View source for Natural parameter

''on a rectifiable curve''

A parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661001.png" /> for a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661002.png" /> with parametric representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661003.png" /> such that the arc length on the curve between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661005.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661006.png" />. The parametrization of a curve by the natural parameter is known as its natural parametrization. The natural parametrization of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661007.png" />-times differentiable (analytic) curve with no singular points is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066100/n0661008.png" /> times differentiable (analytic).



====Comments====
See also (the references to) [[Natural equation|Natural equation]].