Curves and surfaces

Abstract

If x = x(t), y = y(t), z = z(t), a≤ t ≤ b are the equations of a curve C, then under the usual classroom assumptions the length l(C) of C is given by the formula

$$l\left( C \right) = \int\limits_a^b {{{\left[ {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left({\frac{{dz}}{{dt}}} \right)}^2}} \right]}^{\frac{1}{2}}}dt.} $$

((1.1))

If C reduces to a straight segment of length /, then the formula (1.1) reduces to l = $l = \left( {l_1^2 + l_2^2 + l_3^2} \right)\frac{1}{2}$ where l ₁, l ₂, l ₃ denote the lengths of the orthogonal projections of the segment upon the axes x, y, z (the coordinate system will always be supposed to be rectangular). The formula (1.1) is equally evident geometrically if C is a polygon. It is then clear that for a general curve C the formula results by approximating C by polygons¹. As a matter of fact, (1.1) follows immediately by approximating C by an inscribed polygon.