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Positions, Azimuths, Headings, Distances, Length, and Ranges Working with Length and Distance Units Working with Angles: Units and Representations |

When using spherical coordinates, distances are expressed as
angles, not lengths. As there is an infinity of arcs that can connect
two points on a sphere or spheroid, by convention the shortest one
(the *great circle* distance) is used to measure
how close two points are. As is explained in Working with Distances on the Sphere, you can convert angular
distance on a sphere to linear distance. This is different from working
on an ellipsoid, where one can only speak of linear distances between
points, and to compute them one must specify which reference ellipsoid
to use.

In spherical or geodetic coordinates, a *position* is
a latitude taken together with a longitude, e.g., `(lat,lon)`,
which defines the horizontal coordinates of a point on the surface
of a planet. When we consider two points, e.g.,`(lat1,lon1)` and `(lat2,lon2)`,
there are several ways in which their 2–D spatial relationships
are typically quantified:

The azimuth (also called heading) to take to get from

`(lat1,lon1)`to`(lat2,lon2)`The back azimuth (also called heading) from

`(lat2,lon2)`to`(lat1,lon1)`The spherical distance separating

`(lat1,lon1)`from`(lat2,lon2)`The linear distance (range) separating

`(lat1,lon1)`from`(lat2,lon2)`

The first three are angular quantities, while the last is a length. Mapping Toolbox™ functions exist for computing these quantities. For more information, see Directions and Areas on the Sphere and Spheroid and also Navigation for additional examples.

There is no single default unit of distance measurement in the
toolbox. Navigation functions use nautical miles as a default and
the `distance` function uses degrees of arc length.
For many functions, the default unit for distances and positions is
degrees, but you need to verify the default assumptions before using
any of these functions.

Linear measurements of lengths and distances on spheres and spheroids can use the same units they do on the plane, such as feet, meters, miles, and kilometers. They can be used for

Absolute positions, such as map coordinates or terrain elevations

Dimensions, such as a planet's radius or its semimajor and semiminor axes

Distances between points or along routes, in 2-D or 3-D space or across terrain

Length units are needed to describe

The dimensions of a reference sphere or ellipsoid

The line-of-sight distance between points

Distances along great circle or rhumb line curves on an ellipsoid or sphere

*X-Y*locations in a projected coordinate system or map gridOffsets from a map origin (false eastings and northings)

*X-Y-Z*locations in Earth-centered Earth-fixed (ECEF) or local vertical systemsHeights of various types (terrain elevations above a geoid, an ellipsoid, or other reference surface)

Using the toolbox effectively depends on being consistent about units of length. Depending on the specific function and the way you are calling it, when you specify lengths, you could be

Explicitly specifying a radius, reference spheroid object, or ellipsoid vector

Relying on the function itself to specify a default radius or ellipsoid

Relying on the reference ellipsoid associated with a map projection structure (mstruct)

Whenever you are doing a computation that involves a reference sphere or ellipsoid, make sure that the units of length you are using are the same units used to define the radius of the sphere or semimajor axis of the ellipsoid. These considerations are discussed below.

The following Mapping Toolbox functions convert between different units of length:

`unitsratio`computes multiplicative factors for converting between 12 different units of length as well as between degrees and radians. You can use`unistratio`to perform conversions when neither the input units of length nor the output units of length are known until run time. See Converting Angle Units that Vary at Run Time for more information.`km2nm`,`km2sm`,`nm2km`,`nm2sm`,`sm2km`, and`sm2nm`perform simple and convenient conversions between kilometers, nautical miles, and statute miles.

These utility functions accept scalars, vectors, and matrices, or any shape. For an overview of these functions and angle conversion functions, see Summary: Available Distance and Angle Conversion Functions.

The `unitsratio` function
can compute the ratio between any of the following units of length:

Microns

Millimeters

Centimeters

Meters

Kilometers

Inches

International feet

U.S. survey feet

Yards

International miles

U.S. survey (statute) miles

The syntax for `unitsratio` is

ratio = unitsratio(,to-unit)from-unit

You can use the output from `unitsratio` as
a multiplicative conversion factor.

For example, the following shows that 4 inches span just over 10 centimeters:

cmPerInch = unitsratio('cm','inch') cm = cmPerInch * 4 cmPerInch = 2.5400 cm = 10.1600

To convert this number of centimeters back to inches, type

inch = unitsratio('in','centimeter') * cmPerInch inch = 1

Note that

`unitsratio`supports various abbreviations for units of length.

The `unitsratio` function also lets you convert
angles between degrees and radians.

Angular measurements have many distinct roles in geospatial data handling. For example, they are used to specify

Absolute positions — latitudes and longitudes

Relative positions — azimuths, bearings, and elevation angles

Spherical distances between point locations

Absolute positions are expressed in *geodetic coordinates*,
which are actually angles between lines or planes on a reference sphere
or ellipsoid. Relative positions use units of angle to express the
direction between one place on the reference body from another one.
Spherical distances quantify how far two places are from one another
in terms of the angle subtended along a great-circle arc. On nonspherical
reference bodies, distances are usually given in linear units such
as kilometers (because on them, arc lengths are no longer proportional
to subtended angle).

The basic unit for angles in MATLAB^{®} is the radian. For
example, if the variable theta represents an angle and you want to
take its sine, you can use `sin(theta)` if and only
if the value of theta is expressed in radians. If a variable represents
the value of an angle in degrees, then you must convert the value
to radians before taking the sine. For example,

thetaInDegrees = 30; thetaInRadians = thetaInDegrees * (pi/180) sinTheta = sin(thetaInRadians)

As shown above, you can scale degrees to radians
by multiplying by `pi/180`. However, you should consider
using the Mapping Toolbox function `degtorad` for
this purpose:

thetaInRadians = degtorad(thetaInDegrees)

Likewise, you can perform the opposite conversion by applying the inverse factor,

thetaInDegrees = thetaInRadians * (180/pi)

or by using `radtodeg`,

thetaInDegrees = radtodeg(thetaInRadians)

The practice of using these functions has two significant advantages:

It reduces the likelihood of human error (e.g., you might type "pi/108" by mistake)

It signals clearly your intent—important to do should others ever read, modify, or debug your code

The functions `radtodeg` and `degtorad` are
very simple and efficient, and operate on vector and higher-dimensioned
input as well as scalars.

Unlike MATLAB trigonometric functions, Mapping Toolbox functions do not always assume that angular arguments are in units of radians.

The low-level utility functions intended as building blocks
of more complex features or applications work only in units of radians.
Examples include the functions `unwrapMultipart` and `meridianarc`.

Many high-level functions, including `distance`,
can work in either degrees or radians. Their interpretation of angles
is controlled by a string-valued `'angleunits'` input
argument. (`angleunits` can be either `'degrees'` or `'radians'`,
and can generally be abbreviated.) This flexibility balances convenience
and efficiency, although it means that you must take care to check
what assumptions each function is making about its inputs.

In all Mapping Toolbox computations that involve angles in degrees, floating-point numbers (generally MATLAB class double) are used, which allows for integer and fractional values and rational approximations to irrational numbers. However, several traditional notations, which are still in wide use, represent angles as pairs or triplets of numbers, using minutes of arc (1/60 of degree) and seconds of arc (1/60 of a minute):

Degrees-minutes notation (DM), e.g., 35° 15', equal to 35.25°

Degrees-minutes-seconds notation (DMS) , e.g., 35° 15' 45'', equal to 35.2625°

In degrees-minutes representation, an angle is split into three separate parts:

A sign

A nonnegative, integer-valued degrees component

A nonnegative minutes component, real-valued and in the half-open interval [0 60)

For example, -1 radians is represented by a minus sign (-) and the numbers [57, 17.7468...]. (The fraction in the minutes part approximates an irrational number and is rounded here for display purposes. This subtle point is revisited in the following section.)

The toolbox includes the function `degrees2dm` to
perform conversions of this sort. You can use this function to export
data in DM form, either for display purposes or for use by another
application. For example,

degrees2dm(radtodeg(-1)) ans = -57.0000 17.7468

More generally, `degrees2dm` converts
a single-columned input to a pair of columns. Rather than storing
the sign in a separate element, `degrees2dm` applies
to the first nonzero element in each row. Function `dm2degrees` converts
in the opposite direction, producing a real-valued column vector of
degrees from a two-column array having an integer degrees and real-valued
minutes column. Thus,

dm2degrees(degrees2dm(pi)) == pi ans = 1

Similarly, in degrees-minutes-seconds representation, an angle is split into four separate parts:

A sign

A nonnegative integer-valued degrees component

A minutes component which can be any integer from 0 through 59

A nonnegative minutes component, real-valued and in the half-open interval [0 60)

For example, -1 radians is represented by a minus sign
(-) and the numbers [57, 17, 44.8062...], which can be seen using Mapping Toolbox function `degrees2dms`,

degrees2dms(radtodeg(-1)) ans = -57.0000 17.0000 44.8062

`degrees2dms` works
like `degrees2dm`; it converts single-columned
input to three-column form, applying the sign to the first nonzero
element in each row.

A fourth function, `dms2degrees`, is similar
to `dm2degrees` and supports data import by producing
a real-valued column vector of degrees from an array with an integer-valued
degrees column, an integer-value minutes column, and a real-valued
seconds column. As noted, the four functions, `degrees2dm`, `degrees2dms`, `dm2degrees`,
and `dms2degrees`, are particular about the shape
of their inputs; in this regard they are distinct from the other angle-conversion
functions in the toolbox.

The toolbox makes no internal use of DM or DMS representation.
The conversion functions `dm2degrees` and `dms2degrees` are
provided only as tools for data import. Likewise, `degrees2dm` and `degrees2dms` are
only useful for displaying geographic coordinates on maps, publishing
coordinate values, and for formatting data to be exported to other
applications. Methods for accomplishing this are discussed below,
in Formatting Latitudes and Longitudes as Strings.

Functions `degtorad` and `radtodeg` are
simple to use and efficient, but how do you write code to convert
angles if you do not know ahead of time what units the data will use?
The toolbox provides a set of utility functions that help you deal
with such situations at run time.

In almost all cases—even at the time you are coding—you know either the input or destination angle units. When you do, you can use one of these functions:

For example, you might wish to implement a very simple sinusoidal projection on the unit sphere, but allow the input latitudes and longitudes to be in either degrees or radians. You can accomplish this as follows:

function [x, y] = sinusoidal(lat, lon, angleunits) [lat, lon] = toRadians(angleunits, lat, lon); x = lon .* cos(lat); y = lat;

Whenever `angleunits` turns
out to be

In the very rare instances when you must code an application
or MATLAB function in which the units of both input angles and
output angles remain unknown until run time, you can still accomplish
the conversion by using the `unitsratio` function.
For example,

fromUnits = 'radians'; toUnits = 'degrees'; piInDegrees = unitsratio(toUnits, fromUnits) * pi piInDegrees = 180

Many geospatial domains (seismology, for example) describe distances between points on the surface of the earth as angles. This is simply the result of dividing the length of the shortest great-circle arc connecting a pair points by the radius of the Earth (or whatever planet one is measuring). This gives the angle (in radians) subtended by rays from each point that join at the center of the Earth (or other planet). This is sometimes called a "spherical distance." You can thus call the resulting number a "distance in radians." You could also call the same number a "distance in earth radii." When you work with transformations of geodata, keep this in mind.

You can easily convert that angle from radians to degrees. For
example, you can call `distance` to compute the
distance in meters from London to Kuala Lumpur:

latL = 51.5188; lonL = -0.1300; latK = 2.9519; lonK = 101.8200; earthRadiusInMeters = 6371000; distInMeters = distance(latL, lonL,... latK, lonK, earthRadiusInMeters) distInMeters = 1.0571e+007

Then convert the result to an angle in radians:

distInRadians = distInMeters / earthRadiusInMeters distInRadians = 1.6593

Finally, convert to an angle in degrees:

distInDegrees = radtodeg(distInRadians) distInDegrees = 95.0692

This really only makes sense and produces accurate results when we approximate the Earth (or planet) as a sphere. On an ellipsoid, one can only describe the distance along a geodesic curve using a unit of length.

Mapping Toolbox software includes a set of six functions to conveniently convert distances along the surface of the Earth (or another planet) from units of kilometers (km), nautical miles (nm), or statue miles (sm) to spherical distances in degrees (deg) or radians (rad):

`km2deg`,`nm2deg`, and`sm2deg`go from length to angle in degrees`km2rad`,`nm2rad`, and`sm2rad`go from length to angle in radians

You could replace the final two steps in the preceding example with

distInKilometers = distInMeters/1000; earthRadiusInKm = 6371; km2deg(distInKilometers, earthRadiusInKm) ans = 95.0692

Because these conversion can be reversed, the toolbox includes another six convenience functions that convert an angle subtended at the center of a sphere, in degrees or radians, to a great-circle distance along the surface of that sphere:

`deg2km`,`deg2nm`, and`deg2sm`go from angle in degrees to length`rad2km`,`rad2nm`, and`rad2sm`go from angle in radians to length

When given a single input argument, all 12 functions assume a radius of 6,371,000 meters (6371 km, 3440.065 nm, or 3958.748 sm), which is widely-used as an estimate of the average radius of the Earth. An optional second parameter can be used to specify a planetary radius (in output length units) or the name of an object in the Solar System.

On the Earth, a degree of arc length at the equator is about 60 nautical miles:

nauticalmiles = deg2nm(1) nauticalmiles = 60.0405

The Earth is the default assumption for these conversion functions. You can use other radii, however:

moon = referenceSphere('moon','km'); nauticalmiles = deg2nm(1,moon.Radius) nauticalmiles = 30.3338

The function `deg2sm` returns distances in
statute, rather than nautical, miles:

deg2sm(1) ans = 69.0932

Certain syntaxes of the `distance` and `reckon` functions
use angles to denote distances in the way described above. In the
following statements, the range argument, `arclen`,
is in degrees (along with all the other inputs and outputs):

[arclen, az] = distance(lat1, lon1, lat2, lon2) [latout, lonout] = reckon(lat, lon, arclen, az)

By adding the optional `units` argument, you can
use radians instead:

[arclen, az] = distance(lat1, lon1, lat2, lon2, 'radians') [latout, lonout] = reckon(lat, lon, arclen, az, 'radians')

If
an `ellipsoid` argument is provided, however, then `arclen` has
units of length, and they match the units of the semimajor axis length
of the reference ellipsoid. If you specify `ellipsoid = [1
0]` (the unit sphere), `arclen` can be considered
to be either an angle in radians or a length defined in units of earth
radii. It has the same value either way. Thus, in the following computation, `lat1`, `lon1`, `lat2`, `lon2`,
and `az` are in degrees, but `arclen` will
appear to be in radians:

[arclen, az] = distance(lat1, lon1, lat2, lon2, [1 0])

The following table shows the Mapping Toolbox unit-to-unit distance and arc conversion functions. They all accept scalar, vector, and higher-dimension inputs. The first two columns and rows involve angle units, the last three involve distance units:

**Functions that Directly Convert Angles, Lengths, and Spherical
Distances**

Convert | To Degrees | To Radians | To Kilometers | To Nautical Miles | To Statute Miles |
---|---|---|---|---|---|

Degrees | toDegreesfromDegrees | degtoradtoRadiansfromDegrees | deg2km | deg2nm | deg2sm |

Radians | radtodegtoDegreesfromRadians | toRadiansfromRadians | rad2km | rad2nm | rad2sm |

Kilometers | km2deg | km2rad | km2nm | km2sm | |

Nautical Miles | nm2deg | nm2rad | nm2km | nm2sm | |

Statute Miles | sm2deg | sm2rad | sm2km | sm2nm |

The angle conversion functions along the major diagonal, `toDegrees`, `toRadians`, `fromDegrees`,
and `fromRadians`, can have no-op results. They
are intended for use in applications that have no prior knowledge
of what angle units might be input or desired as output.

The terms *decimal degrees* and *decimal
minutes* are often used in geospatial data handling and
navigation. The preceding section avoided using them because its focus
was on the representation of angles within MATLAB, where they can
be arbitrary binary floating-point numbers.

However, once an angle in degrees is converted to a string, it is often helpful to describe that string as representing the angle in decimal degrees. Thus,

num2str(radtodeg(1)) ans = 57.2958

gives a value in decimal degrees. In casual
communication it is common to refer to a quantity such as `radtodeg(1)` as
being in decimal degrees, but strictly speaking, that is not true
until it is somehow converted to a string in base 10. That is, a binary
floating-point number is not a decimal number, whether it represents
an angle in degrees or not. If it does represent an angle and that
number is then formatted and displayed as having a fractional part,
only then is it appropriate to speak of "decimal degrees."
Likewise, the term "decimal minutes" applies when you
convert a degrees-minutes representation to a string, as in

num2str(degrees2dm(radtodeg(1))) ans = 57 17.7468

When a DM or DMS representation of an angle is expressed as
a string, it is traditional to tag the different components with
the special characters `d`, `m`,
and `s`, or °, ', and ".

When the angle is a latitude or longitude, a letter often designates the sign of the angle:

`N`for positive latitudes`S`for negative latitudes`E`for positive longitudes`W`for negative longitudes

For example, 123 degrees, 30 minutes, 12.7 seconds west of Greenwich can be written as 123d30m12.7sW, 123° 30° 12.7" W, or -123° 30° 12.7".

Use the function `str2angle` to
import latitude and longitude data formatted as such strings. Conversely,
you can format numeric degree data for display or export with `angl2str`, or combine `degrees2dms` or `degrees2dm` with `sprintf` to
customize formatting.

See Degrees, Minutes, and Seconds for more details about DM and DMS representation.

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