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Unformatted text preview: math1 math1 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, AkenineMoller is more relevant to the class. Appendix A in T. AkenineMoller, E. Haines, N. Hoffman, RealTime 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. math1 EE 77001 Lecture Transparency. Formatted 10:16, 29 January 2010 from set1math. math1 math2 math2 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! math2 EE 77001 Lecture Transparency. Formatted 10:16, 29 January 2010 from set1math. math2 math3 math3 PointRelated 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. math3 EE 77001 Lecture Transparency. Formatted 10:16, 29 January 2010 from set1math. math3 math4 math4 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 . math4 EE 77001 Lecture Transparency. Formatted 10:16, 29 January 2010 from set1math. math4 math5 math5 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 nonzero. 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 ). math5 EE 77001 Lecture Transparency. Formatted 10:16, 29 January 2010 from set1math. math5 math6 math6 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.
 Spring '08
 Staff

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