Theorem:

Considerthe equation:  on the complex plane，where z is the complexvariable x+iy，a and b are real，and ，the the equation is equivalent to the rectangular equation of the ellipse:

Pf:

∴，

We get:

Use for linkage:

OACB is a parallelogram with OA=a, OB=b. In order to produce elliptic motion, we must arrange matters so that OA and OB make equal angles at all time with some direction OO'. This may be accomplished by attaching two crossed-parallelograms OO'BB' and OO'DD', as shown. It is obvious that either a or b (or both) may be altered to produce ellipses of different size. Thus every point, P of BCor Q of AC, describes an ellipse.