Definition:Intrinsic Equation
Definition
Let $C$ be a curve.
An intrinsic equation of $C$ is an equation that defines $C$ using a relation between the intrinsic properties of $C$.
Thus an intrinsic equation defines the shape of $C$ without specifying its position relative to an arbitrarily defined coordinate system.
Natural Equation
A natural equation is an intrinsic equation for a space curve $C$ such that $C$ is expressed in terms of its intrinsic properties:
Cesàro Equation
A Cesàro equation is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties in one of either of the following two forms:
Formulation 1
A Cesàro equation is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties:
- Arc length $s$
- Curvature $\kappa$
Formulation 2
A Cesàro equation is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties:
- Arc length $s$
- Radius of curvature $\rho$
Whewell Equation
A Whewell equation is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties:
- Arc length $s$
- Turning angle $\psi$
Also known as
Some sources refer to the general intrinsic equation as a natural equation.
However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve this term for the specific example where the intrinsic properties used are curvature and torsion.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): intrinsic equation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): intrinsic equation
- Weisstein, Eric W. "Natural Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalEquation.html