Differential Geometry

A plane curve Γ in a Cartesian coordinate system can be defined by equations in the following form:

Explicitly, y = f (x). (9.1.1.1)

Implicitly, F (x, y) = 0.

Parametrically, x = x(t), y = y(t). (9.1.1.2)

In vector form, r = r(t), where r(t) = x(t)i + y(t)j .