SRV-26

Traverse latitude and departure

ΔN = L cos θ, ΔE = L sin θ. Whole-circle bearing from north.

Reading speed

TraverseTraverse ΔE ΔN

Governing equation

\Delta N=L\cos\theta,\quad \Delta E=L\sin\theta

$\theta \in \left[0, 360\right]$

where

L: Length (m)
\theta: Azimuth from north (°)
\Delta N: Latitude ΔN (m)
\Delta E: Departure ΔE (m)

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

\Delta N=L\cos\theta,\quad \Delta E=L\sin\theta

$\theta \in \left[0, 360\right]$

$\theta = 0$ $\Delta N = 0$

Along the curve

②

Domain of the variable — blue

\theta_{\min} = 0,\quad \theta_{\max} = 360

$\theta \in \left[0, 360\right]$

$\theta = 0$ $\theta = 0$

Along the curve

③

Response curve — green

\Delta N=L\cos\theta,\quad \Delta E=L\sin\theta

$\theta \in \left[0, 360\right]$

$\theta = 0$ $\Delta N = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta N}{\mathrm{d}\theta}

$\theta \in \left[0, 360\right]$

$\theta = 2.264$ $\Delta N^{\prime} = 2.532$

Along the curve

Lecture brief

Historical brief

From chain and theodolite through Bowditch’s compass-rule (1802) to GPS PDOP, surveying is the propagation of small errors across a network. These sheets close traverses, reduce stadia and state map scale. This sheet (SRV-26 — Traverse latitude and departure) is the form associated with Traverse ΔE ΔN. Working symbols: $L$ , $\theta$ $\rightarrow$ $\Delta N$ , $\Delta E$ . Plane surveying on a projected grid. Azimuth clockwise from north, so easting takes the sine.

Purpose

Purpose: compute

\Delta N

\Delta E

from

L

\theta

in Surveying via

\Delta N=L\cos\theta,\quad \Delta E=L\sin\theta

ΔN = L cos θ, ΔE = L sin θ. Whole-circle bearing from north. Use it when a real surveying question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

L = 145.200\,\mathrm{m}

\theta = 42.000\,\mathrm{^{\circ}}

, the governing relation

\Delta N=L\cos\theta,\quad \Delta E=L\sin\theta

yields

\Delta N = 107.905\,\mathrm{m}

\Delta E = 97.158\,\mathrm{m}

. A course, a bearing, a ΔE and ΔN. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Length L145.2 mAzimuth from north \theta42 °

Outputs

Latitude ΔN \Delta N107.905 m
Departure ΔE \Delta E97.158 m

Reading speed

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Narration of this film

A course, a bearing, a ΔE and ΔN.

Plane surveying on a projected grid. Azimuth clockwise from north, so easting takes the sine.

Reading speed

Watch on YouTube