Download presentation
Presentation is loading. Please wait.
Published byWillis Cook Modified over 9 years ago
1
1 Math Review Coordinate systems 2-D, 3-D Vectors Matrices Matrix operations
2
CS-321 Dr. Mark L. Hornick 2 Math Review Cornerstone of graphics Basis for most algorithms Systematic notation Simplifying communication Organizing ideas Compact representation
3
CS-321 Dr. Mark L. Hornick 3 2-D Coordinate Systems 0x y P y P 0x Rectangular or Cartesian
4
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
5
CS-321 Dr. Mark L. Hornick 5 2-D Vector Representations Cartesian components Magnitude and direction angle VxVx VyVy V |V|
6
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
7
CS-321 Dr. Mark L. Hornick 7 2-D Vector Operations Scalar multiply VxVx VyVy V aV x aV y aVaV
8
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
9
CS-321 Dr. Mark L. Hornick 9 Direction cosines α β VxVx VyVy V
10
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.
11
CS-321 Dr. Mark L. Hornick 11 3-D Vector Operations Addition Scalar multiply
12
CS-321 Dr. Mark L. Hornick 12 3-D Vector Representations Cartesian components Magnitude and direction cosines x V z y α β γ
13
CS-321 Dr. Mark L. Hornick 13 3-D Vector Operations Inner (dot) product θ V1V1 V2V2 Portion of in direction Projections
14
CS-321 Dr. Mark L. Hornick 14 3-D Vector Cross Product “Right-hand rule” θ V1V1 V2V2 V 1 x V 2
15
CS-321 Dr. Mark L. Hornick 15 Matrices Rectangular matrix (m x n) (rows x cols) Row vector Column vector
16
CS-321 Dr. Mark L. Hornick 16 Scalar Matrix Multiplication
17
CS-321 Dr. Mark L. Hornick 17 Matrix Addition Matrices must have same dimensions
18
CS-321 Dr. Mark L. Hornick 18 Matrix Transpose
19
CS-321 Dr. Mark L. Hornick 19 Matrix Multiplication Matrices must be conformable (n=p)
20
CS-321 Dr. Mark L. Hornick 20 Matrix Multiplication Example 123456123456 213457213457 = ABC C row = A row, C column = B column 23 6653 30
21
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
22
CS-321 Dr. Mark L. Hornick 22 Determinants 1 A -1 = adj(A) det(A)
23
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.