Coordinate Transformation Calculator

Forward Intersection · Helmert Transformation · Site Plan · Surveying · Geodesy

Transformation Calculator

Select calculation method:

Forward Intersection (Stations → New Point)

Helmert Transformation (4-Parameter: Translation + Rotation + Scale)

Polar to Cartesian (Direction + Distance → Coordinates)

Station 1 (Known)

X₁ [m]

Y₁ [m]

Direction Angle to Target [gon] Bearing in gradians (gon) or decimal degrees

Distance 1 to Target [m]

Station 2 (Known)

X₂ [m]

Y₂ [m]

Direction Angle to Target [gon]

Distance 2 to Target [m]

Source System (Old Coordinates) - Point 1

X₁ old [m]

Y₁ old [m]

X₁ new [m]

Y₁ new [m]

Source System - Point 2

X₂ old [m]

Y₂ old [m]

X₂ new [m]

Y₂ new [m]

Point to Transform (old)

X old [m]

Y old [m]

Formulas & References

Forward Intersection:
X = X₁ + s₁ · cos(α₁) = X₂ + s₂ · cos(α₂)
Y = Y₁ + s₁ · sin(α₁) = Y₂ + s₂ · sin(α₂)
New point from two known stations with measured bearings and distances

Helmert Transformation (4-Parameter):
X_new = c_x + s·cos(ω)·X_old − s·sin(ω)·Y_old
Y_new = c_y + s·sin(ω)·X_old + s·cos(ω)·Y_old
s = Scale factor, ω = Rotation angle
c_x, c_y = Translations; computed from minimum 2 control points

Polar → Cartesian:
X = X₀ + s · cos(α)
Y = Y₀ + s · sin(α)
α = Direction angle [gon], s = Distance [m]
Conversion of polar coordinates (bearing, distance) to Cartesian (X, Y)

Practical Applications:

Forward Intersection: Survey of inaccessible points (building corners, forests, buildings) from two known stations. Standard in property surveying and building surveys.

Helmert Transformation: Alignment of survey data between different reference systems (e.g., local → UTM, or old survey → new survey). Standard method for coordinate system conversion.

Polar to Cartesian: Daily application in tachymetric surveying. Instruments provide bearing + distance; coordinates are calculated from these polar values.

Symbols & Units:

X, Y	Cartesian coordinates [m]
α, ω	Direction angle / Azimuth [gon] or [°]
s, d	Distance / Range [m]
c_x, c_y	Translations [m]
s_m	Scale factor [dimensionless or ppm]
gon	Gradian (400 gon = 360°, 1° ≈ 1.111 gon)

Technical Background

Coordinate transformation is a core process in surveying, geodesy, and technical planning. Several application scenarios exist:

Forward Intersection: A point is determined from at least two known stations using bearing and distance measurements. This is a classical surveying method and is used when a point cannot be directly measured from the instrument (e.g., measuring a building corner from two street control points).
Helmert Transformation: An affine transformation with 4 parameters (2 translations, 1 rotation, 1 scale). Used to convert between different coordinate systems, such as converting old local surveys to modern UTM or Gauß-Krüger coordinates.
Polar to Cartesian: Total stations and theodolites deliver measurements as bearing (azimuth) and distance. These must be converted to Cartesian coordinates (X, Y) for use in CAD or GIS software.

Angular Units: In surveying, angles are often given in gon (gradians) or decimal degrees. Conversion: 1 full circle = 400 gon = 360°. Correct units are essential.

Accuracy: Transformations require accurate measurement data and correct computation. Rounding errors can accumulate, especially with large coordinate values; double precision provides approximately 7–8 significant decimal digits.

Is this page helpful?

Thank you for your feedback!

Sorry about that
How can we improve it?