Stereographic projection calculator

It uses the spherical azimuthal stereographic formula, preserves local angles, and reports X toward the east and Y toward the north. Set the sphere radius to control the output unit; the default mean Earth radius produces meters. The calculation is deterministic, runs without network access, and rejects the single antipodal point that cannot have a finite stereographic image. Use the browser tool interactively, or call the same calculation through the API for $0.002 per successful request.

What the centered stereographic projection calculates

A stereographic projection maps points on a sphere to a plane while preserving angles at sufficiently small scales. This calculator uses an oblique spherical form, meaning the tangent point can be any latitude and longitude rather than being fixed at a pole or the equator. The latitude and longitude fields identify the point being mapped, while center latitude and center longitude define the origin of the plane. The center itself always returns X zero and Y zero. Positive X points east at the center and positive Y points north, giving a practical local coordinate frame for plotting and comparison. The scale factor grows with angular distance from the center. Nearby shapes retain their angles, but distances and areas increasingly expand toward the far side of the sphere. The exact antipode of the center lies at infinity and is therefore rejected rather than represented by an arbitrary large number. Results also echo every coordinate, the sphere radius, chosen precision, projection name, and axis convention so downstream code does not have to guess which orientation or model produced the numbers. This is a spherical calculation, not an ellipsoidal datum transformation and not an EPSG coordinate reference system lookup. It does not apply false eastings, false northings, grid shifts, or a scale factor selected by a national mapping agency. Those parameters belong in a fully specified cartographic CRS workflow; this tool deliberately answers the direct mathematical question about stereographic projection around a supplied center.

How to choose the center, radius, and precision

Enter all four angles in decimal degrees. Target and center latitudes must fall from negative ninety to positive ninety, and longitudes must fall from negative one hundred eighty to positive one hundred eighty. Choose the center near the region you intend to display because stereographic scale distortion is smallest at the center and grows monotonically away from it. For a city map, a survey sketch, or a regional visualization, a center near the middle of the observations normally yields more useful coordinates than a distant conventional origin. Radius controls the unit and overall scale. The default is 6,371,008.8, the IUGG mean Earth radius in meters, so X and Y are returned in meters. Supplying a radius in kilometers produces kilometers; supplying one creates dimensionless coordinates on a unit sphere. This flexibility is useful for geometry lessons and normalized graphics, but all points in one dataset must use the same radius and center if they are to share a plane. Precision controls only output rounding, not the internal trigonometric calculation. Six decimal places is the default, and values from zero through twelve are accepted. More decimals can help create reproducible software fixtures, although they do not make a spherical Earth model more physically accurate. The implementation converts degrees to radians, computes the standard azimuthal denominator, derives a common scale, and then calculates eastward X and northward Y. It never wraps invalid longitudes or clips latitudes silently, because hidden normalization can conceal swapped fields or malformed source data.

Interpreting results and recognizing the projection limits

Use the returned X and Y as coordinates on one plane only when every source point shares the same center and radius. Subtracting two projected positions can be convenient for local graphics or approximate planar analysis, but the resulting straight-line distance is not generally the same as a geodesic distance on Earth. Stereographic projection is conformal: it preserves angles locally, not lengths or areas globally. This makes it valuable for directional diagrams, polar charts, educational maps, local plotting, and transformations where angular fidelity matters. It is less suitable for comparing region areas or measuring long routes across a continent without a distortion study. As the target approaches the antipode of the chosen center, the denominator approaches zero, scale tends toward infinity, and small changes in geographic coordinates create enormous changes on the plane. The calculator rejects the antipodal singularity with an invalid-input error. Large finite coordinates near that point may still be mathematically correct but impractical for a display, which is a reason to recenter the projection around the actual area of interest. The method treats longitude differences through sine and cosine, so crossing the antimeridian does not require a special branch when both supplied longitude values remain in the documented range. At either geographic pole, longitude labels refer to the same physical point; tiny floating-point orientation differences can nevertheless reflect the supplied meridian, so applications that need a single canonical pole representation should normalize that convention before projection. Interactive use is free in the browser. Automated calls cost $0.002 for each successful request and return the same deterministic calculation without storing the coordinates.