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

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1 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: Where things are (e.g., coordinates) Geometr = changed Topolog = same (----) Geometr = same Topolog = changed (----) Topolog: How things are connected transformations.pptx D Coordinate Sstems Transformations P P Suppose ou have a point P and ou want to move it over b units in how would ou change P s coordinates? Left-Handed Right-Handed P( P, P) ( P., P ) x x This is known as a coordinate transformation General Form of D Linear Transformations 5 Translation Equations 6 x Ax B Cz D ExFGzH z IxJKzL It s called a Linear Transformation because all of the coordinates are raised to the st power, that is, there are no x, x, etc. terms. x xt x T z zt z T Transform the geometr leave the topolog as is T x

2 Scaling About the Origin 7 D Rotation About the Origin 8 x = xs x = S x xcos sin xsin cos z = zs z Linear Equations in Matrix Form 9 Identit Matrix ( [ I ] ) 0 x Ax B Cz D ExFGzH z IxJKzL x producing row producing row z producing row x consuming column consuming column z consuming column constant column x A B C D x E F G H z I J K L z x x z z x x z z [ I ] signifies that Nothing has changed Matrix Inverse Translation Matrix [M] [M] - = [I] [M] [M] - = Nothing has changed Whatever [M] does, [M] - undoes x 0 0 Tx x 0 0 T z 0 0 T Quick! What is the inverse of this matrix?

3 Scaling Matrix D Rotation Matrix About Right-handed coordinates Right-handed positive rotation rule x Sx x 0 S 0 0 z 0 0 Sz 0 z Quick! What is the inverse of this matrix? +90º rotation gives: = x x cos sin 0 0 x sin cos 0 0 z z D Rotation Matrix About 5 D Rotation Matrix About 6 +90º rotation gives: z = +90º rotation gives: x = z x cos 0 sin 0 x z sin 0 cos 0 z x x 0 cos sin 0 z 0 sin cos 0 z How it Reall Works :-) 7 B Compound Transformations Q: Our rotation matrices onl work around the origin? What if we want to rotate about an arbitrar point (A,B)? 8 A: Use more than one matrix. A Write it x x TA, B R T A, B z z Sa it

4 Matrix Multiplication is not Commutative 9 Matrix Multiplication is Associative 0 Rotate, then translate x x TA, B R T A, B z z Translate, then rotate x x TA, BR TA, B z z } One matrix the Current Transformation Matrix, or CTM Can Multipl All Geometr b One Matrix! OpenGL Will Do the Transformation Compounding for ou! B A x x [ M] z z Graphics hardware can do this ver quickl! glrotatef( (GLfloat)rot, 0.,., 0. ); glrotatef( (GLfloat)rot,., 0., 0. ); Tpicall objects are modeled around their own local origin, so the gltranslate( -A, -B, -C ) step is unnecessar. glrotatef( (GLfloat)rot, 0.,., 0. ); glrotatef( (GLfloat)rot,., 0., 0. ); OpenGL Will Do the Transformation Compounding for ou! The Funk Rotation Matrix for an Arbitrar Axis and Angle for( ; ; ) { } << Turn mouse position into rot and rot >> glloadidentit( ); glrotatef( (GLfloat)rot, 0.,., 0. ); glrotatef( (GLfloat)rot,., 0., 0. ); P Â Q P P P P Aˆ( AˆP) P P P P Aˆ( Aˆ P) Q A ˆ P A ˆ P 0 A ˆ P A ˆ P A ˆ P P A ˆ P

5 The Funk Rotation Matrix for an Arbitrar Axis and Angle 5 The Funk Rotation Matrix for an Arbitrar Axis and Angle 6 P ' P P ' P cos Qsin P ' P' P' x x x ˆ( ˆ P) x z P z P ' ˆ( ˆP) cos P ˆ( ˆP) sin AˆP 0 Az A ˆ AP Az 0 Ax P A Ax 0 The Funk Rotation Matrix for an Arbitrar Axis and Angle 7 The Funk Rotation Matrix for an Arbitrar Axis and Angle 8 P ' ˆ( ˆP) cos P ˆ( ˆP) sin AˆP x x x x x x 0 Az A P ' x zcos I x zsin Az 0 AxP z z A Ax 0 x x x x x x 0 Az A M AAx AA AAzcos AAx AA AAz sin Az 0 Ax z z A Ax 0 x x x x x x 0 Az A M AAx AA AAzcos AAx AA AAz sin Az 0 Ax z z A A 0 x x x cos x x x cos x sinaz cos sina M x cos xsinaz cos z cos zsina x cos sina z cos z sinax cos For this to be correct, A must be a unit vector 5