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

Guarding Art Gallery

Visibility Problem

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

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?

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

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)

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

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

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

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

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

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

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

Intersection of Planes

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

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

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

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

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

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

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

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

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

Linear Equation and its Normal

Normal of a Plane

Plane and Affine Combination in 3D u v

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

Affine Combination in n-D

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

Linear Equation and its Normal

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