CAL-12

Plane curvature

κ = |x' y'' − y' x''| / (x'² + y'²)^{3/2}. Turning of a parametric curve.

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CurvesCurvature

Governing equation

\kappa=\dfrac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}

$y'' \in \left[-10, 10\right]$

where

x': x' (—)
y': y' (—)
x'': x'' (—)
y'': y'' (—)
\kappa: Curvature (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\kappa=\dfrac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}

$y'' \in \left[-10, 10\right]$

$y'' = -10$ $\kappa = 10$

Along the curve

②

Domain of the variable — blue

y''_{\min} = -10,\quad y''_{\max} = 10

$y'' \in \left[-10, 10\right]$

$y'' = -10$ $y'' = -10$

Along the curve

③

Response curve — green

\kappa=\dfrac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}

$y'' \in \left[-10, 10\right]$

$y'' = -10$ $\kappa = 10$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\kappa}{\mathrm{d}y''}

$y'' \in \left[-10, 10\right]$

$y'' = -9.874$ $\kappa^{\prime} = -1$

Along the curve

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-12 — Plane curvature) is the form associated with Curvature. Working symbols: $x'$ , $y'$ , $x''$ , $y''$ $\rightarrow$ $\kappa$ . κ = |dθ/ds|. The osculating circle has radius 1/κ.

Purpose

Purpose: compute

\kappa

from

x'

y'

x''

y''

in Calculus via

\kappa=\dfrac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}

κ = |x' y'' − y' x''| / (x'² + y'²)^{3/2}. Turning of a parametric curve. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

x' = 1.000\,\mathrm{—}

y' = 0.000\,\mathrm{—}

x'' = 0.000\,\mathrm{—}

y'' = 1.000\,\mathrm{—}

, the governing relation

\kappa=\dfrac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}

yields

\kappa = 1.00000\,\mathrm{—}

. First and second parametric derivatives at a point. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

x' x'1 —y' y'0 —x'' x''0 —y'' y''1 —

Outputs

Curvature \kappa1.00000 —

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Narration of this film

First and second parametric derivatives at a point.

κ = |dθ/ds|. The osculating circle has radius 1/κ.

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Watch on YouTube