For many calculations that one might want to perform on a "reasonable" curve (e.g., one coordinatized by functions that are differentiable, or piecewise differentiable/smooth/analytic/nice), taking a finely spaced mesh of points on the curve and computing the result on the polygon joining those points, instead of the curve, will produce an answer close to the one for the curve. The smaller the spacing, the closer the result will be to the result for the curve.
Quantities approximable in this way include area enclosed by a curve, arc length, integrals of given functions along the curve, winding number around a point, splines, parametrizations, and others. Quantities not approximable in this way include curvature, which will be zero on the sides of any polygon used as a substitute for the curve, and integer-valued "global" quantities such as number of tangent lines.