Goal: To transform a unit square into a parallelogram in which (a) the diagonals are parallel to the unit square's diagonals, (b) the longest diagonal is equal in length to either of the unit square's diagonals, and (c) the diagonals intersect at the midpoint of the longest diagonal.
Using basic algebra, not trigonometry or matrix format, what is/are the transformation equation(s)?
The matrix for the system is rooted in the basic algebraic form. If you labeled your vertices clockwise starting at the left top in the following way:
For the square:
$$(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$$
For the parallelogram:
$$(x'_1,y'_1),(x'_2,y'_2),(x'_3,y'_3),(x'_4,y'_4)$$
you would get the following system of equations:
$$a x_n + b y_n = x'_n $$ $$c x_n + d y_n = y'_n$$
For each vertex, where $n$ specifies the vertex and $a,b,c,d$ are the coefficients to be solved for. Assuming the transforms are linear.