Authalic latitude calculator

Enter a latitude and, when needed, the inverse flattening of your reference ellipsoid. The result includes the authalic angle in degrees and radians, the normalized authalic function, and the eccentricity squared used in the calculation. WGS 84 is the default, making the tool immediately useful for global geographic data and equal-area projection work.

Why authalic latitude is used

A geographic latitude measured on an ellipsoid cannot simply be copied onto a sphere when area must remain correct. Ellipsoidal meridians and parallels distribute surface area differently from their spherical counterparts, especially as latitude increases. Authalic latitude supplies the needed transformation: a zone extending from the equator to a geodetic latitude on the ellipsoid has the same area as the corresponding zone extending to the authalic latitude on an equal-area sphere. This property makes the angle valuable in the derivation and implementation of equal-area map projections. The conversion changes latitude but does not describe a new physical position, distance, or bearing. It is an auxiliary coordinate used inside projection mathematics. At the equator and poles, authalic and geodetic latitude agree. At intermediate latitudes on an oblate Earth ellipsoid, the authalic magnitude is slightly smaller. That difference may look modest in degrees, yet ignoring it can introduce systematic area error when a spherical projection formula is applied to ellipsoidal data.

How the conversion is calculated

The calculator first derives flattening from the supplied inverse flattening, then obtains the ellipsoid's first eccentricity squared. It evaluates the standard authalic function q at the requested geodetic latitude and divides that value by q at the pole. The authalic latitude is the inverse sine of this normalized ratio. Normalization is the key step because it maps the complete range from the south pole to the north pole onto minus one through plus one before the inverse sine is taken. The implementation uses natural logarithms and double-precision arithmetic, clamps only the final ratio against tiny floating-point excursions outside that interval, and rounds the published fields to the requested number of decimal places. The returned q_ratio is also the sine of the authalic latitude, so it can be useful for checking another implementation. No semi-major axis is required: scale cancels when q is divided by its polar value. Only the ellipsoid's shape, represented by inverse flattening, affects this angular transformation.

Choosing inputs and interpreting results

Supply lat_deg as a signed decimal angle between minus 90 and 90. Positive values are north of the equator and negative values are south. If f_inv is omitted, the calculator uses the WGS 84 inverse flattening of 298.257223563, appropriate for GPS coordinates and most worldwide datasets. Provide another inverse flattening when your source coordinates are explicitly tied to a different reference ellipsoid; mixing a latitude from one datum with another ellipsoid's shape parameter makes the transformation internally inconsistent. The decimals option controls presentation rather than the internal calculation, which always uses full JavaScript numeric precision. Use the degree result for reports and projection workflows expressed in degrees, or the radian result when feeding trigonometric code. The transformation is antisymmetric, so changing only the sign of the input changes only the signs of the latitude-dependent outputs. Remember that authalic latitude alone does not complete an equal-area projection. A projection may additionally require the authalic sphere radius, central meridian, standard parallels, false origins, or other parameters defined by its equations.