g. x, then y, then z: heading, attitude, bank (common for airplanes) roll, pitch, yaw (common for vehicles) pan, tilt, roll (common for cameras) R( x, y, z )=R z ( z )R y ( y )R x ( x ) 2017 Steve Marschner 31
43 Euler angles in applications Wikipedia Ishikawa Watanabe Laboratory 2017 Steve Marschner 32
44 Affine transformation gallery Reflection can consider it a special case of nonuniform scale 2017 Steve Marschner 33
45 Affine transformation gallery Shear 2017 Steve Marschner 34
46 Properties of Matrices Translations: linear part is the identity Scales: linear part is diagonal Rotations: linear part is orthogonal Columns of R are mutually orthonormal: RR T =R T R=I Also, determinant of R is 1.0 [ det(r) = 1 ] 2017 Steve Marschner 35
47 General affine transformations The previous slides showed canonical examples of the types of affine transformations Generally, transformations contain elements of multiple types often define them as products of canonical transforms sometimes work with their properties more directly 2017 Steve Marschner 36
48 Composite affine transformations In general not commutative: order matters! rotate, then translate translate, then rotate 2017 Steve Marschner 37
49 Composite affine transformations In general not commutative: order matters! rotate, then translate translate, then rotate 2017 Steve Marschner 37
50 Composite affine transformations In general not commutative: order matters! rotate, then translate translate, then rotate 2017 Steve Marschner 37
51 Composite affine transformations In general not commutative: order matters! rotate, then translate translate, then rotate 2017 Steve Marschner 37
52 Composite affine transformations Another example scale, then rotate rotate, then scale 2017 Steve Marschner 38
53 Composite affine transformations Another example scale, then rotate rotate, then scale 2017 Steve Marschner 38
54 Composite affine transformations Another example scale, then rotate rotate, then scale 2017 Steve Marschner 38
55 Composite affine transformations Another example scale, then rotate rotate, then scale 2017 Steve Marschner 38
56 Rigid motions A transform made up of only translation and rotation is a rigid motion or a rigid body transformation The linear part is an orthonormal matrix Inverse of orthonormal matrix is transpose so inverse of rigid motion is easy: 2017 Steve Marschner 39
57 Composing to change axes Want to rotate about a particular point could work out formulas directly Know how to rotate about the origin so translate that point to the origin 2017 Steve Marschner 40
58 Composing to change axes Want to rotate about a particular point could work out formulas directly Know how to rotate about the origin so translate that point to the origin 2017 Steve Marschner 40
59 Composing to change axes Want to rotate about a particular point could work out formulas directly Know how to rotate about the origin so translate that point to the origin 2017 Steve Marschner 40
60 Composing to change axes Want to rotate about a particular point could work out formulas directly Know how to rotate about the origin so translate that point to the origin 2017 Steve Marschner 40
61 Composing to change axes Want to scale along a particular axis and point Know how to scale along the y axis at the origin so translate to the origin and rotate to align axes 2017 Steve Marschner 41
62 Composing to change axes Want to scale along a particular axis and point Know how to scale along the y axis at the origin so translate to the origin and rotate to align axes 2017 Steve Marschner 41
63 Composing to change axes Want to scale along a particular axis and point Know how to scale along the y axis at the origin so translate to the origin and rotate to align axes 2017 Steve Marschner 41
64 Composing to change axes Want to scale along a particular axis and point Know how to scale along the y axis at the origin so translate to the origin and rotate to align axes 2017 Steve Marschner 41
65 Composing to change axes Want to scale along a particular axis and point Know how to scale along the y axis at the origin so translate to the origin and rotate to align axes 2017 Steve Marschner 41
66 Composing to change axes Want to scale along a particular axis and point Know how to scale along the y axis at the origin so translate to the origin and rotate to align axes 2017 Steve Marschner 41
67 Transforming points and vectors Recall distinction points vs. vectors vectors are just offsets (differences between points) points have a location represented by vector offset from a fixed origin Points and vectors transform differently points respond to translation; vectors do not 2017 Steve Marschner 42
68 Transforming points and vectors Homogeneous coords. let us exclude translation just put 0 rather than 1 in the last place and note that subtracting two points cancels the extra coordinate, resulting in a vector! Preview: projective transformations what s really going on with this last coordinate? think of R 2 embedded in R 3 : all affine xfs. preserve z=1 plane could have other transforms; project back to z= Steve Marschner 43
69 Transforming normal vectors Transforming surface normals differences of points (and therefore tangents) transform OK normals do not; therefore use inverse transpose matrix 2017 Steve Marschner 44
70 Transforming normal vectors Transforming surface normals differences of points (and therefore tangents) transform OK normals do not; therefore use inverse transpose matrix 2017 Steve Marschner 44
71 More math background Coordinate systems Expressing vectors with respect to bases Linear transformations as changes of basis 2017 Steve Marschner 45
72 Affine change of coordinates Six degrees of freedom or 2017 Steve Marschner 46
73 Affine change of coordinates Coordinate frame: point plus basis Interpretation: transformation changes representation of point from one basis to another Frame to canonical matrix has frame in columns takes points represented in frame represents them in canonical basis e.g. [0 0], [1 0], [0 1] Seems backward but bears thinking about 2017 Steve Marschner 47
74 Affine change of coordinates A new way to read off the matrix e.g. shear from earlier can look at picture, see effect on basis vectors, write down matrix Also an easy way to construct transforms e. g. scale by 2 across direction (1,2) 2017 Steve Marschner 48
75 Affine change of coordinates When we move an object to the canonical frame to apply a transformation, we are changing coordinates the transformation is easy to express in object s frame so define it there and transform it T e is the transformation expressed wrt. {e 1, e 2 } T F is the transformation expressed in natural frame F is the frame-to-canonical matrix [u v p] This is a similarity transformation 2017 Steve Marschner 49
76 Coordinate frame summary Frame = point plus basis Frame matrix (frame-to-canonical) is Move points to and from frame by multiplying with F Move transformations using similarity transforms 2017 Steve Marschner 50
77 Building transforms from points 2D affine transformation has 6 degrees of freedom (DOFs) this is the number of knobs we have to set to define one So, 6 constraints suffice to define the transformation handy kind of constraint: point p maps to point q (2 constraints at once) three point constraints add up to constrain all 6 DOFs (i.e. can map any triangle to any other triangle) 3D affine transformation has 12 degrees of freedom count them from the matrix entries we re allowed to change So, 12 constraints suffice to define the transformation in 3D, this is 4 point constraints (i.e. can map any tetrahedron to any other tetrahedron) 2017 Steve Marschner 51
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