Hotine Oblique Mercator Projection Calculator
This calculator applies the Hotine Oblique Mercator projection on the ellipsoid to a single point. The central line is defined by two points, as in Snyder's alternative A, and the calculation runs forward only: from the latitude and longitude of a point to its x and y coordinates and the point scale factor k.
It follows equations 9-11 to 9-34 of J. P. Snyder, Map Projections - A Working Manual (USGS Professional Paper 1395, 1987), in double precision with no rounding inside. Besides x, y and k it shows the intermediate values v and u, the longitude λ0 and the angles γ0 and αc, and it writes out every formula with the values you typed.
Calculator
Formulas with your values
- e = √e²
- t(p) = tan(π/4 − p/2) / ((1 − e sin p)/(1 + e sin p))^(e/2)
- B = √(1 + e² cos⁴φ0 / (1 − e²))
- A = a B k0 √(1 − e²) / (1 − e² sin²φ0)
- D = B √(1 − e²) / (cos φ0 √(1 − e² sin²φ0))
- E = (D + √(D² − 1)) t(φ0)^B
- H = t(φ1)^B
- L = t(φ2)^B
- F = E / H
- G = (F − 1/F) / 2
- J = (E² − L H) / (E² + L H)
- P = (L − H) / (L + H)
- λ0 = (λ1 + λ2)/2 − atan(J tan(B (λ1 − λ2)/2) / P) / B
- γ0 = atan(sin(B (λ1 − λ0)) / G)
- αc = asin(D sin γ0)
- Q = E / t(φ)^B
- S = (Q − 1/Q) / 2
- T = (Q + 1/Q) / 2
- V = sin(B (λ − λ0))
- U = (−V cos γ0 + S sin γ0) / T
- v = A ln((1 − U)/(1 + U)) / (2B)
- u = A atan((S cos γ0 + V sin γ0) / cos(B (λ − λ0))) / B
- k = A cos(B u/A) √(1 − e² sin²φ) / (a cos φ cos(B (λ − λ0)))
- x = v cos αc + u sin αc + x0
- y = u cos αc − v sin αc + y0
How to use the calculator
- Enter the ellipsoid: the semi-major axis a in any length unit and the eccentricity squared e², which is dimensionless and between 0 and 1.
- Enter the scale factor k0 (dimensionless, above 0) and the centre latitude φ0 in degrees, from 1 to 89.
- Enter the two points that define the central line, φ1, λ1 and φ2, λ2 in degrees with west negative: latitudes from 1 to 89, φ1 above φ2, and λ1 below λ2.
- Enter the latitude φ and longitude λ of the point in degrees, and the false easting x0 and false northing y0 in the unit of a.
- Results update as you type, and the Calculate button recomputes them. Read x, y and k, then check each step in the formulas listed with your values.
Note: only a north-west to south-east line is supported, with the centre and the line points in the north. This is Snyder alternative A (line defined by two points), not EPSG 9812 or 9815. No datum transformation is performed.
Source: J. P. Snyder, Map Projections - A Working Manual, USGS Professional Paper 1395, 1987.
Frequently asked questions
Which version of the Hotine Oblique Mercator projection does this calculator use?
It uses the ellipsoidal form with the central line defined by two points, Snyder alternative A, following equations 9-11 to 9-34 of USGS Professional Paper 1395 (1987). It is not EPSG method 9812 or 9815. It works forward only, from latitude and longitude to x, y and the point scale factor k.
What units do the inputs and results use?
The semi-major axis a can be in any length unit, and x, y, v, u, the false easting and the false northing are in that same unit. Latitudes, longitudes and the angles λ0, γ0 and αc are in degrees, with west longitudes negative. The eccentricity squared e², the scale factor k0 and the point scale factor k are dimensionless.
Which line positions are supported?
Only a north-west to south-east line, with the centre and the line points in the north. The centre latitude φ0 and the line latitudes φ1 and φ2 must be from 1 to 89 degrees, with φ1 above φ2 and λ1 below λ2, longitudes from −180 to 180 degrees with no wrapping, and |B (λ − λ0)| below 90 degrees. Anything else shows a message instead of a number.
Does the calculator convert between datums?
No. No datum transformation is performed. The latitude and longitude are used as entered, on the ellipsoid described by a and e².