1. Mathematics for Computer Graphics Chun-Yuan Lin CG

2. Coordinate Reference Frames • See the powerpoint: Coordinate Reference Frames.ppt CG

3. Points and Vectors (1) • There is a fundamental difference between the concept of a geometric point and that of a vector. • A point is a position specified with coordinate values in some reference frame. (depend on the choice for the frame of refernece) • A vector has properties that are independent of any particular coordinate system. • Point Properties P y Frame B x Frame A CG

4. Points and Vectors (2) • Vector Properties • We can define a vector as the difference between two point positions. • Vx and Vy are the projection V onto the x and the y axes. • We can obtain these same vector components using two other point positions in the same coordinate reference frames. • A vector has no fixed position within a coordinate system. • We can describe a vector as a directed line segmentthat has two fundamental properties: magnitudeand direction. P2 V P1 CG

5. Points and Vectors (3) • Magnitude: • We can specify the vector direction in various ways, such as • A vector has the same magnitude and direction within a single coordinate system. • If we transform the vector to another reference frame, the value for its components and direction within that reference frame may change. • For a three-dimensional Cartesian vector representation CG

6. Points and Vectors (4) • We can give the vector direction in terms of the direction angles, α, β, γ. • The values cosα, cos β, cos γ are called the direction cosines of the vector. • Vectors are used to represent any quantities that have the properties of magnitude and direction. (force and velocity) z V γ β y α x CG

7. Points and Vectors (5) • Vector Addition and Scalar Multiplication V2 V1+V2 V2 V1 V1 CG

8. Points and Vectors (6) • Scalar Product of two Vectors • This multiplication scheme is called the scalar product or dot product. (inner product) • is the projection of vector V2 in the direction of V1. • In addition to the coordinate-independent form of the scalar product. V2 V1 θ CG

9. Points and Vectors (7) • The scalar product of two vectors is zero if and only if the two vectors are perpendicular (orthogonal) CG

10. Points and Vectors (8) • Vector Product of Two Vectors V1 × V2 V2 V1 u Cross product CG

11. Points and Vectors (9) CG

12. Matrices (1) • A matrix is a rectangular array of quantities, called the elements of the matrix. • We identify matrices according to the number of rows and number of columns. When the number of rows is the same as the number of columns, this matrix is called a square matrix. An r by c matrix Row vector Column vector CG

13. Matrices (2) • The matrix representation for a three-dimensional vector in Cartesian coordinates as • We use this standard matrix representation for both points and vectors. CG

14. Scalar Multiplication and Matrix Addition CG

15. Matrix Multiplication(1) • The product of two matrices is defined as a generalization of the vector dot product. CG

16. Matrix Multiplication(2) AB≠BA A(B+C)=AB+AC CG

17. Matrix Transpose • The transpose MT of a matrix is obtained by interchanging rows and columns. (M1M2)T=M2TM1T CG

18. Determinant of a Matrix • If we have a square matrix, we can combine the matrix elements to produce a single number called the determinant of the matrix. CG

19. Matrix Inverse • With square matrices, we can obtain an inverse matrix if and only of the determinant of the matrix is nonzero. Identity matrix CG