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Angle Representations and Angular Units

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).

Radians and Degrees

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 deg2rad for this purpose:
thetaInRadians = deg2rad(thetaInDegrees)
Likewise, you can perform the opposite conversion by applying the inverse factor,
thetaInDegrees = thetaInRadians * (180/pi)
or by using rad2deg,
thetaInDegrees = rad2deg(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 rad2deg and deg2rad are very simple and efficient, and operate on vector and higher-dimensioned input as well as scalars.

Default and Variable Angle Units

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 the '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.

Degrees, Minutes, and Seconds

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:

  1. A sign

  2. A nonnegative, integer-valued degrees component

  3. 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(rad2deg(-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:

  1. A sign

  2. A nonnegative integer-valued degrees component

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

  4. 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(rad2deg(-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.

Converting Angle Units that Vary at Run Time

Functions deg2rad and rad2deg 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 'radians' at run time, the toRadians function has no real work to do; all the functions in this group handle such "no-op" situations efficiently.

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