Motivation. General Idea. Goals. (Nonuniform) Scale. Outline. Foundations of Computer Graphics. s x Scale(s x. ,s y. 0 s y. 0 0 s z.

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1 Fondations of Compter Graphics Online Lectre 3: Transformations 1 Basic 2D Transforms Motivation Man different coordinate sstems in graphics World, model, bod, arms, To relate them, we mst transform between them Also, for modeling objects. I have a teapot, bt Want to place it at correct location in the world Want to view it from different angles (HW 1) Want to scale it to make it bigger or smaller Demo of HW 1 Goals This nit is abot the math for these transformations Represent transformations sing matrices and matrixvector mltiplications. Demos throghot lectre: HW 1 and Applet Transformations Game Applet Brown Universit Exploratories of Software Credit: Andries Van Dam and Jean Lalef General Idea Object in model coordinates Transform into world coordinates Represent points on object as vectors Mltipl b matrices Demos with applet Otline Translation: Homogeneos Coordinates (next time) Transforming Normals (next time) (Nonniform) Scale s x 0 Scale(s x,s ) S 0 s s x s x 0 0 x s x x 0 s 0 s 0 0 s z z s z z 0 0 s 1

2 Shear Shear 1 a 0 1 S 1 a 0 1 2D simple, 3D complicated. [Derivation? Examples?] 2D? Linear Commtative 2D 2D simple, 3D complicated. [Derivation? Examples?] 2D? x' ' cos θ sin θ sin θ cos θ x Linear Commtative Fondations of Compter Graphics Online Lectre 3: Transformations 1 Composing Transforms Otline Translation: Homogeneos Coordinates Transforming Normals 2

3 Composing Transforms E.g. Composing rotations, scales Often want to combine transforms E.g. first scale b 2, then rotate b 45 degrees Advantage of matrix formlation: All still a matrix Not commtative!! Order matters Rx 2 x 2 Sx 1 R(Sx 1 ) (RS)x 1 SRx 1 Inverting Composite Transforms Sa I want to invert a combination of 3 transforms Option 1: Find composite matrix, invert Option 2: Invert each transform and swap order Obvios from properties of matrices, demo M M 1 M 2 M 3 M M 3 M 2 M 1 Fondations of Compter Graphics Online Lectre 3: Transformations 1 3D M M M 3 (M 2 (M 1 M 1 )M 2 )M 3 Otline Review of 2D case x' ' cos θ sin θ sin θ cos θ x Translation: Homogeneos Coordinates Transforming Normals Orthogonal?, R T R I 3

4 in 3D abot coordinate axes simple R z R cosθ sinθ 0 sinθ cosθ cosθ 0 sinθ sinθ 0 cosθ Alwas linear, orthogonal Rows/cols orthonormal R x cosθ sinθ 0 sinθ cosθ R T R I Geometric Interpretation 3D Rows of matrix are 3 nit vectors of new coord frame Can constrct rotation matrix from 3 orthonormal vectors x z R vw x v x w z w x z Rp x v x w z w x p p z p x X + Y + z Z? Geometric Interpretation 3D Rows of matrix are 3 nit vectors of new coord frame Can constrct rotation matrix from 3 orthonormal vectors x z R vw x v x w z w x z Rp x v x w z w x p p z p x X + Y + z Z? p v p w p Geometric Interpretation 3D x z x p p Rp x v p v p x w z w z p w p Rows of matrix are 3 nit vectors of new coord frame Can constrct rotation matrix from 3 orthonormal vectors Effectivel, projections of point into new coord frame New coord frame vw taken to cartesian components xz Inverse or transpose takes xz cartesian to vw Non-Commtativit Not Commtative (nlike in 2D)!! Rotate b x, then is not same as then x Order of appling rotations does matter Follows from matrix mltiplication not commtative R1 * R2 is not the same as R2 * R1 Demo: HW1, order of right or p will matter Arbitrar rotation formla Rotate b an angle θ abot arbitrar axis a Homework 1: mst rotate ee, p direction Somewhat mathematical derivation bt sefl formla Problem setp: Rotate vector b b θ abot a Helpfl to relate b to X, a to Z, verif does right thing For HW1, o probabl jst need final formla 4

5 Axis-Angle formla Step 1: b has components parallel to a, perpendiclar Parallel component nchanged (rotating abot an axis leaves that axis nchanged after rotation, e.g. rot abot z) Axis-Angle formla Step 2: Define c orthogonal to both a and b Analogos to defining Y axis Use cross prodcts and matrix formla for that Axis-Angle formla Step 3: With respect to the perpendiclar comp of b Cos θ of it remains nchanged Sin θ of it projects onto vector c Axis-Angle: Ptting it together (b \ a) ROT (I 3 3 cosθ aa T cosθ)b + (A * sinθ)b (b a) ROT (aa T )b R(a,θ) I 3 3 cosθ + aa T (1 cosθ) + A * sinθ Unchanged (cosine) Component along a Perpendiclar (rotated comp) (hence nchanged) Axis-Angle: Ptting it together (b \ a) ROT (I 3 3 cosθ aa T cosθ)b + (A * sinθ)b (b a) ROT (aa T )b R(a,θ) I 3 3 cosθ + aa T (1 cosθ) + A * sinθ R(a,θ) cosθ (1 cosθ) x 2 x xz x 2 z + sinθ xz z z 2 0 z z 0 x x 0 (x z) are cartesian components of a 5