Shear matrix simple explanation

Question

I can understand translation, dilation and rotation matrices, but the shear one is still obscure to me (despite understanding what shearing means graphically).

This is the matrix:

1 -cos($)/sin($) 0
0    1/sin($)    0
0       0        1

Can you explain, in simple terms (and maybe graphically), the meaning of the sin and cos functions? As far as I understand there is some line with slope $ on which the points to be transformed are 'projected' (to obtain the shear effect).

Answer 1

For sake of having an answer, what the OP describes is not a shear matrix, but the product of a shear matrix and a scaler matrix: $$ \pmatrix{1&-\cos(t)&0\\ 0&1&0\\ 0&0&1}\pmatrix{1\\ &\frac1{\sin(t)}\\ &&1}. $$ The diagonal matrix first performs scaling in the direction of the $y$-axis by a factor $\frac1{\sin(t)}$. Then the shear matrix performs a shear parallel to the $x$-axis that maps every point $(x,y,z)^T$ to $\left(x-y\cos(t),\,y,\,z\right)^T$.