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SRV-01

Bowditch compass rule

Distribute latitude/departure closures in proportion to length.

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TraversingBowditch 1802

Governing equation

δlat=−L∑L∑lat,δdep=−L∑L∑dep\delta_{\mathrm{lat}}=-\dfrac{L}{\sum L}\sum\mathrm{lat},\quad \delta_{\mathrm{dep}}=-\dfrac{L}{\sum L}\sum\mathrm{dep}

L∈[10,2000]L \in \left[10, 2000\right]

where

L
Side length (m)
\sum L
Perimeter (m)
\sum\mathrm{lat}
Latitude closure (m)
\sum\mathrm{dep}
Departure closure (m)
\delta_{\mathrm{lat}}
Latitude correction (m)
\delta_{\mathrm{dep}}
Departure correction (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.

SRV-01 · curve
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①

Governing law — red

δlat=−L∑L∑lat,δdep=−L∑L∑dep\delta_{\mathrm{lat}}=-\dfrac{L}{\sum L}\sum\mathrm{lat},\quad \delta_{\mathrm{dep}}=-\dfrac{L}{\sum L}\sum\mathrm{dep}

L∈[10,2000]L \in \left[10, 2000\right]

L=10L = 10δlat=−0.005\delta_{\mathrm{lat}} = -0.005

②

Domain of the variable — blue

Lmin⁡=10,Lmax⁡=2000L_{\min} = 10,\quad L_{\max} = 2000

L∈[10,2000]L \in \left[10, 2000\right]

L=10L = 10L=10L = 10

③

Response curve — green

δlat=−L∑L∑lat,δdep=−L∑L∑dep\delta_{\mathrm{lat}}=-\dfrac{L}{\sum L}\sum\mathrm{lat},\quad \delta_{\mathrm{dep}}=-\dfrac{L}{\sum L}\sum\mathrm{dep}

L∈[10,2000]L \in \left[10, 2000\right]

L=10L = 10δlat=−0.005\delta_{\mathrm{lat}} = -0.005

④

Rate of change — amber

dδlatdL\frac{\mathrm{d}\delta_{\mathrm{lat}}}{\mathrm{d}L}

L∈[10,2000]L \in \left[10, 2000\right]

L=22.52L = 22.52δlat′=−5×10−4\delta_{\mathrm{lat}}^{\prime} = -5\times 10^{-4}

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-01 — Bowditch compass rule) is the form associated with Bowditch 1802. Working symbols: LL, ∑L\sum L, ∑lat\sum\mathrm{lat}, ∑dep\sum\mathrm{dep} →\rightarrow δlat\delta_{\mathrm{lat}}, δdep\delta_{\mathrm{dep}}. Bowditch assumed equal precision in angles and distances, so corrections scale with side length: δlat = −(L/ΣL) Σlat.

Purpose

Purpose: compute δlat\delta_{\mathrm{lat}}, δdep\delta_{\mathrm{dep}} from LL, ∑L\sum L, ∑lat\sum\mathrm{lat}, ∑dep\sum\mathrm{dep} in Surveying via δlat=−L∑L∑lat,δdep=−L∑L∑dep\delta_{\mathrm{lat}}=-\dfrac{L}{\sum L}\sum\mathrm{lat},\quad \delta_{\mathrm{dep}}=-\dfrac{L}{\sum L}\sum\mathrm{dep} Distribute latitude/departure closures in proportion to length. 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=120.000 mL = 120.000\,\mathrm{m}, ∑L=800.000 m\sum L = 800.000\,\mathrm{m}, ∑lat=0.400 m\sum\mathrm{lat} = 0.400\,\mathrm{m}, ∑dep=−0.300 m\sum\mathrm{dep} = -0.300\,\mathrm{m}, the governing relation δlat=−L∑L∑lat,δdep=−L∑L∑dep\delta_{\mathrm{lat}}=-\dfrac{L}{\sum L}\sum\mathrm{lat},\quad \delta_{\mathrm{dep}}=-\dfrac{L}{\sum L}\sum\mathrm{dep} yields δlat=−0.0600 m\delta_{\mathrm{lat}} = -0.0600\,\mathrm{m}, δdep=0.0450 m\delta_{\mathrm{dep}} = 0.0450\,\mathrm{m}. One side of a closed traverse. Enter total length and closures. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Latitude correction \delta_{\mathrm{lat}}-0.0600 m
  • Departure correction \delta_{\mathrm{dep}}0.0450 m
Reading speed

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SRV-01 · curve
00:0 / 00:08

Narration of this film

One side of a closed traverse. Enter total length and closures.

Bowditch assumed equal precision in angles and distances, so corrections scale with side length: δlat = −(L/ΣL) Σlat.

Reading speed

Watch on YouTube