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CS 4300 Computer Graphics

CS 4300 Computer Graphics. Prof. Harriet Fell Fall 2012 Lecture 12 – October 1, 2012. Linear Transformations. ala “Foundations of 3D Computer Graphics” by Steven J. Gortler a point in the real world represented by a coordinate vector

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CS 4300 Computer Graphics

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  1. CS 4300Computer Graphics Prof. Harriet Fell Fall 2012 Lecture 12 – October 1, 2012

  2. Linear Transformations • ala “Foundations of 3D Computer Graphics” by Steven J. Gortler • a point in the real world represented by a coordinate vector • x, y, zare numbers that give the position of the point w.r.t an agreed upon coordinate system • with an agreed upon origin • agreed upon directions

  3. Concepts and Notation • point • vector • coordinate vector c • numerical object with real numbers • bold for vertical collection • coordinate system • bold for vertical collection • t makes it horizontal collection • for collection of vectors

  4. Vectors, Coordinate Vectors, Bases • vectoris abstract geometric entity that represents motion between two point in the world • coordinate vector is a set of numbers used to specify a vector in an agreed upon coordinate system. • vector space V is set of vectors that satisfies certain rules – (think actual motions between actual geometric points) • basis is a small set of vectors that can be used to (uniquely) produce the entire set of vectors using vector + and scalar multiplication.

  5. Linear Transformations by 3x3 Matrices • Linear Transformation

  6. Linear Transformations by 3x3 Matrices

  7. Identity and Inverse

  8. Points and Frames • point – fixed place in a geometric world • vector – motion between two points • addition and scalar multiplication make sense for vectors but not for points • other operations that make sense • apply a linear transformation to a point • translate a point

  9. Frames

  10. Affine MatrixAffine Transformation of a Point

  11. Affine TransformationApplied to a Frame

  12. Linear Transformationof a Point

  13. Translations

  14. All Together Now

  15. Transforming Normals

  16. Normals

  17. Computing Transformed Normals

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