Cesàro equation
From Infogalactic: the planetary knowledge core
In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length (
) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (
) to arc length. (These are equivalent because
.) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro.
Examples
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:
- Line:
.
- Circle:
, where
is the radius.
- Logarithmic spiral:
, where
is a constant.
- Circle involute:
, where
is a constant.
- Cornu spiral:
, where
is a constant.
- Catenary:
.
Related parameterizations
The Cesàro equation of a curve is related to its Whewell equation in the following way. If the Whewell equation is then the Cesàro equation is
.
References
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
External links
- Weisstein, Eric W., "Cesàro Equation", MathWorld.
- Weisstein, Eric W., "Natural Equation", MathWorld.
- Curvature Curves at 2dcurves.com.