Definition 1

If point P is not the center O of circle O(r) , the inverse of P with respect to circle O(r) is the point  lying on the line such that

()()=.

Circle O(r) is called the circle of inversion, point O is the center of inversion, r is the radius of inversion.

Definition 2

If r = r(t) is the position function for an object moving along a straight line, then the velocity of the object at time t is

In other words, the velocity function is the derivative of the position function.

Proposition 1

If a smooth curve C is given by the parametric equations x = x(t) and y = y(t), then the velocity of C at (x , y) is

Proof

It is straightforward.