1. Lecture 3 Hyper-planes, Matrices, and Linear Systems
Scott Russell

2. Guarding Art Gallery

3. Visibility Problem

4. Art Gallery Problem To learn more about this problem, you can
google “Art Gallery Problem” or
google “Art Gallery Problems”

5. Visibility Problems: Intersection of Ray with Line or Plane
How to describe a line passing
a point along a direction?
How to describe a line and a plane?
How to find their intersection?

6. Line in 2D
x=3
y=1
By linear equation

7. Line in 2D By a point and a vector: passing (3,1) along vector (2,1)
x=3
y=1
By a point and a vector: passing (3,1) along vector (2,1)

8. Line in 2D
(3,1)
(0,-1/2)
By two points: passing (3,1) and (0,-1/2)

9. Line and Affine Combination in 2D
The line passing two points or the affine combination of two points is given by

10. System of Linear Equations (2D)
Row Picture[conventional view]:
two lines meets at a point
x=3
y=1

11. System of Linear Equations (2D)
Column Picture: linear combination of the first two vectors produces the third vector

12. And geometrically
Column Picture: linear combination of the first two vector produce the third vector
x=3
y=1

13. Coefficient Matrix and Matrix-Vector Product
A 2 by 2 matrix is a square table of 4 numbers,
two per row and two per column

14. System of Linear Equations (3D)
Row Picture[conventional view]:
Three planes meet at a single point
Row Picture[conventional view]:
Two planes meet at a single line
A line and a plane meet at a single point

15. Intersection of Planes

16. System of Linear Equations (3D)
Column Picture: linear combination of the first three vectors produces the fourth vector

17. Coefficient Matrix and Matrix-Vector Product
A 3 by 3 matrix is a square table of 9 numbers,
three per row and three per column

18. Matrix Vector Product (by row)
If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture

19. Matrix Vector Product (by column)
If A is a 3 by 3 matrix and x is a 3 by 1 vector, then in the row picture

20. More about 3D Geometry Points and distance, Balls and Spheres Lines
0 dimension in 3 dimensions
Lines
1 dimension in 3 dimensions
Plane
2 dimensions in 3 dimensions

21. Line in 3D 2D 3D By linear equation A point and a vector Two points
Affine combination 3D

22. Line in 3D
By a point and a vector: passing p along vector v

23. Line and Affine Combination in 3D
The line passing two points or the affine combination of two points is given by

24. Plane in 3D Line in 2D 3D By linear equation
Affine combination of two points
“Every” two points determine a line 3D
Affine combination of three points
“Every” three points determine a plane

25. Linear Equation and its Normal

26. Normal of a Plane

27. Plane and Affine Combination in 3Du v

28. High Dimensional Geometric Extension
Points and distance, Balls and Spheres
0 dimension in n dimensions
Lines
1 dimension in n dimensions
Plane
2 dimensions in n dimensions
k-flat
k-dimensions in n dimensions
Hyper-plane
(n-1)-dimensions in n dimensions

29. Affine Combination in n-D

30. Hyper-Planes in d-D Line in 2D 3D n-D By linear equation
Affine combination of two points 3D
Affine combination of three points
n-D
Affine combination of n-1 points

31. Linear Equation and its Normal

32. Matrix (Uniform Representation for Any Dimension)
An m by n matrix is a rectangular table of mn numbers