Isometric latitude calculator
Convert a geodetic latitude into the isometric latitude behind the ellipsoidal Mercator projection: 45° gives 0.876634653434599 radians on WGS-84, with the spherical term and the flattening correction shown separately. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Latitude
- 45
- Latitude deg
- 45
- Isometric latitude rad
- 0.8766346534
- Isometric latitude deg
- 50.2274658167
- Spherical term rad
- 0.881373587
Convert a geodetic latitude into isometric latitude, the vertical coordinate used to make the ellipsoidal Mercator projection conformal. This calculator applies the exact WGS-84 eccentricity correction rather than treating Earth as a sphere. Enter a latitude strictly between minus ninety and ninety degrees and receive the dimensionless coordinate in radians, a degree-scaled version for inspection, and the two terms used in the calculation. The result is deterministic and suitable for projection code, geodesy lessons, GIS validation, and reproducible tests.
What isometric latitude means
Isometric latitude is not another angle measured directly on Earth. It is a transformed, dimensionless latitude conventionally written as psi, designed so that a conformal cylindrical projection can use a linear northing coordinate while preserving local angles. For geodetic latitude phi on an ellipsoid, psi equals the inverse hyperbolic sine of tan phi minus the eccentricity multiplied by the inverse hyperbolic tangent of eccentricity times sin phi.
The first part is the familiar spherical Mercator transformation. The second part corrects for the flattening of the reference ellipsoid. This capability fixes that reference to WGS-84, whose flattening is one divided by 298.257223563, and derives the first eccentricity from it.
At the equator, both terms vanish and the isometric latitude is zero. North of the equator it is positive, south of the equator it is negative, and equal-magnitude opposite latitudes produce equal-magnitude opposite results. As geodetic latitude approaches either pole, psi grows without bound.
What you can do with it
Ellipsoidal Mercator implementation
Generate the conformal vertical coordinate before applying a Mercator scale factor, false northing, or coordinate-system-specific units.
GIS formula validation
Compare an application's transformed latitude with a deterministic WGS-84 reference and inspect the spherical and eccentricity terms separately.
FAQ
What formula does the calculator use?
It computes psi = asinh(tan(phi)) - e times atanh(e times sin(phi)), with phi in radians and e derived from the WGS-84 flattening.
Is isometric latitude the same as geodetic latitude?
No. Geodetic latitude is the input angle on the ellipsoid; isometric latitude is an unbounded transformed coordinate used by conformal projections.
Why are plus and minus 90 degrees rejected?
Isometric latitude tends to positive or negative infinity at the poles.
Can I use the output directly as a Mercator northing?
Not by itself. Multiply the radian value by the ellipsoid semi-major axis and the projection scale, then apply any origin and false northing defined by the target coordinate reference system.
Why is the degree version of the result not a real latitude?
Because it is the isometric latitude multiplied by 180 and divided by π, only to make comparison easier. For 45° it reads 50.2274658167161, a value that does not point to any place on Earth and should not replace the latitude.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/geo/isometric-latitude \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"latitude":45}'const res = await fetch("https://api.kit.forhosting.com/geo/isometric-latitude", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"latitude": 45
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geo/isometric-latitude",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"latitude": 45
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geo/isometric-latitude", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"latitude":45}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"latitude":45}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geo/isometric-latitude", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"latitude": 45
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geo.isometric_latitude",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
The browser calculator is free; API processing costs $0.002 per successful item.
What isometric latitude means
For that reason, the calculator requires a finite latitude strictly inside the open interval from minus ninety to ninety degrees and rejects the endpoints instead of returning an infinity that JSON cannot represent reliably.
How to enter latitude and read the response
Provide latitude as decimal degrees in the required latitude field. Results are rounded to a stable significant precision so repeated runs and browser execution agree without hiding the meaningful accuracy of ordinary double-precision arithmetic. No longitude is needed because isometric latitude controls only the projection's vertical coordinate. The aliases lat and phi are also accepted for clients that already use common coordinate notation. Numbers and well-formed numeric strings are handled by the shared implementation, but blank text, non-numeric values, non-finite values, arrays, and missing fields raise an invalid_input error. The response echoes the accepted value as latitude_deg. The main result, isometric_latitude_rad, is psi in its standard dimensionless radian convention. isometric_latitude_deg multiplies psi by 180 over pi; it is convenient for human comparison but should not be mistaken for a new geographic latitude. spherical_term_rad exposes the uncorrected spherical Mercator term, while ellipsoidal_correction_rad shows the positive or negative WGS-84 correction that is subtracted. The response also names the ellipsoid and returns eccentricity, eccentricity_squared, and a stable method identifier.
Using the value in Mercator and geodesy workflows
The exact projection definition may also specify a central scale and origin, so this calculator intentionally returns the reusable transformed latitude rather than claiming to produce a complete projected coordinate. For complete longitude and latitude projection to a named EPSG system, use a dedicated projection tool that applies that system's datum, central meridian, units, and false offsets. The in-browser calculation is free and keeps the input local. In an ellipsoidal Mercator implementation, multiply isometric_latitude_rad by the chosen scale factor and the ellipsoid's semi-major axis to obtain a northing relative to the equator, then add any false northing required by the coordinate reference system. The spherical_term_rad and ellipsoidal_correction_rad fields are particularly useful when a discrepancy must be traced to an omitted eccentricity adjustment. Automated server, SDK, or batch workflows can call the API for $0.002 per successful item. The computation has no network access, randomness, clock dependency, lookup table, or iterative convergence, so the same accepted input always returns the same finite JSON result.
Integration use cases
Geodesy teaching and test fixtures
Demonstrate why ellipsoidal Mercator differs from spherical Mercator and pin reproducible expected values for projection exercises or software tests.
Technical questions
How much does the isometric latitude calculator cost?
The browser calculator is free. API calls cost $0.002 per successful item, while rejected invalid input is not a successful item.