Geometric Transformations Two dimensional Transformations AML CAD LECTURE Representation - PDF document

1 AML710 CAD LECTURE 4 dimensions [x y z] or alternatively by column vectors as [x y] y]Trespectively. x 2 •A geometric object is represposition vectors) •Mathematically a transformation P*=L(P) where P* is called •It can be seen as a mapping from R•Therefore P=L-1 is an inverse operator of L •Given two matrices [A] and [B] find the solution matrix [T] •We know that in the above case the solution works out to e know that in the above case the solution works out to -1is the inverse of the square matrix [A]•An alternate way is to see the matrix [T] as a geometric and the matrices [A] and [T] are assumed known n of position vectors (vertices) w.r.tto some coordinate system that need to be transformed 3 •Consider a 2-D position vector onsider a 2-D position vector •Let us take a 2 x 2 matrix [Tgiven below for studying the eftransformed coordinates of the point [x y] [][][] Transformation of Points and Lines •Let us consider some typical cases•Case 1: a=d=1 and b=c=0 –•Case 2: d=1, b=c=0 –•Case 3: b=c=0 –[][][][][][][][][] 4 Transformation of Points and Lines•Cas�e 4; a=d =|s|1 –•Case 5: 0|1 –Compression of the entity•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[][][][][][][][][] 5 Transformation of Points and Lines•Case 9: a=d=1, c=0 –Shear along y•Case 10: a=d=1, b=0 –•Case 11: a=d=1-Two-dimensional shear[][][][][][][][][] Shear results in material deformations of mechanical problems and it is exploited in motion pictures and animated movies.