GMY-29

Haversine (small)

Great-circle from hav.

Reading speed

GoverningHaversine (small)

Governing equation

d=2R\\arcsin\\sqrt{\\mathrm{hav}}

$hav \in \left[0, 1\right]$

where

hav: hav (—)
R: R (km)
d: Haversine (small) (km)

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

d=2R\\arcsin\\sqrt{\\mathrm{hav}}

$hav \in \left[0, 1\right]$

$hav = 0$ $d = 0$

Along the curve

②

Domain of the variable — blue

hav_{\min} = 0,\quad hav_{\max} = 1

$hav \in \left[0, 1\right]$

$hav = 0$ $hav = 0$

Along the curve

③

Response curve — green

d=2R\\arcsin\\sqrt{\\mathrm{hav}}

$hav \in \left[0, 1\right]$

$hav = 0$ $d = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}d}{\mathrm{d}hav}

$hav \in \left[0, 1\right]$

$hav = 0.006$ $d^{\prime} = 1.14\times 10^{5}$

Along the curve

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-29 — Haversine (small)) is the form associated with Haversine (small). Working symbols: $hav$ , $R$ $\rightarrow$ $d$ . Great-circle from hav. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

d

from

hav

R

in Geometry via

d=2R\\arcsin\\sqrt{\\mathrm{hav}}

Great-circle from hav. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

hav = 0.020\,\mathrm{—}

R = 6371.000\,\mathrm{km}

, the governing relation

d=2R\\arcsin\\sqrt{\\mathrm{hav}}

yields

d = 1808.052\,\mathrm{km}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

hav hav0.02 —R R6371 km

Outputs

Haversine (small) d1808.052 km

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Great-circle from hav. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube