This Demonstration shows a quadrilateral-to-quadrilateral dissection. Each quadrilateral is dissected to its quad elements, which lie in a strip. The two strips intersect in a parallelogram and determine the parts of the dissection. The parameter is the length of a side of the parallelogram, is the angle at basis, and determines the length of the other side. The initial stage of the dissection corresponds to both the "translate" and "rotate" sliders being set to 1.
To get a quad element of a quadrilateral, a diagonal is chosen; it divides the quadrilateral into two triangles. Two lines parallel to the diagonal are drawn to cut off a quarter of each triangle, leaving two small triangles and two trapezoids. The triangular pieces are rotated and together with trapezoids form the quad element, which can be translated along the diagonal in a parallel strip (P-strip).