Math::Trig - trigonometric functions

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The constant <b>pi</b> is also defined as are a few convenience functions for angle conversions.</p> <h1 id="TRIGONOMETRIC-FUNCTIONS"><a class="permalink" href="#TRIGONOMETRIC-FUNCTIONS">#</a><a id="TRIGONOMETRIC"></a>TRIGONOMETRIC FUNCTIONS</h1> <p>The tangent</p> <dl> <dt id="tan"><a class="permalink" href="#tan">#</a><b>tan</b></dt> <dd> </dd> </dl> <p>The cofunctions of the sine, cosine, and tangent (cosec/csc and cotan/cot are aliases)</p> <p><b>csc</b>, <b>cosec</b>, <b>sec</b>, <b>sec</b>, <b>cot</b>, <b>cotan</b></p> <p>The arcus (also known as the inverse) functions of the sine, cosine, and tangent</p> <p><b>asin</b>, <b>acos</b>, <b>atan</b></p> <p>The principal value of the arc tangent of y/x</p> <p><b>atan2</b>(y, x)</p> <p>The arcus cofunctions of the sine, cosine, and tangent (acosec/acsc and acotan/acot are aliases)</p> <p><b>acsc</b>, <b>acosec</b>, <b>asec</b>, <b>acot</b>, <b>acotan</b></p> <p>The hyperbolic sine, cosine, and tangent</p> <p><b>sinh</b>, <b>cosh</b>, <b>tanh</b></p> <p>The cofunctions of the hyperbolic sine, cosine, and tangent (cosech/csch and cotanh/coth are aliases)</p> <p><b>csch</b>, <b>cosech</b>, <b>sech</b>, <b>coth</b>, <b>cotanh</b></p> <p>The arcus (also known as the inverse) functions of the hyperbolic sine, cosine, and tangent</p> <p><b>asinh</b>, <b>acosh</b>, <b>atanh</b></p> <p>The arcus cofunctions of the hyperbolic sine, cosine, and tangent (acsch/acosech and acoth/acotanh are aliases)</p> <p><b>acsch</b>, <b>acosech</b>, <b>asech</b>, <b>acoth</b>, <b>acotanh</b></p> <p>The trigonometric constant <b>pi</b> is also defined.</p> <p>$pi2 = 2 * <b>pi</b>;</p> <h2 id="ERRORS-DUE-TO-DIVISION-BY-ZERO"><a class="permalink" href="#ERRORS-DUE-TO-DIVISION-BY-ZERO">#</a><a id="ERRORS"></a>ERRORS DUE TO DIVISION BY ZERO</h2> <p>The following functions</p> <pre><code class="plaintext">acoth acsc acsch asec asech atanh cot coth csc csch sec sech tan tanh</code></pre> <p>cannot be computed for all arguments because that would mean dividing by zero or taking logarithm of zero. These situations cause fatal runtime errors looking like this</p> <pre><code class="plaintext">cot(0): Division by zero. (Because in the definition of cot(0), the divisor sin(0) is 0) Died at ...</code></pre> <p>or</p> <pre><code class="plaintext">atanh(-1): Logarithm of zero. Died at...</code></pre> <p>For the <code>csc</code>, <code>cot</code>, <code>asec</code>, <code>acsc</code>, <code>acot</code>, <code>csch</code>, <code>coth</code>, <code>asech</code>, <code>acsch</code>, the argument cannot be <code>0</code> (zero). For the <code>atanh</code>, <code>acoth</code>, the argument cannot be <code>1</code> (one). For the <code>atanh</code>, <code>acoth</code>, the argument cannot be <code>-1</code> (minus one). For the <code>tan</code>, <code>sec</code>, <code>tanh</code>, <code>sech</code>, the argument cannot be <i>pi/2 + k * pi</i>, where <i>k</i> is any integer.</p> <h2 id="SIMPLE-(REAL)-ARGUMENTS,-COMPLEX-RESULTS"><a class="permalink" href="#SIMPLE-(REAL)-ARGUMENTS,-COMPLEX-RESULTS">#</a><a id="SIMPLE"></a><a id="SIMPLE-REAL-ARGUMENTS-COMPLEX-RESULTS"></a>SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS</h2> <p>Please note that some of the trigonometric functions can break out from the <b>real axis</b> into the <b>complex plane</b>. For example <code>asin(2)</code> has no definition for plain real numbers but it has definition for complex numbers.</p> <p>In Perl terms this means that supplying the usual Perl numbers (also known as scalars, please see <a href="/5.8.2/perldata">perldata</a>) as input for the trigonometric functions might produce as output results that no more are simple real numbers: instead they are complex numbers.</p> <p>The <code>Math::Trig</code> handles this by using the <code>Math::Complex</code> package which knows how to handle complex numbers, please see <a href="/5.8.2/Math::Complex">Math::Complex</a> for more information. In practice you need not to worry about getting complex numbers as results because the <code>Math::Complex</code> takes care of details like for example how to display complex numbers. For example:</p> <pre><code>print asin(2), "\n";</code></pre> <p>should produce something like this (take or leave few last decimals):</p> <pre><code class="plaintext">1.5707963267949-1.31695789692482i</code></pre> <p>That is, a complex number with the real part of approximately <code>1.571</code> and the imaginary part of approximately <code>-1.317</code>.</p> <h1 id="PLANE-ANGLE-CONVERSIONS"><a class="permalink" href="#PLANE-ANGLE-CONVERSIONS">#</a><a id="PLANE"></a>PLANE ANGLE CONVERSIONS</h1> <p>(Plane, 2-dimensional) angles may be converted with the following functions.</p> <pre><code>$radians = deg2rad($degrees); $radians = grad2rad($gradians); $degrees = rad2deg($radians); $degrees = grad2deg($gradians); $gradians = deg2grad($degrees); $gradians = rad2grad($radians);</code></pre> <p>The full circle is 2 <i>pi</i> radians or <i>360</i> degrees or <i>400</i> gradians. The result is by default wrapped to be inside the [0, {2pi,360,400}[ circle. If you don't want this, supply a true second argument:</p> <pre><code>$zillions_of_radians = deg2rad($zillions_of_degrees, 1); $negative_degrees = rad2deg($negative_radians, 1);</code></pre> <p>You can also do the wrapping explicitly by rad2rad(), deg2deg(), and grad2grad().</p> <h1 id="RADIAL-COORDINATE-CONVERSIONS"><a class="permalink" href="#RADIAL-COORDINATE-CONVERSIONS">#</a><a id="RADIAL"></a>RADIAL COORDINATE CONVERSIONS</h1> <p><b>Radial coordinate systems</b> are the <b>spherical</b> and the <b>cylindrical</b> systems, explained shortly in more detail.</p> <p>You can import radial coordinate conversion functions by using the <code>:radial</code> tag:</p> <pre><code>use Math::Trig ':radial'; ($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z); ($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z); ($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z); ($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z); ($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi); ($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);</code></pre> <p><b>All angles are in radians</b>.</p> <h2 id="COORDINATE-SYSTEMS"><a class="permalink" href="#COORDINATE-SYSTEMS">#</a><a id="COORDINATE"></a>COORDINATE SYSTEMS</h2> <p><b>Cartesian</b> coordinates are the usual rectangular <i>(x, y, z)</i>-coordinates.</p> <p>Spherical coordinates, <i>(rho, theta, pi)</i>, are three-dimensional coordinates which define a point in three-dimensional space. They are based on a sphere surface. The radius of the sphere is <b>rho</b>, also known as the <i>radial</i> coordinate. The angle in the <i>xy</i>-plane (around the <i>z</i>-axis) is <b>theta</b>, also known as the <i>azimuthal</i> coordinate. The angle from the <i>z</i>-axis is <b>phi</b>, also known as the <i>polar</i> coordinate. The `North Pole' is therefore <i>0, 0, rho</i>, and the `Bay of Guinea' (think of the missing big chunk of Africa) <i>0, pi/2, rho</i>. In geographical terms <i>phi</i> is latitude (northward positive, southward negative) and <i>theta</i> is longitude (eastward positive, westward negative).</p> <p><b>BEWARE</b>: some texts define <i>theta</i> and <i>phi</i> the other way round, some texts define the <i>phi</i> to start from the horizontal plane, some texts use <i>r</i> in place of <i>rho</i>.</p> <p>Cylindrical coordinates, <i>(rho, theta, z)</i>, are three-dimensional coordinates which define a point in three-dimensional space. They are based on a cylinder surface. The radius of the cylinder is <b>rho</b>, also known as the <i>radial</i> coordinate. The angle in the <i>xy</i>-plane (around the <i>z</i>-axis) is <b>theta</b>, also known as the <i>azimuthal</i> coordinate. The third coordinate is the <i>z</i>, pointing up from the <b>theta</b>-plane.</p> <h2 id="3-D-ANGLE-CONVERSIONS"><a class="permalink" href="#3-D-ANGLE-CONVERSIONS">#</a><a id="3"></a><a id="D-ANGLE-CONVERSIONS"></a>3-D ANGLE CONVERSIONS</h2> <p>Conversions to and from spherical and cylindrical coordinates are available. Please notice that the conversions are not necessarily reversible because of the equalities like <i>pi</i> angles being equal to <i>-pi</i> angles.</p> <dl> <dt id="cartesian_to_cylindrical"><a class="permalink" href="#cartesian_to_cylindrical">#</a>cartesian_to_cylindrical</dt> <dd> <pre><code>($rho, $theta, $z) = cartesian_to_cylindrical($x, $y, $z);</code></pre> </dd> <dt id="cartesian_to_spherical"><a class="permalink" href="#cartesian_to_spherical">#</a>cartesian_to_spherical</dt> <dd> <pre><code>($rho, $theta, $phi) = cartesian_to_spherical($x, $y, $z);</code></pre> </dd> <dt id="cylindrical_to_cartesian"><a class="permalink" href="#cylindrical_to_cartesian">#</a>cylindrical_to_cartesian</dt> <dd> <pre><code>($x, $y, $z) = cylindrical_to_cartesian($rho, $theta, $z);</code></pre> </dd> <dt id="cylindrical_to_spherical"><a class="permalink" href="#cylindrical_to_spherical">#</a>cylindrical_to_spherical</dt> <dd> <pre><code>($rho_s, $theta, $phi) = cylindrical_to_spherical($rho_c, $theta, $z);</code></pre> <p>Notice that when <code>$z</code> is not 0 <code>$rho_s</code> is not equal to <code>$rho_c</code>.</p> </dd> <dt id="spherical_to_cartesian"><a class="permalink" href="#spherical_to_cartesian">#</a>spherical_to_cartesian</dt> <dd> <pre><code>($x, $y, $z) = spherical_to_cartesian($rho, $theta, $phi);</code></pre> </dd> <dt id="spherical_to_cylindrical"><a class="permalink" href="#spherical_to_cylindrical">#</a>spherical_to_cylindrical</dt> <dd> <pre><code>($rho_c, $theta, $z) = spherical_to_cylindrical($rho_s, $theta, $phi);</code></pre> <p>Notice that when <code>$z</code> is not 0 <code>$rho_c</code> is not equal to <code>$rho_s</code>.</p> </dd> </dl> <h1 id="GREAT-CIRCLE-DISTANCES-AND-DIRECTIONS"><a class="permalink" href="#GREAT-CIRCLE-DISTANCES-AND-DIRECTIONS">#</a><a id="GREAT"></a>GREAT CIRCLE DISTANCES AND DIRECTIONS</h1> <p>You can compute spherical distances, called <b>great circle distances</b>, by importing the great_circle_distance() function:</p> <pre><code>use Math::Trig 'great_circle_distance'; $distance = great_circle_distance($theta0, $phi0, $theta1, $phi1, [, $rho]);</code></pre> <p>The <i>great circle distance</i> is the shortest distance between two points on a sphere. The distance is in <code>$rho</code> units. The <code>$rho</code> is optional, it defaults to 1 (the unit sphere), therefore the distance defaults to radians.</p> <p>If you think geographically the <i>theta</i> are longitudes: zero at the Greenwhich meridian, eastward positive, westward negative--and the <i>phi</i> are latitudes: zero at the North Pole, northward positive, southward negative. <b>NOTE</b>: this formula thinks in mathematics, not geographically: the <i>phi</i> zero is at the North Pole, not at the Equator on the west coast of Africa (Bay of Guinea). You need to subtract your geographical coordinates from <i>pi/2</i> (also known as 90 degrees).</p> <pre><code>$distance = great_circle_distance($lon0, pi/2 - $lat0, $lon1, pi/2 - $lat1, $rho);</code></pre> <p>The direction you must follow the great circle can be computed by the great_circle_direction() function:</p> <pre><code>use Math::Trig 'great_circle_direction'; $direction = great_circle_direction($theta0, $phi0, $theta1, $phi1);</code></pre> <p>The result is in radians, zero indicating straight north, pi or -pi straight south, pi/2 straight west, and -pi/2 straight east.</p> <p>Notice that the resulting directions might be somewhat surprising if you are looking at a flat worldmap: in such map projections the great circles quite often do not look like the shortest routes-- but for example the shortest possible routes from Europe or North America to Asia do often cross the polar regions.</p> <h1 id="EXAMPLES"><a class="permalink" href="#EXAMPLES">#</a>EXAMPLES</h1> <p>To calculate the distance between London (51.3N 0.5W) and Tokyo (35.7N 139.8E) in kilometers:</p> <pre><code> use Math::Trig qw(great_circle_distance deg2rad); # Notice the 90 - latitude: phi zero is at the North Pole. @L = (deg2rad(-0.5), deg2rad(90 - 51.3)); @T = (deg2rad(139.8),deg2rad(90 - 35.7)); $km = great_circle_distance(@L, @T, 6378);</code></pre> <p>The direction you would have to go from London to Tokyo</p> <pre><code>use Math::Trig qw(great_circle_direction); $rad = great_circle_direction(@L, @T);</code></pre> <h2 id="CAVEAT-FOR-GREAT-CIRCLE-FORMULAS"><a class="permalink" href="#CAVEAT-FOR-GREAT-CIRCLE-FORMULAS">#</a><a id="CAVEAT"></a>CAVEAT FOR GREAT CIRCLE FORMULAS</h2> <p>The answers may be off by few percentages because of the irregular (slightly aspherical) form of the Earth. The formula used for grear circle distances</p> <pre><code class="plaintext"> lat0 = 90 degrees - phi0 lat1 = 90 degrees - phi1 d = R * arccos(cos(lat0) * cos(lat1) * cos(lon1 - lon01) + sin(lat0) * sin(lat1))</code></pre> <p>is also somewhat unreliable for small distances (for locations separated less than about five degrees) because it uses arc cosine which is rather ill-conditioned for values close to zero.</p> <h1 id="BUGS"><a class="permalink" href="#BUGS">#</a>BUGS</h1> <p>Saying <code>use Math::Trig;</code> exports many mathematical routines in the caller environment and even overrides some (<code>sin</code>, <code>cos</code>). This is construed as a feature by the Authors, actually... ;-)</p> <p>The code is not optimized for speed, especially because we use <code>Math::Complex</code> and thus go quite near complex numbers while doing the computations even when the arguments are not. This, however, cannot be completely avoided if we want things like <code>asin(2)</code> to give an answer instead of giving a fatal runtime error.</p> <h1 id="AUTHORS"><a class="permalink" href="#AUTHORS">#</a>AUTHORS</h1> <p>Jarkko Hietaniemi <<i>jhi@iki.fi</i>> and Raphael Manfredi <<i>Raphael_Manfredi@pobox.com</i>>.</p> </div> <div id="footer"> <p>Perldoc Browser is maintained by Dan Book (<a href="https://metacpan.org/author/DBOOK">DBOOK</a>). Please contact him via the <a href="https://github.com/Grinnz/perldoc-browser/issues">GitHub issue tracker</a> or <a href="mailto:dbook@cpan.org">email</a> regarding any issues with the site itself, search, or rendering of documentation.</p> <p>The Perl documentation is maintained by the Perl 5 Porters in the development of Perl. 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Math::Trig - trigonometric functions - Perldoc Browser