SRV-08

Plane grid distance

D = √(ΔE² + ΔN²) on a projected grid.

Reading speed

CoordinatesUTMISO 19111

Governing equation

D=\sqrt{(\Delta E)^2+(\Delta N)^2}

$\Delta E \in \left[-5000, 5000\right]$

where

\Delta E: Easting difference (m)
\Delta N: Northing difference (m)
D: Grid distance (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.

00:0 / 00:08

①

Governing law — red

D=\sqrt{(\Delta E)^2+(\Delta N)^2}

$\Delta E \in \left[-5000, 5000\right]$

$\Delta E = -5000$ $D = 5000.6$

Along the curve

②

Domain of the variable — blue

\Delta E_{\min} = -5000,\quad \Delta E_{\max} = 5000

$\Delta E \in \left[-5000, 5000\right]$

$\Delta E = -5000$ $\Delta E = -5000$

Along the curve

③

Response curve — green

D=\sqrt{(\Delta E)^2+(\Delta N)^2}

$\Delta E \in \left[-5000, 5000\right]$

$\Delta E = -5000$ $D = 5000.6$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}D}{\mathrm{d}\Delta E}

$\Delta E \in \left[-5000, 5000\right]$

$\Delta E = -4937.1$ $D^{\prime} = -1$

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-08 — Plane grid distance) is the form associated with UTM · ISO 19111. Working symbols: $\Delta E$ , $\Delta N$ $\rightarrow$ $D$ . On a conformal projection the grid distance is Euclidean in easting and northing; a scale factor later recovers the ellipsoid.

Purpose

Purpose: compute

D

from

\Delta E

\Delta N

in Surveying via

D=\sqrt{(\Delta E)^2+(\Delta N)^2}

D = √(ΔE² + ΔN²) on a projected grid. Use it when a real surveying question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Delta E = 150.000\,\mathrm{m}

\Delta N = 80.000\,\mathrm{m}

, the governing relation

D=\sqrt{(\Delta E)^2+(\Delta N)^2}

yields

D = 170.000\,\mathrm{m}

. No scale factor applied (grid, not ground). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Easting difference \Delta E150 mNorthing difference \Delta N80 m

Outputs

Grid distance D170.000 m

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

No scale factor applied (grid, not ground).

On a conformal projection the grid distance is Euclidean in easting and northing; a scale factor later recovers the ellipsoid.

Reading speed

Watch on YouTube