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This chapter discuss the purpose of the components required for successful communications; describe these uses of computer communications: wireless messaging services, wireless Internet access points, cybercafés, global positioning systems, collaboration, groupware, voice mail, and Web services;
Computer Graphics Lecture 15 Fasih ur Rehman Last Class • Combining Transformations • Affine versus Rigid body Transformations • Homogenous Transformations Today’s Agenda • Homogeneous transformations • Types of Transformations – Linear Transformations – Affine Transformations – Projective Transformations • Homogenous Coordinate System in 3D are Basic 2D transformations – Translation x' y' 1 a x b y 0 1 Scaling as 3x3 x' y' a 0 x b y 0 1 Rotation as 3x3 x' y' cos sin sin cos 0 x y 1 Shear as 3x3 x' hx x y' hy y 0 1 Linear Transformations • Combination of Scaling, rotation and shear are linear transformations x' y' • a b c d x y Linear Transformations satisfy following – Origin maps origin – Lines maps lines Affine Transformations • • Affine transformations are combinations of linear transformations and Translation x' a b c x y' w d e 0 f y w Affine transformations obey the following – Origin does not necessarily map the origin – Lines maps lines Projective Transformations • • Affine Transformations and projective warps form projective transformation x' a b c x y' w d g f i y w e h Projective Transformations obey the following – Origin does not necessarily map the origin Matrix Composition • Homogenous transformations can also be combined by Matrix multiplication Example Example Multiplication Order • Scaling, Rotation and then translation Summary • Homogeneous transformations • Types of Transformations – Linear Transformations – Affine Transformations – Projective Transformations References • • Fundamentals of Computer Graphics Third Edition by Peter Shirley and Steve Marschner Interactive Computer Graphics, A Topdown Approach with OpenGL (Sixth Edition) by Edward Angel ... Projective Transformations References • • Fundamentals of Computer Graphics Third Edition by Peter Shirley and Steve Marschner Interactive Computer Graphics, A Topdown Approach with OpenGL (Sixth