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
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
1) In French: courbe.