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AML710 CAD LECTURE 4 Geometric Transformations Two dimensional Transformations Representation of Points and Lines A vertex or point denotes location A point is represented as a position vector In two dimensions as [x y] and in three dimensions [x y z] or alternatively by column vectors as [x y]T and [x y z]T respectively. y P(x,y) x 1 GeometricTransformations • A geometric object is represented by its vertices (as position vectors) A geometric transformation is an operation that modifies its shape, size, position, orientation etc with respect to its current configuration operating on the vertices (position vectors). • Mathematically a transformation P*=L(P) where P* is called the image of P • It can be seen as a mapping from R2 to R2 • Therefore P=L-1(P*), where L-1 is an inverse operator of L Transformation as Matrix multiplication • Given two matrices [A] and [B] find the solution matrix [T] such that [ B] = [ A][T ] • We know that in the above case the solution works out to be: [T ] = [ A]−1[ B] where [A]-1 is the inverse of the square matrix [A] • An alternate way is to see the matrix [T] as a geometric operator and the matrices [A] and [T] are assumed known where matrix [A] contains set of position vectors (vertices) w.r.t to some coordinate system that need to be transformed 2 Geometric Transformation • Consider a 2-D position vector of an arbitrary point as [x y] • Let us take a 2 x 2 matrix [T] (a geometrical operator) as given below for studying the effect of each element on the transformed coordinates of the point [x y] [ X *] = [ X ][T ] [x * y *] = [x ⎡a b ⎤ y ]⎢ = [ax + cy bx + dy ] ⎥ ⎣c d ⎦ Transformation of Points and Lines • • • • Let us consider some typical cases Case 1: a=d=1 and b=c=0 – No Change (identity) Case 2: d=1, b=c=0 – Scaling in x coordinate Case 3: b=c=0 – Scaling in both x and y coordinates [x * y *] = [x [x * y *] = [x [x * y *] = [x ⎡1 y ]⎢ ⎣0 ⎡a y ]⎢ ⎣0 0⎤ = [x y ] 1⎥⎦ 0⎤ = [ax y ] 1⎥⎦ ⎡ a 0⎤ y ]⎢ ⎥ = [ax by ] 0 b ⎣ ⎦ 3 Transformation of Points and Lines • Case 4; a=d =|s|>1 – Enlargement of the original entity • Case 5: 0<a=d=|s|<1 – Compression of the entity [x * y *] = [x ⎡ a 0⎤ y ]⎢ ⎥ = [ax by ] 0 b ⎣ ⎦ • Note that scaling with respect to origin involves translation Transformation of Points and Lines • Case 6: b=c=0, a=1,d=-1 – Reflection about x- axis • Case 7: b=c=0, a=-1,d=1 – Reflection about y- axis • Case 8: b=c=0, a=d<0 – Reflection about the origin ⎡1 0 ⎤ y ]⎢ ⎥ = [x − y ] 0 − 1 ⎣ ⎦ − 1 0⎤ [x * y *] = [x y ]⎡⎢ ⎥ = [− x y ] 0 1 ⎣ ⎦ −1 0 ⎤ [x * y *] = [x y ]⎡⎢ ⎥ = [− x − y ] 0 1 − ⎣ ⎦ [x * y *] = [x 4 Transformation of Points and Lines • Case 9: a=d=1, c=0 – Shear along y • Case 10: a=d=1, b=0 – Shear along x • Case 11: a=d=1- Two-dimensional shear ⎡1 y ]⎢ ⎣0 1 [x * y *] = [x y ]⎡⎢ ⎣c 1 [x * y *] = [x y ]⎡⎢ ⎣c [x * y *] = [x b⎤ = [x (bx + y )] 1⎥⎦ 0⎤ = [( x + cy ) y ] 1⎥⎦ b⎤ = [x + cy bx + y ] 1⎥⎦ Morphing: An Application of Shear Shear results in material deformations of mechanical problems and it is exploited in motion pictures and animated movies. 5