TRN-29

Clothoid spiral Ls = A² / R

A² = R Ls, θs = Ls /(2R). Curvature grows linearly from 0 to 1/R.

Reading speed

Geometric designClothoid

Governing equation

A^2=R L_s,\quad \theta_s=L_s/(2R)

$L_s \in \left[20, 250\right]$

where

R: Circular radius (m)
L_s: Spiral length (m)
A: Clothoid parameter (m)
\theta_s: Spiral angle (°)

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

A^2=R L_s,\quad \theta_s=L_s/(2R)

$L_s \in \left[20, 250\right]$

$L_s = 20$ $A = 89.44$

Along the curve

②

Domain of the variable — blue

L_s_{\min} = 20,\quad L_s_{\max} = 250

$L_s \in \left[20, 250\right]$

$L_s = 20$ $L_s = 20$

Along the curve

③

Response curve — green

A^2=R L_s,\quad \theta_s=L_s/(2R)

$L_s \in \left[20, 250\right]$

$L_s = 20$ $A = 89.44$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}A}{\mathrm{d}L_s}

$L_s \in \left[20, 250\right]$

$L_s = 21.45$ $A^{\prime} = 2.161$

Along the curve

Lecture brief

Historical brief

Highway and traffic theory crystallized with Greenshields’ 1935 speed–density line, then AASHTO stopping sight and equivalent axles. The lab is that operational arithmetic of flow, braking and pavement load. This sheet (TRN-29 — Clothoid spiral Ls = A² / R) is the form associated with Clothoid. Working symbols: $R$ , $L_s$ $\rightarrow$ $A$ , $\theta_s$ . Euler's spiral. Lateral jerk is limited by picking Ls from v³/(C R) with C a comfort rate.

Purpose

Purpose: compute

A

\theta_s

from

R

L_s

in Transport via

A^2=R L_s,\quad \theta_s=L_s/(2R)

A² = R Ls, θs = Ls /(2R). Curvature grows linearly from 0 to 1/R. Use it when a real transport question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 400.000\,\mathrm{m}

L_s = 80.000\,\mathrm{m}

, the governing relation

A^2=R L_s,\quad \theta_s=L_s/(2R)

yields

A = 178.9\,\mathrm{m}

\theta_s = 5.73\,\mathrm{^{\circ}}

. A tangent, a clothoid, a circular arc. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Circular radius R400 mSpiral length L_s80 m

Outputs

Clothoid parameter A178.9 m
Spiral angle \theta_s5.73 °

Reading speed

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Narration of this film

A tangent, a clothoid, a circular arc.

Euler's spiral. Lateral jerk is limited by picking Ls from v³/(C R) with C a comfort rate.

Reading speed

Watch on YouTube