Geometric Transformations Geometric Transformations Translate X X dx

Geometric Transformations

Geometric Transformations • Translate – X’ = X + dx – Y’ = Y + dy dy – X’ = X + (-6) – Y’ = Y + (-4) dx

Geometric Transformations • Scale – X’ = X * Sx – Y’ = Y * Sy – X’ = X * 2 – Y’ = Y * 0. 5 • Only origin is stable

Geometric Transformations • Scale – X’ = X * Sx – Y’ = Y * Sy – X’ = X * -1 – Y’ = Y * 1 • Only origin is stable

Geometric Transformations • Rotate – X’ = cos(a)X – sin(a)Y – Y’ = sin(a)X + cos(a)Y – X’ = cos(45)X – sin(45) Y – Y’ = sin(45)X + cos(45)Y • Only origin is stable

General Form • Translate – X’ = 0 * X + 0 * Y + dx – Y’ = 0 * X + 0 * Y + dy • Scale – X’ = Sx * X + 0 * Y + 0 – Y’ = 0 * X + Sy * Y + 0 • Rotate – X’ = cos(a)*X – sin(a)*Y + 0 – Y’ = sin(a)*X + cos(a)*Y + 0

Homogenous Coordinates • Translate – X’ = 1 * X + 0 * Y + dx = [1, 0, dx] * [ X, Y, 1] – Y’ = 0 * X + 1 * Y + dy = [0, 1, dy] * [ X, Y, 1] • Scale – X’ = Sx * X + 0 * Y + 0 = [Sx, 0, 0] * [ X, Y, 1] – Y’ = 0 * X + Sy * Y + 0 = [0, Sy, 0] * [ X, Y, 1] • Rotate – X’ = cos(a)*X – sin(a)*Y + 0 = [cos(a), -sin(a), 0] * [ X, Y, 1] – Y’ = sin(a)*X + cos(a)*Y + 0 = [sin(a), cos(a), 0] * [ X, Y, 1]

Matrix form [ X Y 1] * a d 0 b e 0 c f 1 a b c d e f 0 0 1 * = [X’ Y’ 1] X Y 1 = X’ Y’ 1

Matrix form * X Y 1 = T(dx, dy) 1 0 dx 0 1 dy 0 0 1 X’ Y’ 1 * X Y 1 = S(Sx, Sy) Sx 0 0 0 Sy 0 0 0 1 X’ Y’ 1 cos(a) -sin(a) 0 * R(a) sin(a) cos(a) 0 R(sin(a), cos(a)) 0 0 1 X Y 1 = X’ Y’ 1

T(2, 2) S(3/4, 1/3) T(-2, -2) 1 0 2 0 1 2 0 0 1 3/4 0 0 0 1/3 0 0 0 1 1 0 -2 0 1 -2 0 0 1 X Y 1 = X’ Y’ 1

T(2, 2) S(3/4, 1/3) T(-2, -2) 1 0 2 0 1 2 0 0 1 3/4 0 0 0 1/3 0 0 0 1 1 0 -2 0 1 -2 0 0 1 X Y 1 = X’ Y’ 1

R(45) S(2, 1) R(45)