CS232

# CS232 - Lecture 5 Hyper-planes Matrices and Linear Systems...

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Lecture 5 Hyper-planes, Matrices, and Linear Systems Shang-Hua Teng

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Guarding Art Gallery
Visibility Problem

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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?

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Line in 2D By linear equation 1 2 = - y x 1 2 = - y x x=3 y=1
Line in 2D By a point and a vector: passing (3,1) along vector (2,1) { } t y t x t t y x + = + = 2200 + = 1 2 3 : ) 1 , 2 ( ) 1 , 3 ( ) , ( set ed Parametriz 1 2 have we , g eliminatin By = - y x t x=3 y=1

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Line in 2D By two points: passing (3,1) and (0,-1/2) 2200 - + - = 2200 - - + = t t t y x t t y x : 5 . 0 0 1 3 ) 1 ( have we term, same the Combine : 1 3 5 . 0 0 1 3 set ed Parametriz 1 2 have we , g eliminatin By = - y x t ( 3,1 ) ( 0,-1/2 )
Line and Affine Combination in 2D The line passing two points or the affine combination of two points is given by 2200 + - = = t b a t b a t b a b a b a b a : ) 1 ( , affine , line points two of n Combinatio Affine or Line 2 2 1 1 2 2 1 1 2 2 1 1

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System of Linear Equations (2D) Row Picture[conventional view]: two lines meets at a point 11 2 3 1 2 = + = - y x y x 11 2 3 = + y x 1 2 = - y x x=3 y=1
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CS232 - Lecture 5 Hyper-planes Matrices and Linear Systems...

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