# Equivalence of Definitions of Curvature

## Theorem

The following definitions of the concept of **Curvature** are equivalent:

### Whewell Form

The **curvature** $\kappa$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as:

- $\kappa = \dfrac {\d \psi} {\d s}$

where:

- $\psi$ is the turning angle of $C$
- $s$ is the arc length of $C$.

### Cartesian Form

Let $C$ be embedded in a cartesian plane.

The **curvature** $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by:

- $\kappa = \dfrac {y''} {\paren {1 + y'^2}^{3/2} }$

where:

- $y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ with respect to $x$ at $P$
- $y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ with respect to $x$ at $P$.

### Parametric Form

#### Cartesian Coordinates

Let $C$ be embedded in a cartesian plane and defined by the parametric equations:

- $\begin{cases} x = \map x t \\ y = \map y t \end{cases}$

The **curvature** $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by:

- $\kappa = \dfrac {x' y'' - y' x''} {\tuple {x'^2 + y'^2}^{3/2} }$

where:

- $x' = \dfrac {\d x} {\d t}$ is the derivative of $x$ with respect to $t$ at $P$
- $y' = \dfrac {\d y} {\d t}$ is the derivative of $y$ with respect to $t$ at $P$
- $x''$ and $y''$ are the second derivatives of $x$ and $y$ with respect to $t$ at $P$.

#### Polar Coordinates

Let $C$ be embedded in a polar plane and defined by the parametric equations:

- $\begin{cases} r = \map r t \\ \theta = \map \theta t \end{cases}$

The **curvature** $\kappa$ of $C$ at a point $P = \polar {r, \theta}$ is given by:

- $\kappa = \dfrac {2 r'^2 \theta' + r r'' \theta' + r r' \theta'' + r^2 \theta'^3} {\paren {r'^2 + \paren {r \theta'}^2}^{1/2} }$

where:

- $r' = \dfrac {\d r} {\d t}$ is the derivative of $r$ with respect to $t$ at $P$
- $\theta' = \dfrac {\d \theta} {\d t}$ is the derivative of $\theta$ with respect to $t$ at $P$
- $r''$ and $\theta''$ are the second derivatives of $r$ and $y$ with respect to $t$ at $P$.

## Proof

### Whewell Form to Cartesian Form

Consider the curvature of a curve $C$ at a point $P$ expressed as a Whewell equation:

- $\kappa = \dfrac {\d \psi} {\d s}$

where:

- $\psi$ is the turning angle of $C$
- $s$ is the arc length of $C$.

The derivative of the tangent of the turning angle $\psi$ at a point $P$ with respect to $\psi$ is also the derivative of the tangent to $C$ at $P$, again with respect to $\psi$.

That is:

\(\ds \frac {\d} {\d \psi} \tan \psi\) | \(=\) | \(\ds \frac {\d} {\d \psi} \frac {\d y} {\d x}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \sec^2 \psi\) | \(=\) | \(\ds \frac {\d y'} {\d \psi}\) | Derivative of Tangent Function | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds 1 + \tan^2 \psi\) | \(=\) | \(\ds \frac {\d y'} {\d \psi}\) | Difference of Squares of Secant and Tangent | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds 1 + y'^2\) | \(=\) | \(\ds \frac {\d y'} {\d \psi}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \frac 1 {1 + y'^2}\) | \(=\) | \(\ds \frac {\d \psi} {\d y'}\) |

This article, or a section of it, needs explaining.In particular: Clarify the relationship between the tangent and turning angle. The latter is still inadequately defined.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

We also have that:

\(\ds \d s\) | \(=\) | \(\ds \sqrt {\d x^2 + \d y^2}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \frac {\d s} {\d x}\) | \(=\) | \(\ds \sqrt {\paren {\frac {\d x} {\d x} }^2 + \paren {\frac {\d y} {\d x} }^2}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \sqrt {1 + y'^2}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d s}\) | \(=\) | \(\ds \frac 1 {\paren {1 + y'^2}^{1/2} }\) |

Then:

\(\ds \kappa\) | \(=\) | \(\ds \dfrac {\d \psi} {\d s}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {\d \psi} {\d y'} \dfrac {\d y'} {\d x} \dfrac {\d x} {\d s}\) | Chain Rule for Derivatives | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 {1 + y'^2} y'' \frac 1 {\paren {1 + y'^2}^{1/2} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {y''} {\paren {1 + y'^2}^{3/2} }\) |

which is the Cartesian form of curvature as required.

$\blacksquare$

### Whewell Form to Parametric Polar Form

Consider the curvature of a curve $C$ at a point $P$ expressed as a Whewell equation:

- $\kappa = \dfrac {\d \psi} {\d s}$

where:

- $\psi$ is the turning angle of $C$
- $s$ is the arc length of $C$.

Let us consider $C$ expressed in cartesian form:

\(\ds x\) | \(=\) | \(\ds r \cos \theta\) | ||||||||||||

\(\ds y\) | \(=\) | \(\ds r \sin \theta\) |

Then:

\(\ds \dfrac {\d x} {\d t}\) | \(=\) | \(\ds r' \cos \theta - r \theta' \sin \theta\) | Chain Rule for Derivatives, Derivative of Cosine Function | |||||||||||

\(\ds \dfrac {\d y} {\d t}\) | \(=\) | \(\ds r' \sin \theta + r \theta' \cos \theta\) | Chain Rule for Derivatives, Derivative of Sine Function |

In Whewell form:

\(\ds \tan \psi\) | \(=\) | \(\ds \dfrac {y'} {x'}\) | ||||||||||||

\(\ds s^2\) | \(=\) | \(\ds x'^2 + y'^2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds r^2 \theta'^2 + r'2\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map \kappa t\) | \(=\) | \(\ds \dfrac {\map \arctan {\dfrac {y'} {x'} }' } {s'}\) |

Let:

\(\ds g\) | \(=\) | \(\ds \dfrac {y'} {x'}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \tan \psi\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \dfrac \d {\d \psi}\) | \(=\) | \(\ds \dfrac {\d g} {\d \psi}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d \psi} {\d g}\) | \(=\) | \(\ds \dfrac 1 {1 + g^2}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {x'^2} {x'^2 + y'^2}\) |

We have:

- $\map \kappa t = \dfrac {\d \psi} {\d g} \dfrac {\d g} {\d t} \dfrac {\d t} {\d s}$

Then:

\(\ds \dfrac {\d t} {\d s}\) | \(=\) | \(\ds \dfrac 1 {\paren {x'^2 + y'^2}^{1/2} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac 1 {\paren {r^2 \theta'^2 + r'2}^{1/2} }\) |

\(\ds \dfrac {\d g} {\d t}\) | \(=\) | \(\ds \map {\dfrac \d {\d t} } {\dfrac {y'} {x'} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {\dfrac \d {\d t} } {\dfrac {r' \sin \theta + r \theta' \cos \theta} {r' \cos \theta - r \theta' \sin \theta} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {x' y'' - y' x''} {\paren {x'}^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {2 r'^2 \theta' + r r' \theta'' - r r'' \theta' + r^2 \theta'^3} {\paren {x'}^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {r' \paren {r \theta'}' - r r'' \theta' + \paren {r'^2 + \paren {r \theta'}^2} \theta'} {\paren {x'}^2}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map \kappa t\) | \(=\) | \(\ds \dfrac {x'^2} {s'^2} \cdot \dfrac {2 r'^2 \theta' + r r' \theta'' - r r'' \theta' + r^2 \theta'^3} {\paren {x'}^2} \cdot \dfrac 1 {\paren {r'^2 + \paren {r \theta'}^2}^{1/2} }\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {2 r'^2 \theta' + r r'' \theta' + r r' \theta'' + r^2 \theta'^3} {\paren {r'^2 + \paren {r \theta'}^2}^{1/2} }\) |

$\blacksquare$

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