Notations

(φ, λ, h) - geographic coordinates where:
• φ - latitude;
• λ - longitude;
• h - altitude from the ellipsoid.

(X, Y, Z) - geocentric Cartesian coordinates with the X axes always orientated on the Greenwich meridian.

(N, E, H) - Cartesian coordinates in the projection plane where:
• N - coordinate on the north direction (y);
• E - coordinate on the east direction (x);
• H - orthometric height (elevation from geoid).

The geometrical parameters of the rotational ellipsoid

a - semi-major axes

f - flattening

b - semi-minor axes, b = a(1 - f)

N_e - radius of curvature in the prime vertical, N_e = a / [1 - e ² sin ² (φ)] ^½

e - the first eccentricity, e = [(a ² - b ²) / a ²] ^½

e'- the second eccentricity, e'= [(a ² - b ²) / b ²] ^½

Transforming the geographic coordinates (φ, λ, h) into geocentric Cartesian coordinates (X, Y, Z)

X = (N_e + h) cos(φ) cos(λ)

Y = (N_e + h) cos(φ) sin(λ)

Z = [N_e (1 - e ²) + h) sin(φ)

Transforming the geocentric Cartesian coordinates (X, Y, Z) into geographic coordinates (φ, λ, h)

The value of φ is calculated iteratively:

d = (X² + Y²) ^½

φ_ini = arctan[Z(1 + e' ²) / d]

φ_i+1 = arctan{[Z + N_e e ² sin(φ_i)] / d} (iterative)

λ = arctan(Y / X)

h = d / cos(φ) - N_e

1D Transformations

1. 3D plane rotation (5 parameters)

XS YS ZD    =  R(α1, α2, 0.0)  ×    XS YS ZS    +    TX TY TZ

where:
• (X_S, Y_S, Z_S) - coordinates in the projection system point source;
• (Z_D) - the coordinate on the elevation point destination;
• (T_X, T_Y, T_Z) - translations;
• R(α1, α2, 0.0) - the rotation matrix, α1, α2 around the X, Y axes and α3 = 0.0;

Note: To determine the parameters from the common points, there was applied iteratively the method of the minimum square.

2. Translation on the elevation (1 parameter)

T_Z = [ ∑_i=1..n (Z_D - Z_S)] / n

where:
• (Z_S) - the coordinate on the elevation point source;
• (Z_D) - the coordinate on the elevation point destination;
• (T_Z) - translation on the elevation;

Note: In the case of searching for the best combination the standard deviation is calculated as a deviation from the average.

2D Transformation

1. Conformal 2D Helmert transformation (4 parameters)

XD YD    =   m   ×    cos(R) -sin(R) sin(R) cos(R)    ×    XS YS    +    TX TY

where:
• (X_S, Y_S) - source point coordinate;
• (X_D, Y_D) - destination point coordinate;
• (T_X, T_Y) - translations;
• R - the rotation;
• m - scale factor, m = 1 + S / 10⁶ (S - scale factor in ppm).

Note: To determine the parameters from common points, the method of the minimum squares was applied.

2. Conformal 2D Helmert transformation with rotation origin (6 parameters)

XD YD    =   m   ×    cos(R) -sin(R) sin(R) cos(R)    ×    XS - XO YS - YO    +    TX + XO TY + YO

where:
• (X_S, Y_S) - source point coordinates;
• (X_D, Y_D) - destination point coordinates;
• (T_X, T_Y) - translations;
• (X_O, Y_O) - origin point of rotation X_O = ∑ X_S(i) / n, Y_O = ∑ Y_S(i) / n;
• R - the rotation;
• m - scale factor, m = 1 + S / 10⁶ (S - scale factor in ppm).

Note: To determine the parameters from common points, the method of the minimum squares was applied.

3. Affine 2D orthogonal transformation (5 parameters)

XD YD    =    mX 0 0 mY    ×    cos(R) -sin(R) sin(R) cos(R)    ×    XS YS    +    TX TY

where:
• (X_S, Y_S) - source point coordinates;
• (X_D, Y_D) - destination point coordinates;
• (T_X, T_Y) - translations;
• R - the rotation;
• (m_X, m_Y) - scale factors on X, Y.

Note: To determine the parameters from the common points, there was applied iteratively the method of the minimum square.

4. Affine 2D non-orthogonal transformation (6 parameters)

XD YD    =    1 ε 0 1    ×    mX 0 0 mY    ×    cos(R) -sin(R) sin(R) cos(R)    ×    XS YS    +    TX TY

where:
• (X_S, Y_S) - source point coordinates;
• (X_D, Y_D) - destination point coordinates;
• (T_X, T_Y) - translations;
• ε - non-orthogonality coefficient;
• R - the rotation;
• (m_X, m_Y) - scale factors on X, Y.

Note: To determine the parameters from common points, the method of the minimum squares was applied.

3D Transformations

1. Conformal 3D Helmert transformation the Bursa-Wolf method (7 or 15 parameters)

This method is the most used in the geodetic applications. This method was adopted in geodesy because the rotation angles between the ellipsoids are very small. Generally the producers of the GPS equipment use this method to transform the coordinates from an ellipsoid to another.

Because of the approximations made by this method the rotation angles are accepted only for values between -10” and +10”. If you use the transformation between two coordinate systems that calls for rotations bigger than ±10”, we recommend using the transformation method 3D Helmert with 7 parameters.

The approximations made by this method:

sin(dΘ) ≈ dΘ

cos(dΘ)≈ 1.0

available for rotation angles smaller than ±10''.

XD YD ZD    =  m  ×    1 -RZ +RY +RZ 1 -RX -RY +RX 1    ×    XS YS ZS    +    TX TY TZ

where:
• (X_D, Y_D, Z_D) - destination point coordinates;
• (X_S, Y_S, Z_S) - source point coordinates;
• (T_X, T_Y, T_Z) - translations;
• (R_X, R_Y, R_Z) - rotations;
• m - scale factor, m = 1 + S / 10⁶ (S - scale factor in ppm).

 

In the case of the 15-parametre time-dependant transformation the values of offsets and rotations and the scale factor are previously adjusted by the time value.

   T_X = T_X + δT_X (t - t₀)

   T_Y = T_Y + δT_Y (t - t₀)

   T_Z = T_Z + δT_Z (t - t₀)

   R_X = R_X + δR_X (t - t₀)

   R_Y = R_Y + δR_Y (t - t₀)

   R_Z = R_Z + δR_Z (t - t₀)

   S = S + δS (t - t₀), m = 1 + S / 10⁶ (S - scale factor in ppm).

where:
• t₀ - the transformation reference epoch;
• t - the epoch of the dynamic coordinates.

Note: To determine the parameters from common points the method of the minimum squares was applied.

2. Conformal 3D Helmert transformation the Molodenski-Badekas method (10 parameters)

This transformation is similar to the Bursa-Wolf method with the difference that the origin of rotation is moved near the transformation points. By reducing the distance between the rotation origin and the points to be transformed through this method the accuracy of the unknown T_X, T_Y, T_Z (translations) are obtained much better. The total precision of the transformation is better than the one obtained using the Bursa-Wolf method but from the practical point of view there are no differences between the transformed points with the two methods.

The approximations made by this method:

sin(dΘ) ≈ dΘ

cos(dΘ)≈ 1.0

available for rotation angles smaller than ±10''.

XD YD ZD    =  m  ×    1 -RZ +RY +RZ 1 -RX -RY +RX 1    ×    XS - XO YS - YO ZS - ZO    +    TX + XO TY + YO TZ + ZO

where:
• (X_S, Y_S, Z_S) - source point coordinates;
• (X_D, Y_D, Z_D) - destination point coordinates;
• (X_O, Y_O, Z_O) - origin point of rotation X_O = ∑ X_S(i) / n, Y_O = ∑ Y_S(i) / n, Z_O = ∑ Z_S(i) / n;
• (T_X, T_Y, T_Z) - translations;
• (R_X, R_Y, R_Z) - rotations;
• m - scale factor, m = 1 + S / 10⁶ (S - scale factor in ppm).

For X_O = 0.0; Y_O = 0.0; Z_O = 0.0 0 the transformation becomes a 7 parameter Bursa-Wolf parameter transformation.

Note: To determine the parameters from common points, the method of the minimum squares was applied.

3. Affine 3D transformation with 7, 8, 9 or 15 parameters

XD YD ZD    =  M(mX,mY,mZ)  ×  R(α1,α2,α3)  ×    XS YS ZS    +    TX TY TZ

where:
• (X_S, Y_S, Z_S) - source point coordinates;
• (X_D, Y_D, Z_D) - destination point coordinates;
• (T_X, T_Y, T_Z) - translations;
• R(α1, α2, α3) - the rotation matrix, α1, α2, α3 rotations around the axes X, Y, Z;
• M(m_X, m_Y, m_Z) - diagonal matrix of the scale factors.

For the 7 parameter transformation m_X = m_Y = m_Z, for the 8 parameter transformation m_X = m_Y.

This transformation isn’t limited to rotation angles of ±10” as the Bursa-Wolf and Molodenski-Badekas methods are.

In the case of the 15-parametre time-dependant transformation the values of offsets and rotations and the scale factor are previously adjusted by the time value.

   T_X = T_X + δT_X (t - t₀)

   T_Y = T_Y + δT_Y (t - t₀)

   T_Z = T_Z + δT_Z (t - t₀)

   R_X = R_X + δR_X (t - t₀)

   R_Y = R_Y + δR_Y (t - t₀)

   R_Z = R_Z + δR_Z (t - t₀)

   S = S + δS (t - t₀), m_X = m_Y = m_Z = 1 + S / 10⁶ (S - scale factor in ppm).

where:
• t₀ - the transformation reference epoch;
• t - the epoch of the dynamic coordinates.

Note: To determine the parameters from the common points, there was applied iteratively the method of the minimum square.

4. Affine 3D transformation with the rotation origin (12 parameters)

XD YD ZD    =  M(mX,mY,mZ)  ×  R(α1,α2,α3)  ×    XS - XO YS - YO ZS - ZO    +    TX + XO TY + YO TZ + ZO

where:
• (X_S, Y_S, Z_S) - source point coordinates;
• (X_D, Y_D, Z_D) - destination point coordinates;
• (X_O, Y_O, Z_O) - origin point of rotation X_O = ∑ X_S(i) / n, Y_O = ∑ Y_S(i) / n, Z_O = ∑ Z_S(i) / n;
• (T_X, T_Y, T_Z) - translations;
• R(α1, α2, α3) - the rotation matrix, α1, α2, α3 rotations around the axes X, Y, Z;
• M(m_X, m_Y, m_Z) - the diagonal matrix of the scale factors.

Note: To determine the parameters from the common points, there was applied iteratively the method of the minimum square.

The calculation of transformation coefficients A0, A1, A2, B0, B1, B2 for 2D affine transformation with 6 parameters

A0 = T_X

B0 = T_Y

B1 = m_Y * sin(R)

B2 = m_Y * cos(R)

A1 = m_X*cos(R) + ε * B1

A2 = -m_X*sin(R) + ε * B2

where:
• (T_X, T_Y) - translations;
• ε - non-orthogonality coefficient;
• R - the rotation;
• (m_X, m_Y) - scale factors.

Rotation directions

1. For the 2D transformations

2. For the 3D transformations

Note: If you use the parameters determined with other programs check, if the rotation angles correspond to these rotation directions, otherwise change the sign of the rotation angles.

Projections

The formulas with the projections used in TransLT are not presented in this document because of the large volume of information. Those interested can find them in Map Projections - A Working Manual author John P. Snyder published in the year 1987 by the United States Government Printing Office and in Guidance Note Number 7, part 2 - Coordinate Conversions and Transformations including Formulas revised in July 2012, published by the O.G.P.

The formulas were verified according to the tests presented in these two books.

Particular case for the Oblique Stereographic projection – calculation with constant coefficients

The direct conversion (φ, λ) to (N, E)

f  = 10 ^-4 (φ - φ₀)''

g = 10 ^-4 (λ - λ₀)''

F  =    f0 f1 f2 f3 f4 f5 f6    ;  Ga  =    g0 g2 g4 g6    ;  Gb  =    g1 g3 g5 g7

ΔN = ∑_j=0..3 {∑_i=0..6 [F(i) × a(i, j)] × G_a(j)}

ΔE = ∑_j=0..3 {∑_i=0..6[F(i) × b(i, j)] × G_b(j)}

N = F_N + ΔN × k

E = F_E + ΔE × k

The inverse conversion (N, E) to (φ, λ)

f  = 10^-5(N - F_N) / k

g = 10^-5(E - F_E) / k

F  =    f0 f1 f2 f3 f4 f5 f6    ;  Ga  =    g0 g2 g4 g6    ;  Gb  =    g1 g3 g5 g7

Δφ'' = Σ_j=0..3 {Σ_i=0..6 [F(i) × a(i, j)] × G_a(j)}

Δλ'' = Σ_j=0..3 {Σ_i=0..6 [F(i) × b(i, j)] × G_b(j)}

φ = φ₀ + Δφ''/3600

λ = λ₀ + Δλ''/3600

where:
• (φ₀, λ₀) - the geographic coordinates of natural origin
• k - the scale factor used to pass from a tangent plane to a secant plane
• F_N - translation on the north (false north)
• F_E - translation on the east (false east)
• a(i, j) - the matrix of constant coefficients
• b(i, j) - the matrix of constant coefficients

Polynomial transformations

The formulas for all types of polynomial transformations used in TransLT can be found in Guidance Note Number 7, part 2 - Coordinate Conversions and Transformations including Formulas revised in July 2012, published by O.G.P.

The formulas have been verified according to the tests presented in this document.

Coordinate offsets

The formulas for all methods of coordinate offset used in TransLT can be found in Guidance Note Number 7, part 2 - Coordinate Conversions and Transformations including Formulas revised in July 2012, published by O.G.P.

The formulas have been verified according to the tests presented in this document.