set-1-math

# set-1-math - math-1 math-1 Mathematics for 3D Graphics...

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Unformatted text preview: math-1 math-1 Mathematics for 3D Graphics Topics Points, Vectors, Vertices, Coordinates Dot Products, Cross Products Lines, Planes, Intercepts References Many texts cover the linear algebra used for 3D graphics ... ... the texts below are good references, Akenine-Moller is more relevant to the class. Appendix A in T. Akenine-Moller, E. Haines, N. Hoffman, Real-Time Rendering, Third Edition, A. K. Peters Ltd. Appendix A in Foley, van Dam, Feiner, Huges, Computer Graphics: Principles and Prac- tice, Second Edition, Addison Wesley. math-1 EE 7700-1 Lecture Transparency. Formatted 10:16, 29 January 2010 from set-1-math. math-1 math-2 math-2 Points and Vectors Point: Indivisible location in space. E.g. , P 1 = bracketleftBigg 1 2 3 bracketrightBigg , P 2 = bracketleftBigg 4 5 6 bracketrightBigg Vector: Difference between two points. E.g. , V = P 2 P 1 = P 1 P 2 = bracketleftBigg 4 1 5 2 6 3 bracketrightBigg = bracketleftBigg 3 3 3 bracketrightBigg . Equivalently: P 2 = P 1 + V . Dont confuse points and vectors! math-2 EE 7700-1 Lecture Transparency. Formatted 10:16, 29 January 2010 from set-1-math. math-2 math-3 math-3 Point-Related Terminology Will define several terms related to points. At times they may be used interchangeably. Point: A location in space. Coordinate: A representation of location. Vertex: Term may mean point, coordinate, or part of graphical object. As used in class, vertex is a less formal term. It might refer to a point, its coordinate, and other info like color. math-3 EE 7700-1 Lecture Transparency. Formatted 10:16, 29 January 2010 from set-1-math. math-3 math-4 math-4 Coordinate: A representation of where a point is located. Familiar representations: 3D Cartesian P = ( x,y,z ). 2D Polar P = ( r, ). In class we will use 3D homogeneous coordinates . math-4 EE 7700-1 Lecture Transparency. Formatted 10:16, 29 January 2010 from set-1-math. math-4 math-5 math-5 Homogeneous Coordinates Homogeneous Coordinate: A coordinate representation for points in 3D space consisting of four components ... ... each component is a real number ... ... and the last component is non-zero. Representation: P = x y z w , where w negationslash = 0. P refers to same point as Cartesian coordinate ( x/w, y/w, z/w ). To save paper sometimes written as ( x, y, z, w ). math-5 EE 7700-1 Lecture Transparency. Formatted 10:16, 29 January 2010 from set-1-math. math-5 math-6 math-6 Homogeneous Coordinates Each point can be described by many homogeneous coordinates . . . . . . for example, (10 , 20 , 30) = 10 20 30 1 = 5 10 15 . 5 = 20 40 60 2 = 10 w 20 w 30 w w = . . ....
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## This note was uploaded on 01/03/2012 for the course EE 7700 taught by Professor Staff during the Spring '08 term at LSU.

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set-1-math - math-1 math-1 Mathematics for 3D Graphics...

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