1 Math Review Coordinate systems 2-D, 3-D Vectors Matrices Matrix operations

CS-321 Dr. Mark L. Hornick 2 Math Review Cornerstone of graphics Basis for most algorithms Systematic notation Simplifying communication Organizing ideas Compact representation

CS-321 Dr. Mark L. Hornick 3 2-D Coordinate Systems 0x y P y P 0x Rectangular or Cartesian

CS-321 Dr. Mark L. Hornick 4 Points and Vectors A point at (x,y) can be represented by vector P from the origin (0,0). 0x y P In general, a vector represents the difference (directed distance) between two points. 0x y P1P1 P2P2 V

CS-321 Dr. Mark L. Hornick 5 2-D Vector Representations Cartesian components Magnitude and direction angle VxVx VyVy V  |V|

CS-321 Dr. Mark L. Hornick 6 2-D Vector Operations Addition V1V1 V2V2 V 1 +V 2 Subtraction V1V1 -V 2 V 1 -V 2 V2V2

CS-321 Dr. Mark L. Hornick 7 2-D Vector Operations Scalar multiply VxVx VyVy V aV x aV y aVaV

CS-321 Dr. Mark L. Hornick 8 2-D Unit Vector For any vector V, V can also be written as av VxVx VyVy V

CS-321 Dr. Mark L. Hornick 9 Direction cosines α β VxVx VyVy V

CS-321 Dr. Mark L. Hornick 10 3-D Coordinate Systems z x y P In a Right-handed coordinate system, the z axis defined by the vector cross product of the x and y axes.

CS-321 Dr. Mark L. Hornick 11 3-D Vector Operations Addition Scalar multiply

CS-321 Dr. Mark L. Hornick 12 3-D Vector Representations Cartesian components Magnitude and direction cosines x V z y α β γ

CS-321 Dr. Mark L. Hornick 13 3-D Vector Operations Inner (dot) product θ V1V1 V2V2 Portion of in direction Projections

CS-321 Dr. Mark L. Hornick 14 3-D Vector Cross Product “Right-hand rule” θ V1V1 V2V2 V 1 x V 2

CS-321 Dr. Mark L. Hornick 15 Matrices Rectangular matrix (m x n) (rows x cols) Row vector Column vector

CS-321 Dr. Mark L. Hornick 16 Scalar Matrix Multiplication

CS-321 Dr. Mark L. Hornick 17 Matrix Addition Matrices must have same dimensions

CS-321 Dr. Mark L. Hornick 18 Matrix Transpose

CS-321 Dr. Mark L. Hornick 19 Matrix Multiplication Matrices must be conformable (n=p)

CS-321 Dr. Mark L. Hornick 20 Matrix Multiplication Example = ABC C row = A row, C column = B column

CS-321 Dr. Mark L. Hornick 21 Identity Matrix and Inverse Inverse computed by Gaussian elimination, determinants, or other methods; used directly or indirectly to solve sets of linear equations

CS-321 Dr. Mark L. Hornick 22 Determinants 1 A -1 = adj(A) det(A)

CS-321 Dr. Mark L. Hornick 23 Determinants Gaussian elimination is the best method Swapping two rows changes the sign of Multiplying a row by s, multiplies by s Adding row multiples has no effect Only on square matrices For an upper triangular matrix