1. ©College of Computer and Information Science, Northeastern University CS 4300 Computer Graphics Prof. Harriet Fell Fall 2012 Lecture 12 – October 1, 2012

2. ©College of Computer and Information Science, Northeastern University 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, z are 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. ©College of Computer and Information Science, Northeastern University 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. ©College of Computer and Information Science, Northeastern University Vectors, Coordinate Vectors, Bases vector is 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. ©College of Computer and Information Science, Northeastern University Linear Transformations by 3x3 Matrices Linear Transformation

6. ©College of Computer and Information Science, Northeastern University Linear Transformations by 3x3 Matrices

7. ©College of Computer and Information Science, Northeastern University Identity and Inverse

8. ©College of Computer and Information Science, Northeastern University Points and Frames point – fixed place in a geometric world vector – motion between two points o addition and scalar multiplication make sense for vectors but not for points other operations that make sense o apply a linear transformation to a point o translate a point

9. ©College of Computer and Information Science, Northeastern University Frames

10. ©College of Computer and Information Science, Northeastern University Affine Matrix Affine Transformation of a Point

11. ©College of Computer and Information Science, Northeastern University Affine Transformation Applied to a Frame

12. ©College of Computer and Information Science, Northeastern University Linear Transformation of a Point

13. ©College of Computer and Information Science, Northeastern University Translations

14. ©College of Computer and Information Science, Northeastern University All Together Now

15. ©College of Computer and Information Science, Northeastern University Transforming Normals

16. ©College of Computer and Information Science, Northeastern University Normals

17. ©College of Computer and Information Science, Northeastern University Computing Transformed Normals