Geographic and Astronomical Positioning | Malone Math

The Central Idea

Large-scale positioning depends on a reference surface, a coordinate system, and a method for connecting angular or distance measurements to location.

A simplified pattern is

\[ \boxed{ \text{reference frame} + \text{angles and distances} \longrightarrow \text{position} }. \]

On Earth, the reference is commonly geographic. In astronomy, the reference may be tied to Earth's rotation, the celestial sphere, or another observation frame.

Why Flat Coordinates Are Not Always Enough

A small local region can often be approximated by a plane.

Over large distances, however, Earth's curvature becomes important.

Similarly, celestial directions are naturally described using angles on a sphere surrounding the observer.

This makes spherical and angular coordinate systems especially useful.

Latitude

Latitude describes angular position north or south of the equator.

The equator has latitude

\[ 0^\circ. \]

The North Pole is at

\[ 90^\circ\text{ N}, \]

and the South Pole is at

\[ 90^\circ\text{ S}. \]

Longitude

Longitude describes angular position east or west of a chosen prime meridian.

Longitude is commonly measured from

\[ 0^\circ \]

at the prime meridian toward

\[ 180^\circ \]

east or west.

Geographic Coordinates

A geographic position can be written as a pair

\[ \boxed{(\phi,\lambda)}, \]

where

\[ \phi=\text{latitude} \]

and

\[ \lambda=\text{longitude}. \]

Elevation or height may be added when three-dimensional position is needed.

Signed Angle Convention

One convenient mathematical convention uses signed coordinates.

Latitude may be represented as positive north of the equator and negative south:

\[ -90^\circ\le\phi\le90^\circ. \]

Longitude may be represented as positive east and negative west:

\[ -180^\circ<\lambda\le180^\circ. \]

Other conventions exist, so the coordinate convention should always be identified.

Degrees, Minutes, and Seconds

Angular coordinates may be written in degrees, minutes, and seconds.

One degree contains

\[ 60 \]

arcminutes:

\[ 1^\circ=60'. \]

One arcminute contains

\[ 60 \]

arcseconds:

\[ 1'=60''. \]

Converting DMS to Decimal Degrees

If an angle is

\[ D^\circ M' S'', \]

its decimal-degree magnitude is

\[ \boxed{ D+\frac{M}{60}+\frac{S}{3600} }. \]

The appropriate sign is then attached according to north, south, east, or west.

Example: DMS to Decimal Degrees

Convert

\[ 47^\circ30'00'' \]

to decimal degrees.

\[ 47+\frac{30}{60}+\frac{0}{3600} = 47.5. \]

Therefore,

\[ \boxed{47.5^\circ}. \]

Converting Decimal Degrees to DMS

Suppose an angle is

\[ 122.375^\circ. \]

The whole-degree part is

\[ 122^\circ. \]

Convert the decimal remainder:

\[ 0.375(60)=22.5'. \]

Therefore,

\[ 0.5(60)=30''. \]

Thus,

\[ \boxed{122^\circ22'30''}. \]

Earth as a Sphere: A First Model

Earth is not a perfect sphere, but a spherical model provides a useful first approximation for many large-scale geometric ideas.

In the spherical model, Earth's surface is represented by

\[ \boxed{x^2+y^2+z^2=R^2}, \]

where \(R\) is an assumed Earth radius.

Spherical Coordinates

A point can be located by:

distance \(r\) from the origin;
an angular position around an axis;
an angular position above or below a reference plane.

Different fields use different symbols and angle conventions, so the definitions must be stated explicitly.

From Latitude and Longitude to Cartesian Coordinates

In a simple spherical Earth model of radius \(R\), let latitude be \(\phi\) and longitude be \(\lambda\).

Then an Earth-centered coordinate representation is

\[ \boxed{ x=R\cos\phi\cos\lambda }, \] \[ \boxed{ y=R\cos\phi\sin\lambda }, \] \[ \boxed{ z=R\sin\phi }. \]

Example: Point on the Equator

Suppose

\[ \phi=0^\circ, \qquad \lambda=0^\circ. \]

Then

\[ x=R, \] \[ y=0, \] \[ z=0. \]

Therefore, the point is

\[ \boxed{(R,0,0)}. \]

Example: North Pole

At the North Pole,

\[ \phi=90^\circ. \]

Since

\[ \cos90^\circ=0 \]

and

\[ \sin90^\circ=1, \]

the coordinates become

\[ \boxed{(0,0,R)}. \]

Great Circles

A great circle is the intersection of a sphere with a plane passing through the sphere's center.

Great circles play the spherical role that straight lines play in a plane.

The shortest surface path between two non-antipodal points on a sphere lies along the shorter arc of a great circle.

Central Angle and Surface Distance

Suppose two points on a sphere of radius \(R\) subtend central angle \(\theta\) radians.

The great-circle arc length is

\[ \boxed{s=R\theta}. \]

This is the same arc-length relationship used for circles, applied to a great circle on the sphere.

Example: Quarter-Circle Surface Distance

If two points are separated by a central angle of

\[ 90^\circ=\frac{\pi}{2} \]

radians on a sphere of radius \(R\), then

\[ s=R\frac{\pi}{2}. \]

Therefore,

\[ \boxed{s=\frac{\pi R}{2}}. \]

Central Angle from Latitude and Longitude

For two spherical positions

\[ (\phi_1,\lambda_1) \]

and

\[ (\phi_2,\lambda_2), \]

the spherical law of cosines gives central angle \(\theta\):

\[ \boxed{ \cos\theta = \sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2 \cos(\lambda_2-\lambda_1) }. \]

Great-Circle Distance

Once the central angle is known,

\[ \boxed{d=R\theta}. \]

This gives a spherical surface-distance approximation.

Practical geodesy uses more accurate Earth models when higher precision is required.

Same Longitude

If two points share the same longitude, their north-south angular separation in the spherical model is the difference in latitude:

\[ \boxed{ \theta=|\phi_2-\phi_1| }. \]

Convert \(\theta\) to radians before using

\[ d=R\theta. \]

Same Latitude Is Different

Two points at the same latitude generally do not lie on a great circle, except at the equator.

Their east-west separation along the latitude circle becomes smaller as the latitude approaches either pole.

In a spherical model, the radius of a latitude circle is

\[ \boxed{R\cos\phi}. \]

East-West Distance Along a Parallel

At latitude \(\phi\), a longitude difference of

\[ \Delta\lambda \]

radians corresponds to parallel-circle arc length

\[ \boxed{ s=R\cos\phi\,|\Delta\lambda| }. \]

This is not generally the shortest surface route between the two points, but it measures distance along that latitude circle.

Elevation and Height

Horizontal geographic position is often supplemented by a vertical coordinate.

A location may therefore be represented conceptually as

\[ \boxed{ (\text{latitude},\text{longitude},\text{height}) }. \]

The meaning of height depends on the chosen vertical reference.

Reference Surfaces and Datums

Coordinates require a reference system.

In geographic work, a datum defines how coordinates relate to a mathematical Earth model and physical reference framework.

Two coordinate pairs that look numerically similar can refer to slightly different physical positions if they use different reference systems.

Ellipsoidal Earth Models

A more realistic global model treats Earth approximately as a flattened ellipsoid rather than a perfect sphere.

One common ellipsoid form is

\[ \boxed{ \frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{b^2} = 1 }, \]

where \(a\) is an equatorial semiaxis and \(b\) is a polar semiaxis.

Why the Ellipsoid Matters

A spherical model is often adequate for conceptual reasoning.

Higher-precision positioning requires a more accurate representation of Earth's shape.

This changes the detailed relationship between latitude, longitude, Cartesian coordinates, and surface distance.

Map Coordinates and Geographic Coordinates

Geographic coordinates are angular.

Maps often use planar coordinates such as

\[ (x,y) \]

after applying a map projection.

Therefore,

\[ \boxed{ (\phi,\lambda) \longrightarrow (x,y) } \]

requires a projection rule.

Projection Distortion

A curved surface cannot generally be flattened without distortion.

A map projection may distort some combination of:

distance;
area;
shape;
direction;
scale.

Geographic positioning calculations should account for the intended projection properties when precision matters.

Geographic Distance from Cartesian Coordinates

If two positions are represented in a common three-dimensional Cartesian frame,

\[ \mathbf{p}_1=(x_1,y_1,z_1) \]

and

\[ \mathbf{p}_2=(x_2,y_2,z_2), \]

their straight-line chord distance is

\[ \boxed{ d_c= \sqrt{ (x_2-x_1)^2 +(y_2-y_1)^2 +(z_2-z_1)^2 } }. \]

Chord distance differs from surface distance.

Chord Distance Versus Surface Distance

Two points on a sphere can be connected by:

a straight chord through three-dimensional space;
a curved surface path along the sphere.

For distinct non-antipodal surface points, the chord is shorter than the corresponding great-circle arc.

Triangulation in Geographic Positioning

A position can be inferred from measured angles to known reference points.

In a local planar approximation,

\[ \boxed{ \text{known baseline} + \text{measured angles} \longrightarrow \text{unknown position} }. \]

This is the same geometric principle used in surveying.

Trilateration in Positioning Systems

Trilateration determines position from distances to known reference points.

In two dimensions, a distance from known point \((a,b)\) gives a circle:

\[ \boxed{ (x-a)^2+(y-b)^2=r^2 }. \]

Multiple distance constraints can determine the unknown position.

Three-Dimensional Trilateration

In three dimensions, a measured distance from a known reference point

\[ (a,b,c) \]

defines a sphere:

\[ \boxed{ (x-a)^2+(y-b)^2+(z-c)^2=r^2 }. \]

Several such constraints can be combined to locate an unknown point in space.

Position from Time-of-Travel Measurements

If a signal travels at known speed \(v\) for measured time \(t\), then the inferred travel distance is

\[ \boxed{d=vt}. \]

Positioning systems can convert signal travel times into distance-like constraints and then combine those constraints geometrically.

Clock Error and Distance Error

If a travel-time measurement has error

\[ \Delta t, \]

then the associated distance error is approximately

\[ \boxed{\Delta d=v\Delta t}. \]

When signals travel very quickly, even small timing errors can correspond to substantial distance errors.

Local Tangent Coordinates

Near a chosen geographic location, a curved Earth surface can often be approximated by a local tangent plane.

Local coordinates may be organized as:

east;
north;
up.

This produces a convenient local three-dimensional frame for engineering, surveying, robotics, and navigation.

Earth-Centered and Local Frames

A global Earth-centered coordinate can be transformed into a local east-north-up frame around a selected reference location.

This is another example of a coordinate transformation:

\[ \boxed{ \text{global frame} \longrightarrow \text{local frame} }. \]

The underlying point is unchanged; only its coordinate description changes.

Astronomical Position Is Often Directional

Many astronomical observations do not immediately provide a full three-dimensional position.

Instead, they provide a direction in the sky.

This direction can be represented by two angular coordinates on an imagined celestial sphere surrounding the observer.

The Celestial Sphere

The celestial sphere is a conceptual sphere used to organize directions to astronomical objects.

Objects at very different physical distances can appear at nearby angular positions on this sphere.

The celestial sphere therefore describes apparent direction, not necessarily physical distance.

Altitude and Azimuth

A local sky position can be described by altitude and azimuth.

Altitude measures angular height above the horizon.

Azimuth measures direction around the horizon relative to a chosen north convention.

Together they specify a direction from the observer.

Altitude

The horizon has altitude

\[ 0^\circ. \]

The point directly overhead, the zenith, has altitude

\[ \boxed{90^\circ}. \]

Objects below the geometric horizon have negative altitude in this angular convention.

Zenith Angle

The zenith angle \(z\) is measured from the upward vertical direction.

If altitude is \(h\), then

\[ \boxed{ z=90^\circ-h }. \]

Thus, an object at altitude \(30^\circ\) has zenith angle

\[ 60^\circ. \]

Equatorial Celestial Coordinates

Another astronomical coordinate system uses declination and right ascension.

Declination behaves similarly to celestial latitude.

Right ascension behaves similarly to celestial longitude, though it is commonly expressed using angular time units.

Declination

Declination measures angular position north or south of the celestial equator.

Its range is

\[ \boxed{ -90^\circ\le\delta\le90^\circ }. \]

Right Ascension

Right ascension is commonly expressed in hours, minutes, and seconds.

A full circle corresponds to

\[ 24\text{ h}. \]

Therefore,

\[ \boxed{ 1\text{ h}=15^\circ }. \]

Example: Right Ascension to Degrees

Convert

\[ 6\text{ h} \]

of right ascension to degrees.

\[ 6(15^\circ)=90^\circ. \]

Therefore,

\[ \boxed{90^\circ}. \]

Angular Separation

Two directions in the sky can be compared by the angle between their unit direction vectors.

\[ \hat{\mathbf{u}} \]

and

\[ \hat{\mathbf{v}} \]

are unit vectors, then

\[ \boxed{ \hat{\mathbf{u}}\cdot\hat{\mathbf{v}} = \cos\theta }. \]

Therefore,

\[ \boxed{ \theta = \cos^{-1} \left( \hat{\mathbf{u}}\cdot\hat{\mathbf{v}} \right) }. \]

Angular Size

An object's angular size describes how large it appears from the observer.

For a small object of physical width \(D\) at distance \(r\), a small-angle approximation gives

\[ \boxed{ \theta\approx\frac{D}{r} } \]

when \(\theta\) is measured in radians and \(D\ll r\).

Example: Small-Angle Approximation

An object is

\[ 2\text{ m} \]

wide and

\[ 100\text{ m} \]

away.

Its approximate angular size is

\[ \theta\approx\frac{2}{100}=0.02\text{ rad}. \]

Therefore,

\[ \boxed{\theta\approx0.02\text{ rad}}. \]

Parallax

Parallax is the apparent change in an object's direction when observed from different positions.

A nearby object generally shows a larger parallax shift than a distant object for the same observing baseline.

This converts a geometric baseline and angular change into distance information.

A Simple Parallax Model

In a small-angle approximation, if baseline \(b\) subtends parallax angle \(p\), then distance \(d\) is approximately

\[ \boxed{ d\approx\frac{b}{p} } \]

when \(p\) is measured in radians and the geometry supports the approximation.

Why Baseline Matters

A longer baseline produces a larger angular shift for the same distant object.

Therefore, larger baselines can improve geometric sensitivity to distance.

This principle appears in both terrestrial triangulation and astronomical parallax.

Line of Sight

A line of sight is the ray from an observer toward an observed target.

If observer position is

\[ \mathbf{o} \]

and unit viewing direction is

\[ \hat{\mathbf{d}}, \]

then points along the sight line can be written

\[ \boxed{ \mathbf{r}(t) = \mathbf{o} + t\hat{\mathbf{d}}, \qquad t\ge0 }. \]

Intersecting Lines of Sight

Observations from different positions can produce different sight lines.

In an ideal geometric model, the target lies where the lines intersect.

With measurement error, the lines may fail to intersect exactly, so a best approximate position may be required.

Time Changes Astronomical Coordinates

The apparent sky changes with time because the observer and Earth are moving.

A coordinate system tied to the local horizon therefore depends on:

observer location;
date;
time;
reference convention.

A celestial direction is meaningful only relative to its reference frame and epoch or observation time.

Relative Position in Space

If two objects have position vectors

\[ \mathbf{r}_1 \]

and

\[ \mathbf{r}_2, \]

then the position of object \(2\) relative to object \(1\) is

\[ \boxed{ \mathbf{r}_{2/1} = \mathbf{r}_2-\mathbf{r}_1 }. \]

The same vector idea applies at both terrestrial and astronomical scales.

Three-Dimensional Distance

The Euclidean distance between two Cartesian positions is

\[ \boxed{ d = \sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2 } }. \]

This is appropriate when both positions are represented in the same Cartesian frame and straight-line distance is desired.

Coordinate Frames Must Match

Distances and vector differences should not be computed directly from coordinates expressed in incompatible frames.

First transform the quantities into a common reference system.

Then perform the spatial calculation.

Common Error: Treating Degrees as Linear Distance

Latitude and longitude are angular quantities.

A difference of

\[ 1^\circ \]

is not automatically a fixed linear distance in every direction and at every latitude.

The geometry of the reference surface matters.

Common Error: Using Degrees in an Arc-Length Formula

The formula

\[ s=R\theta \]

requires \(\theta\) in radians.

If \(\theta\) is given in degrees, convert first:

\[ \boxed{ \theta_{\text{rad}} = \theta_{\text{deg}}\frac{\pi}{180^\circ} }. \]

Common Error: Confusing Surface Distance with Straight-Line Distance

Great-circle distance follows the spherical surface.

Chord distance passes through three-dimensional space.

These answer different geometric questions.

Common Error: Ignoring the Coordinate Datum

A coordinate value is tied to a reference system.

Combining positions from different datums or frames without transformation can introduce location error.

Common Error: Treating a Flat Map as the Curved Earth

A map projection converts curved geometry into a plane.

Measurements taken from the map inherit the properties and distortions of that projection.

Large-scale geographic calculations should not assume all planar map relationships are exact on Earth.

Common Error: Confusing Apparent Direction with Physical Distance

Two stars can appear close together in the sky while being separated by a large physical distance.

Celestial angular coordinates describe viewing direction unless additional distance information is supplied.

Common Error: Ignoring Observation Time

Local astronomical coordinates can change as Earth rotates.

A sky direction specified without the relevant time and reference convention may be incomplete.

A Reliable Procedure for Geographic Position Problems

Identify the coordinate reference system.
Determine whether coordinates are angular, planar, or Cartesian.
Convert units and angle formats when necessary.
Choose a spherical, ellipsoidal, or local planar model appropriate to the problem.
Distinguish surface distance from straight-line distance.
Apply the relevant geometric relationship.
Check whether a map projection or datum transformation is required.
Verify that the result has the correct units and scale.

A Reliable Procedure for Astronomical Position Problems

Identify the observation frame.
Identify the angular coordinates being used.
Convert hours, degrees, minutes, seconds, or radians as needed.
Represent sight directions with angles or unit vectors.
Use angular separation, parallax, or line-of-sight geometry as appropriate.
Include observer location and time when the coordinate system requires them.
Distinguish apparent angular position from physical three-dimensional location.

Example: Full Geographic Workflow

Two points lie on the same longitude in a spherical Earth model of radius

\[ R=6371\text{ km}. \]

Their latitudes are

\[ 20^\circ \]

and

\[ 35^\circ. \]

Their central angle is

\[ \theta=15^\circ. \]

Convert to radians:

\[ \theta = 15^\circ \frac{\pi}{180^\circ} = \frac{\pi}{12}. \]

Therefore,

\[ d = 6371\frac{\pi}{12} \approx1667.9\text{ km}. \]

Thus, under the spherical model,

\[ \boxed{d\approx1668\text{ km}}. \]

Example: Full Astronomical Workflow

Two unit direction vectors are

\[ \hat{\mathbf{u}} = \langle1,0,0\rangle \]

and

\[ \hat{\mathbf{v}} = \left\langle \frac{\sqrt2}{2}, \frac{\sqrt2}{2}, 0 \right\rangle. \]

Their dot product is

\[ \hat{\mathbf{u}}\cdot\hat{\mathbf{v}} = \frac{\sqrt2}{2}. \]

Therefore,

\[ \theta = \cos^{-1} \left( \frac{\sqrt2}{2} \right) = 45^\circ. \]

Thus their angular separation is

\[ \boxed{45^\circ}. \]

Practice 1: DMS to Decimal Degrees

Convert

\[ 32^\circ15'30'' \]

to decimal degrees.

Practice 2: Decimal Degrees to DMS

Convert

\[ 18.5^\circ \]

to degrees, minutes, and seconds.

Practice 3: Great-Circle Arc

Two points on a sphere of radius

\[ 10 \]

have central angle

\[ 0.6\text{ rad}. \]

Find their great-circle arc length.

Practice 4: Same-Longitude Distance

Two points have latitudes

\[ 10^\circ \]

and

\[ 25^\circ \]

on the same longitude. Find their central angle in radians.

Practice 5: Latitude-Circle Radius

On a spherical Earth of radius \(R\), find the radius of the latitude circle at

\[ 60^\circ. \]

Practice 6: Right Ascension

Convert

\[ 8\text{ h} \]

of right ascension to degrees.

Practice 7: Zenith Angle

An object has altitude

\[ 35^\circ. \]

Find its zenith angle.

Practice 8: Small-Angle Approximation

An object \(5\) meters wide is \(250\) meters away.

Approximate its angular size in radians.

Practice 9: Three-Dimensional Distance

Find the distance between

\[ P=(1,2,3) \]

and

\[ Q=(4,6,3). \]

Practice 10: Identify the Error

A student uses

\[ s=R\theta \]

with

\[ \theta=30 \]

because the angle is \(30^\circ\).

Explain the error.

Practice Answers

Practice 1

\[ 32+\frac{15}{60}+\frac{30}{3600}. \] \[ 32+0.25+0.008333\ldots = 32.258333\ldots \]

Therefore,

\[ \boxed{32.2583^\circ\text{ approximately}}. \]

Practice 2

The whole-degree part is

\[ 18^\circ. \]

The remaining

\[ 0.5^\circ \]

equals

\[ 0.5(60)=30'. \]

Therefore,

\[ \boxed{18^\circ30'00''}. \]

Practice 3

\[ s=R\theta=10(0.6)=6. \]

Therefore,

\[ \boxed{s=6}. \]

Practice 4

The latitude difference is

\[ 25^\circ-10^\circ=15^\circ. \]

Convert to radians:

\[ 15^\circ \frac{\pi}{180^\circ} = \frac{\pi}{12}. \]

Therefore,

\[ \boxed{\frac{\pi}{12}\text{ rad}}. \]

Practice 5

\[ R_{\text{parallel}} = R\cos60^\circ = \frac{R}{2}. \]

Therefore,

\[ \boxed{\frac{R}{2}}. \]

Practice 6

\[ 8(15^\circ)=120^\circ. \]

Therefore,

\[ \boxed{120^\circ}. \]

Practice 7

\[ z=90^\circ-35^\circ=55^\circ. \]

Therefore,

\[ \boxed{55^\circ}. \]

Practice 8

\[ \theta\approx\frac{5}{250}=0.02. \]

Therefore,

\[ \boxed{0.02\text{ rad}}. \]

Practice 9

\[ d= \sqrt{ (4-1)^2 +(6-2)^2 +(3-3)^2 }. \] \[ d= \sqrt{9+16} = 5. \]

Therefore,

\[ \boxed{5}. \]

Practice 10

The formula

\[ s=R\theta \]

requires \(\theta\) in radians.

Convert:

\[ 30^\circ \frac{\pi}{180^\circ} = \frac{\pi}{6}. \]

The correct substitution is therefore

\[ \boxed{ s=R\frac{\pi}{6} }. \]

What to Remember

Latitude and longitude describe angular position on or near Earth.
Degrees, minutes, seconds, decimal degrees, and radians are different angular representations that can be converted.
A spherical Earth model provides useful first approximations for large-scale spatial reasoning.
Latitude and longitude can be converted into Earth-centered Cartesian coordinates in a spherical model.
Great-circle arcs provide shortest surface routes on a sphere.
The arc-length formula \(s=R\theta\) requires radians.
Surface distance, parallel-circle distance, and straight-line chord distance are different quantities.
Higher-precision geographic positioning uses ellipsoidal models and explicit datums.
Map projections transform angular geographic coordinates into planar coordinates and introduce distortion.
Triangulation uses angles; trilateration uses distances.
Signal travel time can be converted into a distance constraint when propagation speed is known.
Local tangent frames provide convenient east-north-up coordinates near a geographic location.
Astronomical coordinates often describe viewing direction rather than full physical position.
Altitude-azimuth coordinates depend on the observer's local horizon, while equatorial coordinates use celestial reference directions.
Angular separation can be calculated with vector dot products.
Parallax converts baseline and angular change into distance information.

The central relationship is:

\[ \boxed{ \text{angles} + \text{reference geometry} + \text{distance information} \longrightarrow \text{large-scale position} }. \]

Where We Go Next

Geographic and astronomical positioning use geometry to locate objects over large distances. Many spatial systems also involve disturbances that travel through space and interact with surfaces, boundaries, and media.

The next lesson, Waves, Optics, and Acoustics, develops spatial reasoning for wave propagation, reflection, refraction, interference, rays, sound paths, and geometric models of light and vibration.