Geodetic Coordinate Systems

Geodetic Coordinate Systems

• Focuses on conformal two and three dimensional coordinate transformations

Introduction

• Transforming points between coordinate systems is common in Geodesy, Geoinformatics, and GIS.
• Projective Transformations convert 3D shapes to 2D surfaces and vice versa, like geographic coordinates to Transverse Mercator.
• Affine Coordinate Transformations involve scale factor variations in different directions, like rubber sheeting and georeferencing.
• Conformal Coordinate Transformations preserve shapes, maintaining a uniform point scale factor, also known as similarity transformation, for example, datum transformation.

Scope

• Covers introductory procedures using least squares
• Focuses on conformal 2D and 3D transformations.
• Includes applications in GIS, such as ITRF and datum transformations.

2-D Coordinate Transformation (Helmert)

• Also known as four-parameter similarity transformation, maintaining the true shape after transformation
• This can be broken down into three steps: scaling, rotation, and translation.
• Scaling uses 1 parameter to equalize dimensions across coordinate systems.
• Rotation uses 1 parameter to align the reference axes of coordinate systems.
• Translations uses 2 parameters (∆X, ∆Y) to establish a common origin.
• Requires at least two control points shared between systems.
• Uniquely determines the four transformation parameters with a minimum of two points
• Least squares adjustment becomes possible when more than two control points are available.
• Any point in the original system can be transformed into the second system once transformation parameters are known.

Equation Development

• Points A, B, and C have coordinates known in both systems, while points 1-4 are only known in the xy system.
• Scaling: x' = Sx and y' = Sy (eq 1), maintains lengths between points in the xy system when converting to XY.
• Rotation: X' = x' cos 𝜃 – y' sin 𝜃 and Y' = x' sin 𝜃 + y' cos 𝜃 (eq 2).
• Translation: X = X' + TX and Y = Y' + TY (eq 3), is used to align origins, completing the coordinate transformation to XY.
• Combining the scaling, rotation, and translation: X = (S cos 𝜃)x – (S sin 𝜃)y + TX and Y = (S sin 𝜃)x + (S cos 𝜃)y + TY.
• Adding residuals and using S cos 𝜃 = a, S sin 𝜃 = b, TX = c, and TY = d gives: ax − by + c = X + vX and bx + ay + d = Y + vY (eq 4).

Application of Least Squares

• Equation 4 is the 2D conformal coordinate transformation equation, which has 4 unknowns: a, b, c, and d
• These unknowns represent the transformation parameters S (scale), 𝜃 (rotation), TX and TY (translation)
• Two observation equations are needed for each control point; thus, for three control points (A, B, and C), six equations can be written:
  • axa − bya + c = XA + vXA, bxa + aya + d = YA + vYA
  • axb − byb + c = XB + vXB, bxb + ayb + d = YB + vYB
  • axc − byc + c = XC + vXC, bxc + ayc + d = YC + vYC
• Expressing these in matrix form: AX = L + V, where A contains coefficients of the unknowns, X is the vector of unknowns, L is the vector of observations, and V is the vector of residuals.

• Scale factor (S) and rotation angle (𝜃) can be determined after adjustment:

  • 𝜃= tan-1 of (b/a)
  • S= a / cos 𝜃

Example of 2-D Helmert Transformation

• Coordinates provided for points A, B, and C in both E/N and x/y systems, along with x/y coordinates for points 1 and 2
• The observation equation is formed for each point:
  • axa – bya + c = EA +vEA
  • bxa + aya + d = NA + vNA
  • axb – byb + c = EB + vEB
  • bxb + ayb + d = NB + vNB
  • axc – byc + c = EC + vEC
  • bxc + ayc + d = NC + vNC
• Expressed in Matrix form as: AX = L + V, this is solved to find the transformation parameters.
• Using the solution:
  • E = (S cos 𝜃)x – (S sin 𝜃)y + TX
  • N = (S sin 𝜃)x + (S cos 𝜃)y + TY
• Coordinates for points 1 and 2 are transformed from x/y to E/N.

3-D Conformal Transformation

• Involves seven parameters: three rotations, three translations, and one scale factor
• Also know as a seven parameter similarity transformation
• Transfers points from one 3D coordinate system to another.
• Rotation matrix is obtained from 2D rotations about x, y, and z.
• Rotation 𝜃1 about the x-axis in matrix form:
  • X1= RX0, where X1 is the rotated coordinate, R1 is the rotation matrix, and X0 is the original coordinate, note eq a as well with the component matrices
• Rotation 𝜃2 about the y-axis in matrix form:
  • X2 = R2X1, with eq b.
• Rotation 𝜃3 about the z-axis in matrix form:
  • X = R3X2, with eq c
• Substituting a into b and into c gives:
  • X = R3R2r1X0 = RX0
• Three matrices produce single rotation matrix R for the transformation.
• Components = r11, r12, r13, r21, r22, r23, r31, r32, r33

3-D Transformation Mathematical Model

• Rotation matrix R is orthogonal so its inverse is equal to its transpose.
• Terms of matrix X multiplied by scale factor, S, and adding translation factors TX, TY and TZ. Translating to a common origin yields the mathematical model.
  • X = S(r11x + r21y + r31z) + TX,
  • Y = S(r12x + r22y + r32z) + TY,
  • Z = (r13x + r23y+ r33z) + TZ
• The equation above have 7 unknowns which include: S, 𝜃1, 𝜃2, 𝜃3, TX, TY, TZ
• Unique solutions need 7 equations so a minimum of 2 control stations along with known XY coordinates with plus 3 stations that XY coordinates known is needed.
• If there are more control points than needed, a least squares solution may be applied.
• The mathematical model calculation must be linearized to be solvable.