X 1 0 t x cos y 0 1 t y sin

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Unformatted text preview: ransla:on as a 3x3 matrix? ! x ' $ ! 1 0 tx $! x $ ! x + tx $ # &# &# &# # y ' & = # 0 1 ty &# y & = # y + ty # 1 & # 0 0 1 &# 1 & # 1 " %" %" %" & & & % 29 Transla:on •  Example of transla:on ! x ' $ ! 1 0 2 $! x $ ! x + 2 # &# &# &# # y ' & = # 0 1 1 &# y & = # y + 1 # 1 & # 0 0 1 &# 1 & # 1 " %" %" %" $ & & & % tx = 2 ty = 1 30 Homogeneous Coordinates •  Add a 3rd coordinate to every 2D point –  (x, y, w) represents a point at loca:on (x/w, y/w) –  (x, y, 0) represents a point at infinity –  (0, 0, 0) is not allowed y 2 Convenient coordinate system to represent many useful transformations (2,1,1) or (4,2,2) or (6,3,3) 1 1 x 2 31 Vectors? •  Points: represent a posi:on •  Vectors: represent direc:on and magnitude •  Opera:ons: –  Vectors: add, scalar mul:ply –  Points: subtract –  Both: point + vector Represen:ng Vectors ! v $ ! w $ ! v +w x x &# x&# x # v + w = # vy & + # wy & = # vy + wy # &# &# 0 # &# &# "0%"0%" % " p −q '$ x x ' = $ py − q y '...
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