Package maths

Convert nautical miles to meter (m).

Arguments:

• nm - Value in nautical miles (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid nm.

Convert statute miles to meter (m).

Arguments:

• sm - Value in statute miles (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid sm.

Get sin and cos of typed angle.

Arguments:

• x - Angle (Degrees, Radians or radians).

Returns:

2-Tuple (sin(x), cos(x)).

Convert acres to hectare.

Arguments:

• acres - Value in acres (scalar).

Returns:

Value in hectare (float).

Raises:

• ValueError - Invalid acres.

Convert acres to square meter.

Arguments:

• acres - Value in acres (scalar).

Returns:

Value in meter^2 (float).

Raises:

• ValueError - Invalid acres.

Return atan(y / x) angle in degrees [-90..+90] using atan2d for consistency and to avoid ZeroDivisionError.

Return atan2(y, x) in degrees [0..+360].

Return atan2(y, x) in degrees [-180..+180], optionally reversed (by 180 degrees for azimuths).

Return the 4-th, bi-quadratic or quartic root, x**(1 / 4).

Arguments:

• x - Value (scalar).

Returns:

Quartic root (float).

Raises:

• ValueError - Negative x.

Compute the cube root x**(1/3).

Arguments:

• x - Value (scalar).

Returns:

Cubic root (float).

Compute the cube root squared x**(2/3).

Arguments:

• x - Value (scalar).

Returns:

Cube root squared (float).

Convert UK chains to meter.

Arguments:

• chains - Value in chains (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid chains.

Return the radius and center of the inscribed aka Incircle of a (2- or 3-D) triangle.

Arguments:

• point1 - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point2 - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point3 - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• eps - Tolerance for function pygeodesy3.trilaterate3d2 if useZ is True else pygeodesy3.trilaterate2d2.
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Circin6Tuple(radius, center, deltas, cA, cB, cC). The center and contact points cA, cB and cC, each an instance of point1's (sub-)class, are co-planar with the three given points.

Raises:

• ImportError - Package numpy not found, not installed or older than version 1.10 and useZ is True.
• IntersectionError - Near-coincident or -colinear points or a trilateration or numpy issue.
• TypeError - Invalid point1, point2 or point3.

Get the equatorial or a parallel circle of latitude.

Arguments:

• earth - The earth radius, ellipsoid or datum (meter, Ellipsoid, Ellipsoid2, Datum or a_f2Tuple).
• lat - Geodetic latitude (degrees90, str).

Returns:

A Circle4Tuple(radius, height, lat, beta) instance.

Raises:

• RangeError - Latitude lat outside valid range and pygeodesy3.rangerrors set to True.
• TypeError - Invalid earth.
• ValueError - earth or lat.

Return the radius and center of the smallest circle through or containing three (2- or 3-D) points.

Arguments:

• point1 - First point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• point2 - Second point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• point3 - Third point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• circum - If True return the circumradius and circumcenter always, ignoring the Meeus' Type I case (bool).
• eps - Tolerance for function pygeodesy3.trilaterate3d2 if useZ is True else pygeodesy3.trilaterate2d2.
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

A Circum3Tuple(radius, center, deltas). The center, an instance of point1's (sub-)class, is co-planar with the three given points.

Raises:

• ImportError - Package numpy not found, not installed or older than version 1.10 and useZ is True.
• IntersectionError - Near-coincident or -colinear points or a trilateration or numpy issue.
• TypeError - Invalid point1, point2 or point3.

Best-fit a sphere through three or more (3-D) points.

Arguments:

• points - The points (each a Cartesian, Vector3d, Vector3Tuple, or Vector4Tuple).
• useZ_Vector_and_kwds - Keyword arguments useZ=True (bool) to use the Z components, otherwise force all z=INT0, class Vector=None to return the center point with optionally, additional nVector keyword arguments, otherwise the first points' (sub-)class is used.

Returns:

Circum4Tuple(radius, center, rank, residuals) with center an instance of points[0]' (sub-)class or Vector if specified.

Raises:

• ImportError - Package numpy not found, not installed or older than version 1.10.
• NumPyError - Some numpy issue.
• PointsError - Too few points.
• TypeError - One of the points is invalid.

Return the cotangent of an angle in radians.

Arguments:

• rad - Angle (radians).
• error_kwds - Error to raise (ValueError).

Returns:

cot(rad).

Raises:

• ValueError - pygeodesy3.isnear0(sin(rad).

Return the cotangent of angle(s) in radiansresection.

Arguments:

• rads - One or more angles (radians).
• error_kwds - Error to raise (ValueError).

Returns:

Yield the cot(rad) for each angle.

Raises:

• ValueError - See pygeodesy3.cot.

Return the cotangent of an angle in degrees.

Arguments:

• deg - Angle (degrees).
• error_kwds - Error to raise (ValueError).

Returns:

cot(deg).

Raises:

• ValueError - pygeodesy3.isnear0(sin(deg).

Return the cotangent of angle(s) in degrees.

Arguments:

• degs - One or more angles (degrees).
• error_kwds - Error to raise (ValueError).

Returns:

Yield the cot(deg) for each angle.

Raises:

• ValueError - See pygeodesy3.cotd.

Convert radians to degrees and wrap [-180..+180).

Arguments:

• rad - Angle (radians).

Returns:

Angle, wrapped (degrees180).

Convert degrees to grades (aka gons or gradians).

Arguments:

• deg - Angle (degrees).

Returns:

Angle (grades).

Convert an angle to a distance along the equator or along the parallel at an other (geodetic) latitude.

Arguments:

• deg - The angle (degrees).
• radius - Mean earth radius, ellipsoid or datum (meter, Ellipsoid, Ellipsoid2, Datum or a_f2Tuple).
• lat - Parallel latitude (degrees90, str).

Returns:

Distance (meter, same units as radius or equatorial and polar radii) or 0.0 for near-polar lat.

Raises:

• RangeError - Latitude lat outside valid range and pygeodesy3.rangerrors set to True.
• TypeError - Invalid radius.
• ValueError - Invalid deg, radius or lat.

Convert radians to degrees and wrap [0..+360).

Arguments:

• rad - Angle (radians).

Returns:

Angle, wrapped (degrees360).

Convert radians to degrees and wrap [-90..+90).

Arguments:

• rad - Angle (radians).

Returns:

Angle, wrapped (degrees90).

Appoximate the norm sqrt(x**2 + y**2) by max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142....

Arguments:

• x - X component (scalar).
• y - Y component (scalar).

Returns:

Appoximate norm (float).

Appoximate the norm sqrt(sum(x**2 for x in xs)) by cascaded euclid.

Arguments:

• xs - X arguments, positional (scalars).

Returns:

Appoximate norm (float).

Fast approximation of pygeodesy3.acos1(x).

Fast approximation of pygeodesy3.asin1(x).

Fast approximation of atan(x) for 0 <= x <= 1, unchecked.

Fast approximation of atan2(y, x).

Convert Imperial fathom to meter.

Arguments:

• fathoms - Value in fathoms (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid fathoms.

Return the average of two values.

Arguments:

• v1 - One value (scalar).
• v2 - Other value (scalar).
• f - Optional fraction (float).

Returns:

v1 + f * (v2 - v1) (float).

Return the precision dot product sum(a[i] * b[i] for i=0..len(a)).

Arguments:

• a - Iterable, list, tuple, etc. (scalars).
• b - All positional arguments (scalars).

Returns:

Dot product (float).

Raises:

• LenError - Unequal len(a) and len(b).

Return the precision dot product start + sum(a[i] * b[i] * c[i] for i=0..len(a)).

Arguments:

• a - Iterable, list, tuple, etc. (scalars).
• b - Iterable, list, tuple, etc. (scalars).
• c - Iterable, list, tuple, etc. (scalars).
• start - Optional bias (scalar).

Returns:

Dot product (float).

Raises:

• LenError - Unequal len(a), len(b) and/or len(c).
• OverflowError - Partial 2sum overflow.

Evaluate the polynomial sum(cs[i] * x**i for i=0..len(cs)) using the Horner form.

Arguments:

• x - Polynomial argument (scalar).
• cs - Polynomial coeffients (scalars).

Returns:

Horner value (float).

Raises:

• OverflowError - Partial 2sum overflow.
• TypeError - Non-scalar x.
• ValueError - No cs coefficients or x is not finite.

Interpolate using Inverse Distance Weighting (IDW).

Arguments:

• xs - Known values (scalars).
• ds - Non-negative distances (scalars).
• beta - Inverse distance power (int, 0, 1, 2, or 3).

Returns:

Interpolated value x (float).

Raises:

• LenError - Unequal or zero len(ds) and len(xs).
• ValueError - Invalid beta, negative ds value, weighted ds below EPS.

Compute the accurate mean sum(xs[i] for i=0..len(xs)) / len(xs).

Arguments:

• xs - Values (scalar or Fsum instances).

Returns:

Mean value (float).

Raises:

• OverflowError - Partial 2sum overflow.
• ValueError - No xs values.

Compute the accurate mean sum(xs[i] for i=0..len(xs)) / len(xs).

Arguments:

• xs - Values (scalar or Fsum instances).

Returns:

Mean value (float).

Raises:

• OverflowError - Partial 2sum overflow.
• ValueError - No xs values.

Evaluate the polynomial sum(cs[i] * x**i for i=0..len(cs)) [/ over].

Arguments:

• x - Polynomial argument (scalar).
• cs - Polynomial coeffients (scalars), all positional.
• over - Optional, final divisor (scalar

Returns:

Polynomial value (float).

Raises:

• OverflowError - Partial 2sum overflow.
• TypeError - Non-scalar x.
• ValueError - No cs coefficients or x is not finite.

Return a series of powers [x**i for i=1..n].

Arguments:

• x - Value (scalar).
• n - Highest exponent (int).
• alts - Only alternating powers, starting with this exponent (int).

Returns:

Powers of x (floats or ints).

Raises:

• TypeError - Non-scalar x or n not int.
• ValueError - Non-finite x or non-positive n.

Iterable product, like math.prod or numpy.prod.

Arguments:

• xs - Terms to be multiplied, an iterable, list, tuple, etc. (scalars).
• start - Initial term, also the value returned for an empty xs (scalar).

Returns:

The product (float).

Generate a range of floats.

Arguments:

• start - First value (float).
• number - The number of floats to generate (int).
• step - Increment value (float).

Returns:

A generator (floats).

Apply a function of two arguments cumulatively to the items of a sequence, from left to right, so as to reduce the sequence to a single value. For example, reduce(lambda x, y: x+y, [1, 2, 3, 4, 5]) calculates ((((1+2)+3)+4)+5). If initial is present, it is placed before the items of the sequence in the calculation, and serves as a default when the sequence is empty.

Returns: value

Remainder in range [-y / 2, y / 2].

Arguments:

• x - Numerator (scalar).
• y - Modulus, denominator (scalar).

Returns:

Remainder (scalar, preserving signed 0.0) or NAN for any non-finite x.

Raises:

• ValueError - Infinite or near-zero y.

Precision floating point summation based on or like Python's math.fsum.

Arguments:

• xs - Iterable, list, tuple, etc. of values (scalar or Fsum instances).
• floats - Optionally, use floats=True iff all xs are known to be float.

Returns:

Precision fsum (float).

Raises:

• OverflowError - Partial 2sum overflow.
• TypeError - Non-scalar xs value.
• ValueError - Invalid or non-finite xs value.

Precision floating point summation of a few arguments, 1-primed.

Arguments:

• xs - Iterable, list, tuple, etc. of values (scalar or Fsum instances).
• floats - Optionally, use floats=True iff all xs are known to be float.

Returns:

Precision fsum (float).

Precision floating point summation of a few arguments, 1-primed.

Arguments:

• xs - Values to be added (scalar or Fsum instances), all positional.
• floats - Optionally, use floats=True iff all xs are known to be float.

Returns:

Precision fsum (float).

Precision floating point summation of all positional arguments.

Arguments:

• xs - Values to be added (scalar or Fsum instances), all positional.
• floats - Optionally, use floats=True iff all xs are known to be float.

Returns:

Precision fsum (float).

Convert International, US Survey, French or German feet to meter.

Arguments:

• feet - Value in feet (scalar).
• usurvey - If True, convert US Survey foot else ...
• pied - If True, convert French pied-du-Roi else ...
• fuss - If True, convert German Fuss, otherwise International foot to meter.

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid feet.

Convert a furlong to meter.

Arguments:

• furlongs - Value in furlongs (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid furlongs.

Convert radians to grades (aka gons or gradians).

Arguments:

• rad - Angle (radians).

Returns:

Angle (grades).

Convert grades (aka gons or gradians) to degrees.

Arguments:

• gon - Angle (grades).

Returns:

Angle (degrees).

Convert grades (aka gons or gradians) to radians.

Arguments:

• gon - Angle (grades).

Returns:

Angle (radians).

Convert radians to grades (aka gons or gradians) and wrap [0..+400).

Arguments:

• rad - Angle (radians).

Returns:

Angle, wrapped (grades).

Compute the norm sqrt(1 + x**2).

Arguments:

• x - Argument (scalar).

Returns:

Norm (float).

Compute the squared norm x**2 + y**2.

Arguments:

• x - X argument (scalar).
• y - Y argument (scalar).

Returns:

x**2 + y**2 (float).

Compute the squared norm sum(x**2 for x in xs).

Arguments:

• xs - X arguments (scalars), all positional.

Returns:

Squared norm (float).

Raises:

• OverflowError - Partial 2sum overflow.
• ValueError - Invalid or no xs value.

Compute the norm sqrt(sum(x**2 for x in xs)).

Similar to Python 3.8+ n-dimension math.hypot, but exceptions, nan and infinite values are handled differently.

Arguments:

• xs - X arguments (scalars), all positional.

Returns:

Norm (float).

Raises:

• OverflowError - Partial 2sum overflow.
• ValueError - Invalid or no xs values.

Compute the intersection point of two (2- or 3-D) lines, each defined by two points or by a point and a bearing.

Arguments:

• start1 - Start point of the first line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• end1 - End point of the first line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple) or the bearing at start1 (compass degrees).
• start2 - Start point of the second line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• end2 - End point of the second line (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple) or the bearing at start2 (Ccompass degrees).
• eps - Tolerance for skew line distance and length (EPS).
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).
• Vector_and_kwds - Optional class Vector=None to return the intersection points and optional, additional Vector keyword arguments, otherwise start1's (sub-)class.

Returns:

An Intersection3Tuple(point, outside1, outside2) with point an instance of Vector or start1's (sub-)class.

Raises:

• IntersectionError - Invalid, skew, non-co-planar or otherwise non-intersecting lines.

Check whether a point is colinear with two other (2- or 3-D) points.

Arguments:

• point - The point (Vector3d, Vector3Tuple or Vector4Tuple).
• point1 - First point (Vector3d, Vector3Tuple or Vector4Tuple).
• point2 - Second point (Vector3d, Vector3Tuple or Vector4Tuple).
• eps - Tolerance (scalar), same units as x, y and z.
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

True if point is colinear point1 and point2, False otherwise.

Raises:

• TypeError - Invalid point, point1 or point2.

Convert kilo meter to meter (m).

Arguments:

• km - Value in kilo meter (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid km.

Convert meter to nautical miles (NM).

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in NM (float).

Raises:

• ValueError - Invalid meter.

Convert meter to statute miles (SM).

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in SM (float).

Raises:

• ValueError - Invalid meter.

Convert meter to UK chains.

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in chains (float).

Raises:

• ValueError - Invalid meter.

Convert a distance to an angle along the equator or along the parallel at an other (geodetic) latitude.

Arguments:

• distance - Distance (meter, same units as radius).
• radius - Mean earth radius, ellipsoid or datum (meter, an Ellipsoid, Ellipsoid2, Datum or a_f2Tuple).
• lat - Parallel latitude (degrees90, str).

Returns:

Angle (degrees) or INF for near-polar lat.

Raises:

• RangeError - Latitude lat outside valid range and pygeodesy3.rangerrors set to True.
• TypeError - Invalid radius.
• ValueError - Invalid distance, radius or lat.

Convert meter to Imperial fathoms.

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in fathoms (float).

Raises:

• ValueError - Invalid meter.

Convert meter to International, US Survey, French or or German feet (ft).

Arguments:

• meter - Value in meter (scalar).
• usurvey - If True, convert to US Survey foot else ...
• pied - If True, convert to French pied-du-Roi else ...
• fuss - If True, convert to German Fuss, otherwise to International foot.

Returns:

Value in feet (float).

Raises:

• ValueError - Invalid meter.

Convert meter to furlongs.

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in furlongs (float).

Raises:

• ValueError - Invalid meter.

Convert meter to kilo meter (Km).

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in Km (float).

Raises:

• ValueError - Invalid meter.

Convert a distance to an angle along the equator or along the parallel at an other (geodetic) latitude.

Arguments:

• distance - Distance (meter, same units as radius).
• radius - Mean earth radius, ellipsoid or datum (meter, an Ellipsoid, Ellipsoid2, Datum or a_f2Tuple).
• lat - Parallel latitude (degrees90, str).

Returns:

Angle (radians) or INF for near-polar lat.

Raises:

• RangeError - Latitude lat outside valid range and pygeodesy3.rangerrors set to True.
• TypeError - Invalid radius.
• ValueError - Invalid distance, radius or lat.

Convert meter to French toises.

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in toises (float).

Raises:

• ValueError - Invalid meter.

Convert meter to UK yards.

Arguments:

• meter - Value in meter (scalar).

Returns:

Value in yards (float).

Raises:

• ValueError - Invalid meter.

Return the radius and Meeus' Type of the smallest circle through or containing three (2- or 3-D) points.

Arguments:

• point1 - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point2 - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point3 - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• circum - If True return the circumradius and circumcenter always, overriding Meeus' Type II case (bool).
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Meeus2Tuple(radius, Type), with Type the circumcenter iff circum=True.

Raises:

• IntersectionError - Near-coincident or -colinear points, iff circum=True.
• TypeError - Invalid point1, point2 or point3.

Locate the point between two points closest to a reference (2- or 3-D).

Arguments:

• point - Reference point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• point1 - Start point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• point2 - End point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• within - If True return the closest point between both given points, otherwise the closest point on the extended line through both points (bool).
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).
• Vector - Class to return closest point (Cartesian, Vector3d or Vector3Tuple) or None.
• Vector_kwds - Optional, additional Vector keyword arguments, ignored if Vector is None.

Returns:

Closest point, either point1 or point2 or an instance of the point's (sub-)class or Vector if not None.

Raises:

• TypeError - Invalid point, point1 or point2.

Locate the point on a path or polygon closest to a reference point.

The closest point on each polygon edge is either the nearest of that edge's end points or a point in between.

Arguments:

• point - Reference point (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• points - The path or polygon points (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple[]).
• closed - Optionally, close the path or polygon (bool).
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).
• Vector_and_kwds - Optional class Vector=None to return the closest point and optional, additional Vector keyword arguments, otherwise point's (sub-)class.

Returns:

A NearestOn6Tuple(closest, distance, fi, j, start, end) with the closest, the start and the end point each an instance of the Vector keyword argument of if {Vector=None} or not specified, an instance of the reference point's (sub-)class.

Raises:

• PointsError - Insufficient number of points
• TypeError - Non-cartesian point and points.

Normalize a 2-dimensional vector.

Arguments:

• x - X component (scalar).
• y - Y component (scalar).

Returns:

2-Tuple (x, y), normalized.

Raises:

• ValueError - Invalid x or y or zero norm.

Normalize all n-dimensional vector components.

Arguments:

• xs - Components (scalars), all positional.

Returns:

Yield each component, normalized.

Raises:

• ValueError - Invalid or insufficent xs or zero norm.

Parse an "x, y, z" string.

Arguments:

• str3d - X, y and z values (str).
• sep - Optional separator (str).
• Vector - Optional class (Vector3d).
• Vector_kwds - Optional Vector keyword arguments, ignored if Vector is None.

Returns:

A Vector instance or if Vector is None, a named Vector3Tuple(x, y, z).

Raises:

• VectorError - Invalid str3d.

Convert an angle to a distance along the equator or along the parallel at an other (geodetic) latitude.

Arguments:

• rad - The angle (radians).
• radius - Mean earth radius, ellipsoid or datum (meter, Ellipsoid, Ellipsoid2, Datum or a_f2Tuple).
• lat - Parallel latitude (degrees90, str).

Returns:

Distance (meter, same units as radius or equatorial and polar radii) or 0.0 for near-polar lat.

Raises:

• RangeError - Latitude lat outside valid range and pygeodesy3.rangerrors set to True.
• TypeError - Invalid radius.
• ValueError - Invalid rad, radius or lat.

Convert and wrap degrees to radians [-PI..+PI].

Arguments:

• deg - Angle (degrees).

Returns:

Radians, wrapped (radiansPI)

Convert and wrap degrees to radians [0..+2PI).

Arguments:

• deg - Angle (degrees).

Returns:

Radians, wrapped (radiansPI2)

Convert and wrap degrees to radians [-3PI/2..+PI/2].

Arguments:

• deg - Angle (degrees).

Returns:

Radians, wrapped (radiansPI_2)

Return the radii of the In-, Soddy and Tangent circles of a (2- or 3-D) triangle.

Arguments:

• point1 - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point2 - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point3 - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Radii11Tuple(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s).

Raises:

• TriangleError - Near-coincident or -colinear points.
• TypeError - Invalid point1, point2 or point3.

Return the sine and cosine of an angle in radians.

Arguments:

• rad - Angle (radians).

Returns:

2-Tuple (sin(rad), cos(rad)).

Return the sine and cosine of angle(s) in radians.

Arguments:

• rads - One or more angles (radians).

Returns:

Yield the sin(rad) and cos(rad) for each angle.

Return the sine and cosine of an angle in degrees.

Arguments:

• deg - Angle (degrees).
• adeg - Optional correction (degrees).

Returns:

2-Tuple (sin(deg_), cos(deg_), deg_ = deg + adeg).

Return the sine and cosine of angle(s) in degrees.

Arguments:

• degs - One or more angles (degrees).

Returns:

Yield the sin(deg) and cos(deg) for each angle.

Return the sine, cosine and tangent of an angle in radians.

Arguments:

• rad - Angle (radians).

Returns:

3-Tuple (sin(rad), cos(rad), tan(rad)).

Return the radius and center of the inner Soddy circle of a (2- or 3-D) triangle.

Arguments:

• point1 - First point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point2 - Second point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• point3 - Third point (Cartesian, Vector3d, Vector3Tuple, Vector4Tuple or Vector2Tuple if useZ=False).
• eps - Tolerance for function pygeodesy3.trilaterate3d2 if useZ is True otherwise pygeodesy3.trilaterate2d2.
• useZ - If True, use the Z components, otherwise force z=INT0 (bool).

Returns:

Soddy4Tuple(radius, center, deltas, outer). The center, an instance of point1's (sub-)class, is co-planar with the three given points. The outer Soddy radius may be INF.

Raises:

• ImportError - Package numpy not found, not installed or older than version 1.10 and useZ is True.
• IntersectionError - Near-coincident or -colinear points or a trilateration or numpy issue.
• TypeError - Invalid point1, point2 or point3.

Return the square root iff x > EPS02.

Arguments:

• x - Value (scalar).
• Error - Error to raise for negative x.

Returns:

Square root (float) or 0.0.

Return the square root, cubed sqrt(x)**3 or sqrt(x**3).

Arguments:

• x - Value (scalar).

Returns:

Square root cubed (float).

Raises:

• ValueError - Negative x.

Compute a side of a right-angled triangle from sqrt(h**2 - b**2).

Arguments:

• h - Hypotenuse or outer annulus radius (scalar).
• b - Triangle side or inner annulus radius (scalar).

Returns:

copysign(a, h) or unsigned 0.0 (float).

Raises:

• TypeError - Non-scalar h or b.
• ValueError - If abs(h) < abs(b).

Compute the tangent of half angle, 90 degrees rotated.

Arguments:

• rad - Angle (radians).

Returns:

tan((rad + PI/2) / 2) (float).

Compute the tangent of half angle.

Arguments:

• rad - Angle (radians).
• semi - Angle or edge name and index for semi-circular error.

Returns:

tan(rad / 2) (float).

Raises:

• ValueError - If rad is semi-circular and semi is given.

Return the tangent of an angle in degrees.

Arguments:

• deg - Angle (degrees).
• error_kwds - Error to raise (ValueError).

Returns:

tan(deg).

Raises:

• ValueError - If pygeodesy3.isnear0(cos(deg).

Return the tangent of angle(s) in degrees.

Arguments:

• degs - One or more angles (degrees).
• error_kwds - Error to raise (ValueError).

Returns:

Yield the tan(deg) for each angle.

Raises:

• ValueError - See pygeodesy3.tand.

Convert French toises to meter.

Arguments:

• toises - Value in toises (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid toises.

Trilaterate three circles, each given as a (2-D) center and a radius.

Arguments:

• x1 - Center x coordinate of the 1st circle (scalar).
• y1 - Center y coordinate of the 1st circle (scalar).
• radius1 - Radius of the 1st circle (scalar).
• x2 - Center x coordinate of the 2nd circle (scalar).
• y2 - Center y coordinate of the 2nd circle (scalar).
• radius2 - Radius of the 2nd circle (scalar).
• x3 - Center x coordinate of the 3rd circle (scalar).
• y3 - Center y coordinate of the 3rd circle (scalar).
• radius3 - Radius of the 3rd circle (scalar).
• eps - Tolerance to check the trilaterated point delta on all 3 circles (scalar) or None for no checking.
• Vector_and_kwds - Optional class Vector=None to return the trilateration and optional, additional Vector keyword arguments, otherwise (Vector3d).

Returns:

Trilaterated point as Vector(x, y, **Vector_kwds) or Vector2Tuple(x, y) if Vector is None..

Raises:

• IntersectionError - No intersection, near-concentric or -colinear centers, trilateration failed some other way or the trilaterated point is off one circle by more than eps.
• UnitError - Invalid radius1, radius2 or radius3.

Trilaterate three spheres, each given as a (3-D) center and a radius.

Arguments:

• center1 - Center of the 1st sphere (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• radius1 - Radius of the 1st sphere (same units as x, y and z).
• center2 - Center of the 2nd sphere (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• radius2 - Radius of this sphere (same units as x, y and z).
• center3 - Center of the 3rd sphere (Cartesian, Vector3d, Vector3Tuple or Vector4Tuple).
• radius3 - Radius of the 3rd sphere (same units as x, y and z).
• eps - Pertubation tolerance (scalar), same units as x, y and z or None for no pertubations.
• Vector_and_kwds - Optional class Vector=None to return the trilateration and optional, additional Vector keyword arguments, otherwise center1's (sub-)class.

Returns:

2-Tuple with two trilaterated points, each a Vector instance. Both points are the same instance if all three spheres abut/intersect in a single point.

Raises:

• ImportError - Package numpy not found, not installed or older than version 1.10.
• IntersectionError - Near-concentric, -colinear, too distant or non-intersecting spheres.
• NumPyError - Some numpy issue.
• TypeError - Invalid center1, center2 or center3.
• UnitError - Invalid radius1, radius2 or radius3.

Truncate to the given number of digits.

Arguments:

• x - Value to truncate (scalar).
• ndigits - Number of digits (int), aka precision.

Returns:

Truncated x (float).

Unroll longitudinal delta and wrap longitude in degrees.

Arguments:

• lon1 - Start longitude (degrees).
• lon2 - End longitude (degrees).
• wrap - If True, wrap and unroll to the (-180..+180] range (bool).

Returns:

2-Tuple (lon2-lon1, lon2) unrolled (degrees, degrees).

Unroll longitudinal delta and wrap longitude in radians.

Arguments:

• rad1 - Start longitude (radians).
• rad2 - End longitude (radians).
• wrap - If True, wrap and unroll to the (-PI..+PI] range (bool).

Returns:

2-Tuple (rad2-rad1, rad2) unrolled (radians, radians).

Wrap degrees to [-180..+180].

Arguments:

• deg - Angle (degrees).

Returns:

Degrees, wrapped (degrees180).

Wrap degrees to [0..+360).

Arguments:

• deg - Angle (degrees).

Returns:

Degrees, wrapped (degrees360).

Wrap degrees to [-90..+90].

Arguments:

• deg - Angle (degrees).

Returns:

Degrees, wrapped (degrees90).

Wrap radians to [-PI..+PI].

Arguments:

• rad - Angle (radians).

Returns:

Radians, wrapped (radiansPI).

Wrap radians to [0..+2PI).

Arguments:

• rad - Angle (radians).

Returns:

Radians, wrapped (radiansPI2).

Wrap radians to [-PI/2..+PI/2].

Arguments:

• rad - Angle (radians).

Returns:

Radians, wrapped (radiansPI_2).

Define the operation for the keyword argument wrap=True, across pygeodesy3: wrap, normalize or no-op. For backward compatibility, the default is wrap.

Arguments:

• normal - If True, normalize lat- and longitude using normal or normal_, if False, wrap the lat- and longitude individually by wrap90 or wrapPI_2 respectively wrap180, wrapPI or if None, leave lat- and longitude unchanged. Do not supply any value to get the current setting.

Returns:

The previous wrap_normal setting (bool or None).

Convert UK yards to meter.

Arguments:

• yards - Value in yards (scalar).

Returns:

Value in meter (float).

Raises:

• ValueError - Invalid yards.

Return the 6-th, zenzi-cubic root, x**(1 / 6).

Arguments:

• x - Value (scalar).

Returns:

Zenzi-cubic root (float).

Raises:

• ValueError - Negative x.

Return the 8-th, zenzi-quartic or squared-quartic root, x**(1 / 8).

Arguments:

• x - Value (scalar).

Returns:

Zenzi-quartic root (float).

Raises:

• ValueError - Negative x.