curve definition

A curve ¹⁾ can be defined as a (finite) number of arcs, combined together.
Most curves on this site are two-dimensional curves, curves that are lying in a plane, so-called planar curves. Also some three-dimensional curves can be found at this site.

Lawrence defines an arc as a valid one when its parametric equation (x,y) = (f(t), g(t)) - with t in an open interval I - obeys the following conditions:

• f and g are twice continuously differentiable
• for all t in I, at least one of df/dt and dg/dt is unequal zero
• for all points s and t in I, (f(s), g(s)) = (f(t), g(t)) if and only if s = t

However, not all curves I describe are differentiable: see e.g. the blancmange curve.

notes

1) In French: courbe.