Transformations. Mike Bailey. Oregon State University. Oregon State University. Computer Graphics. transformations.

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1 1 Transformations Mike Bailey transformations.pptx

2 Geometry vs. Topology 2 Original Object Geometry: Where things are (e.g., coordinates) Geometry = changed Topology = same ( ) Geometry = same Topology = changed ( ) Topology: How things are connected

3 3D Coordinate Systems 3 Y Y Z X X Left-Handed Z Right-Handed

4 Transformations 4 Y Suppose you have a point P and you want to move it over by 2 units in X how would you change P s coordinates? P P X P ( P, P) ( P 2., P ) x y x y This is known as a coordinate transformation

5 General Form of 3D Linear Transformations 5 x AxByCzD y ExFyGzH z IxJyKzL It s called a Linear Transformation because all of the coordinates are raised to the 1 st power, that is, there are no x 2, x 3, etc. terms. Transform the geometry leave the topology as is

6 Translation Equations 6 x xt x y yt y z zt z T y T x

7 Scaling About the Origin 7 x = x S x y = y S y z = z S z

8 2D Rotation About the Origin 8 x xcos ysin y xsin y cos

9 Linear Equations in Matrix Form 9 x AxByCzD y ExFyGzH z IxJyKzL x consuming column y consuming column z consuming column constant column x producing row y producing row z producing row x A B C D x y E F G H y z I J K L z

10 Identity Matrix ( [ I ] ) 10 x x y y z z x x y y z z [ I ] signifies that Nothing has changed

11 Matrix Inverse 11 [M] [M] -1 [M] [M] -1 = [I] = Nothing has changed Whatever [M] does, [M] -1 undoes

12 Translation Matrix 12 x Tx x y T y y z T z z Quick! What is the inverse of this matrix?

Scaling Matrix 13 x Sx 0 0 0 x y 0 Sy 0 0 y z 0 0 Sz 0

13 Scaling Matrix 13 x Sx x y 0 Sy 0 0 y z 0 0 Sz 0 z Quick! What is the inverse of this matrix?

14 3D Rotation Matrix About Z 14 Y Right-handed coordinates Right-handed positive rotation rule +90º rotation gives: y = x X Z x cos sin 0 0 x y sin cos 0 0 y z z

15 3D Rotation Matrix About Y 15 Y X +90º rotation gives: x = z Z x cos 0 sin 0 x y y z sin 0 cos 0 z

16 3D Rotation Matrix About X 16 Y +90º rotation gives: z = y X Z x x y 0 cos sin 0 y z 0 sin cos 0 z

17 How it Really Works :-) 17

18 B Y Compound Transformations Q: Our rotation matrices only work around the origin? What if we want to rotate about an arbitrary point (A,B)? A: Use more than one matrix. 18 A X Write it 3 2 x x y y T A, B R T A, B z z Say it

19 Matrix Multiplication is not Commutative 19 Y Y Rotate, then translate X X Y Translate, then rotate X

20 Matrix Multiplication is Associative 20 x x y y T A, B R T A, B z z 1 1 x x y y T A, B R T A, B z z 1 1 One matrix the Current Transformation Matrix, or CTM

21 Y Can Multiply All Geometry by One Matrix! 21 B A X x x y [ M] y z z 1 1 Graphics hardware can do this very quickly!

22 OpenGL Will Do the Transformation Compounding for You! 22 gltranslatef( A, B, C ); glrotatef( (GLfloat)Yrot, 0., 1., 0. ); glrotatef( (GLfloat)Xrot, 1., 0., 0. ); glscalef( (GLfloat)Scale, (GLfloat)Scale, (GLfloat)Scale ); glcalllist( BoxList ); Typically objects are modeled around their own local origin, so the gltranslate( -A, -B, -C ) step is unnecessary. gltranslatef( A, B, C ); glrotatef( (GLfloat)Yrot, 0., 1., 0. ); glrotatef( (GLfloat)Xrot, 1., 0., 0. ); glscalef( (GLfloat)Scale, (GLfloat)Scale, (GLfloat)Scale ); glcalllist( BoxList );

23 OpenGL Will Do the Transformation Compounding for You! 23 for( ; ; ) { << Turn mouse position into Xrot and Yrot >> glloadidentity( ); gltranslatef( A, B, C ); glrotatef( (GLfloat)Yrot, 0., 1., 0. ); glrotatef( (GLfloat)Xrot, 1., 0., 0. ); glscalef( (GLfloat)Scale, (GLfloat)Scale, (GLfloat)Scale ); } glcalllist( BoxList );

24 The Rotation Matrix for an Arbitrary Axis and Angle Â 24 P Q P P P P ˆ( ˆ P) P P P P Aˆ( Aˆ P) Q AˆP AˆP 0 AˆP AˆP Aˆ P P AˆP

25 The Rotation Matrix for an Arbitrary Axis and Angle 25 P ' P P ' P cos Qsin P' P' P' P' Aˆ( AˆP) cos PAˆ( AˆP) sin AˆP

26 The Rotation Matrix for an Arbitrary Axis and Angle 26 x x x y x z ˆ( ˆ A AP) y x y y y z P z x z y z z 0 Az A y ˆ AP Az 0 Ax P Ay Ax 0

27 The Rotation Matrix for an Arbitrary Axis and Angle 27 P' Aˆ( AˆP) cos PAˆ( AˆP) sin AˆP x x x y x z x x x y x z 0 Az A y P ' y x y y y zcos I y x y y y zsin Az 0 Ax P z x z y z z z x z y z z Ay Ax 0 x x x y x z 1 x x x y x z 0 Az A y M AyAx AyAy AyAzcos AyAx 1AyAy AyAz sin Az 0 Ax z x z y z z z x z y 1 z z Ay Ax 0

28 The Rotation Matrix for an Arbitrary Axis and Angle 28 x x x y x z 1 x x x y x z 0 Az A y M AyAx AyAy AyAzcos AyAx 1AyAy AyAz sin Az 0 Ax z x z y z z z x z y 1 z z Ay Ax 0 Ax Ax cos 1AxAx AxAy cos AxAy sinaz AxAz cos AxAz sina y M y x cos y x sinaz y y cos 1 y y y z cos y z sina x z x cos z x sinay z y cos z y sinax z z cos 1 z z For this to be correct, A must be a unit vector

T y. x Ax By Cz D. z Ix Jy Kz L. Geometry: Topology: Transformations

T y. x Ax By Cz D. z Ix Jy Kz L. Geometry: Topology: Transformations Transformations Geometr vs. Topolog Original Object This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives.0 International License Mike Baile mjb@cs.oregonstate.edu Geometr: