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February 11th, 2019, 02:49 PM   #1
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Projecting points on a line - 2D

a = (1, 0)
b = (1, 1)
c = (0, 1)

L: x + y = 4/3

$\displaystyle
Proj_{L}(a) = ? \\
Proj_{L}(b) = ? \\
Proj_{L}(c) = ? \\
$

1. How do I express x + y = 4/3 in matrix form for projection matrix multiplication?
2. Is it possible to calculate the projection{a,b,c} on L with matrix multiplication?
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February 11th, 2019, 04:42 PM   #2
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Plain Old Algebra

$\displaystyle
A = (0, 1) \\
L: x + y = 4/3 \\
\perp(L) : x = y \\

Point~A~lies~on~ \perp(L) : ax + b = c, (0) + b = 1, b = 1 \\
L_{1}: x + 1 = y \\
The~intersection~point~between~L~and~L_{1}: x + 1 = 4/3 - x \\
The~intersection~point: (1/6, 7/6)
$

$\displaystyle
B = (1, 1) \\
Point~B~lies~on ~\perp(L): ax + b = c, (1) + b = 1, b = 0 \\
L_{2}: x = y \\
The~intersection~point~between~L~and~L_{2}: x = 4/3 - x \\
The~intersection~point: (2/3, 2/3)
$

$\displaystyle
C = (1, 0) \\
Point~C~lies~on ~\perp(L): ax + b = c, (1) + b = 1, b = -1 \\
L_{3}: x - 1 = y \\
The~intersection~point~between~L~and~L_{3}: x -1 = 4/3 - x \\
The~intersection~point: (7/6, 1/6)
$



Case closed...
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Last edited by zollen; February 11th, 2019 at 04:50 PM.
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February 11th, 2019, 04:52 PM   #3
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I was hoping I could have used projection matrix to solve this problem...
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