Why is translation a shear in 2D-H/3D-H space? I know that after translation (x,y) should be (x+a, y+b), but I am not so clear with what really happens to shearing...
motoole2
Perhaps it is easier to consider what happens in 1D-H space first. Consider the matrix [1 s; 0 1], which shears vectors [x w] along the horizontal dimension. Applying this shear matrix to any point [x 1] (where w = 1) moves the point along the horizontal dimension to produce [x+s 1]. This is a 1D translation operation along the line defined by w = 1. Similarly, for 2D-H space, the shearing matrix T_b shown in the slide above simply translates any point along the plane defined by w = 1 by an amount (b_x,b_y).
Why is translation a shear in 2D-H/3D-H space? I know that after translation (x,y) should be (x+a, y+b), but I am not so clear with what really happens to shearing...
Perhaps it is easier to consider what happens in 1D-H space first. Consider the matrix
[1 s; 0 1], which shears vectors[x w]along the horizontal dimension. Applying this shear matrix to any point[x 1](wherew = 1) moves the point along the horizontal dimension to produce[x+s 1]. This is a 1D translation operation along the line defined byw = 1. Similarly, for 2D-H space, the shearing matrixT_bshown in the slide above simply translates any point along the plane defined byw = 1by an amount(b_x,b_y).