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II 2D Transformation Transformations • Transformations are needed to: – Position objects defined relative to the origin – Build scenes based on hierarchies – Project objects from three to two dimensions • Transformations include: – – – – Translation Scaling Rotation Reflections • Transformations can be represented by matrices and matrix multiplication II-1 x • Representation of Points: x y ; y • Transformation of Points X T x a b * y ( ax cy ) ( bx dy ) x c d y* • Transformation of Straight Lines II-2 Rotation • Consider rotation about the origin by q degrees – radius stays the same, angle increases by q x’ = r cos (f q) y’ = r sin (f q) x’=x cos q –y sin q y’ = x sin q + y cos q x = r cos f y = r sin f II-3 • The transformation for a general rotation about the origin by an arbitrary angle q cos q T sin q sin q cos q X X T x * * cos q T sin q y * x sin q cos q sin q cos q x * cos q X T X y * sin q * cos q y sin q sin q x cos q y II-4 Scaling • Scaling increases or decreases the size of the object • Scaling occurs with respect to the origin – If the object is not centered at the origin, it will move in addition to changing size • In general, this is done with the equations: xn = s x * x yn = s y * y • This can also be done with the matrix multiplication: xn s x y 0 n 0 x s y y II-5 • Example: 2 0 T 0 2 3 0 T 0 1 / 2 II-6 Reflection • corresponds to negative scale factors sx = -1 sy = 1 original sx = -1 sy = -1 sx = 1 sy = -1 II-7 • The reflection about the x-axis (y=0) is obtained by 1 0 T 0 1 • The reflection about the y-axis (x=0) is obtained by 1 0 T 0 1 • The reflection about the y=x is obtained by 0 1 T 1 0 • The reflection about the y=-x is obtained by 0 1 T 1 0 II-8 II-9 II-10 Translations • The amount of the translation is added to or subtracted from the x and y coordinates • In general, this is done with the equations: xn = x + tx y n = y + ty • This can also be done with the matrix multiplication: xn 1 0 t x x y 0 1 t y y n 1 0 0 1 1 xn yn 1 x 1 y 1 0 t x 0 1 ty 0 0 1 II-11 Homogeneous Coordinates • The two dimensional point (x, y) is represented by the homogeneous coordinate (x, y, 1) • Some transformations will alter this third component so it is no longer 1 • In general, the homogeneous coordinate (x, y, w) represents the two dimensional point (x/w, y/w) • General transformation matrix: a b 0 T c d 0 m n 1 a b m T c d n 0 0 1 II-12 Order of Transformations • Matrix multiplication is not commutative so changing the order of transformation can change the result • For example, changing the order of a translation and a rotation produces a different result: II-13 II-14 II-15 II-16 • Rotation about an arbitrary point m n through an angle q x * y * 1 x 0 0 cos q 1 y 1 0 1 0 sin q m n 1 0 sin q cos q 0 0 1 0 0 0 0 1 0 1 m n 1 II-17 II-18 II-19 II-20