Computer Graphics (Fall 2005)To DoCourse OutlineSlide 4MotivationSlide 6General IdeaOutline(Nonuniform) ScaleShearRotationsSlide 12Composing TransformsE.g. Composing rotations, scalesInverting Composite TransformsSlide 16Slide 17Rotations in 3DGeometric Interpretation 3D RotationsSlide 20Non-CommutativityArbitrary rotation formulaAxis-Angle formulaAxis-Angle: Putting it togetherSlide 25Computer Graphics (Fall 2005)Computer Graphics (Fall 2005)COMS 4160, Lecture 3: Transformations 1http://www.cs.columbia.edu/~cs4160To DoTo DoStart (thinking about) assignment 1Much of information you need is in this lecture (slides)Ask TA NOW if compilation problems, visual C++ etc. Not that much coding [solution is approx. 20 lines, but you may need more to implement basic matrix/vector math], but some thinking and debugging likely involvedSpecifics of HW 1Axis-angle rotation and gluLookAt most useful (essential?). These are not covered in text (look at slides).You probably only need final results, but try understanding derivations. Problems in text help understanding material. Usually, we have review sessions per unit, but this one before midtermCourse OutlineCourse Outline3D Graphics Pipeline Rendering(Creating, shading images from geometry, lighting, materials) Modeling(Creating 3D Geometry)Course OutlineCourse Outline3D Graphics Pipeline Rendering(Creating, shading images from geometry, lighting, materials) Modeling(Creating 3D Geometry)Unit 1: TransformationsResizing and placing objects in theworld. Creating perspective images.Weeks 1 and 2 simplestGlut.exeAss 1 due Sep 22 (Demo)MotivationMotivationMany different coordinate systems in graphicsWorld, model, body, arms, …To relate them, we must transform between themAlso, for modeling objects. I have a teapot, but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 smallerMotivationMotivationMany different coordinate systems in graphicsWorld, model, body, arms, …To relate them, we must transform between themAlso, for modeling objects. I have a teapot, but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This unit is about the math for doing all these thingsRepresent transformations using matrices and matrix-vector multiplications. Demo: HW 1, applettransformation_game.jarGeneral IdeaGeneral IdeaObject in model coordinatesTransform into world coordinatesRepresent points on object as vectorsMultiply by matricesDemos with appletChapter 6 in text. We cover most of it essentially as in the book. Worthwhile (but not essential) to read whole chapterOutlineOutline2D transformations: rotation, scale, shearComposing transforms3D rotationsTranslation: Homogeneous Coordinates (next time)Transforming Normals (next time)(Nonuniform) Scale(Nonuniform) Scale11100( , )00xxx yyyssScale s s Sss---� �� �= =� �� �� �� �� �0 00 00 0x xy yz zs x s xs y s ys z s z� �� � � �� �� � � �=� �� � � �� �� � � �� �� � � �transformation_game.jarShearShear11 10 1 0 1a aShear S--� � � �= =� � � �� � � �RotationsRotations2D simple, 3D complicated. [Derivation? Examples?]2D?LinearCommutative' cos sin' sin cosx xy yq qq q-� � � ���=� � � ���� � � ���R(X+Y)=R(X)+R(Y)transformation_game.jarOutlineOutline2D transformations: rotation, scale, shearComposing transforms3D rotationsTranslation: Homogeneous CoordinatesTransforming NormalsComposing TransformsComposing TransformsOften want to combine transformsE.g. first scale by 2, then rotate by 45 degreesAdvantage of matrix formulation: All still a matrixNot commutative!! Order mattersE.g. Composing rotations, scalesE.g. Composing rotations, scales3 2 2 13 1 13 1( ) ( )x Rx x Sxx R Sx RS xx SRx= == =�transformation_game.jarInverting Composite TransformsInverting Composite TransformsSay I want to invert a combination of 3 transformsOption 1: Find composite matrix, invertOption 2: Invert each transform and swap orderObvious from properties of matrices1 2 31 1 1 13 2 11 1 1 13 2 1 1 2 3( ( ) )M M M MM M M MM M M M M M M M- - - -- - - -===transformation_game.jarOutlineOutline2D transformations: rotation, scale, shearComposing transforms3D rotationsTranslation: Homogeneous CoordinatesTransforming NormalsRotationsRotationsReview of 2D caseOrthogonal?, ' cos sin' sin cosx xy yq qq q-� � � ���=� � � ���� � � ���TR R I=Rotations in 3D Rotations in 3D Rotations about coordinate axes simpleAlways linear, orthogonalRows/cols orthonormal TR R I=R(X+Y)=R(X)+R(Y)cos sin 0 1 0 0sin cos 0 0 cos sin0 0 1 0 sin coscos 0 sin0 1 0sin 0 cosz xyR RRq qq q q qq qq qq q-� � � �� � � �= = -� � � �� � � �� � � �� �� �=� �� �-� �Geometric Interpretation 3D RotationsGeometric Interpretation 3D RotationsRows of matrix are 3 unit vectors of new coord frameCan construct rotation matrix from 3 orthonormal vectorsu u uuvw v v v u u uw w wx y zR x y z u x X y Y z Zx y z� �� �= = + +� �� �� �?u u u pv v v pw w w px y z xRp x y z yx y z z� �� �� �� �= =� �� �� �� �� �� �u pv pw p�� �� ��� �� ��� �Geometric Interpretation 3D RotationsGeometric Interpretation 3D RotationsRows of matrix are 3 unit vectors of new coord frameCan construct rotation matrix from 3 orthonormal vectorsEffectively, projections of point into new coord frameNew coord frame uvw taken to cartesian components xyzInverse or transpose takes xyz cartesian to uvwu u u pv v v pw w w px y z x u pRp x y z y v px y z z w p� ��� � � �� �� � � �= = �� �� � � �� � � �� ��� � � �� �Non-CommutativityNon-CommutativityNot Commutative (unlike in 2D)!!Rotate by x, then y is not same as y then xOrder of applying rotations does matterFollows from matrix multiplication not commutativeR1 * R2 is not the same as R2 * R1Demo: HW1, order of right or up will mattersimplestGlut.exeArbitrary rotation formulaArbitrary rotation