Transformations, Cartesian Coordinate

When the coordinates of certain points have been determined in a coordinate system, there is often the need to know the coordinates of the points in another coordinate system. The calculation of the coordinates of the points in the second system based on the coordinates in the first system and the relationship between the two coordinate systems is referred to as coordinate transformation. This entry describes coordinate transformations between Cartesian coordinate systems.

For example, in Figure 1, if the 2D coordinates of points A, B, C, and D in the x-y system are known, they can be transformed into the X-Y system if the relationship between the two systems is known.

In general, coordinate transformations between Cartesian systems can be expressed as

or

where (x, y, z) and (X, Y, Z) are the 3D coordinates of a point in the x-y-z and X-Y-Z systems respectively, and fx, fy, fz, fX, fY and fZ are functions (transformation models) relating the two coordinate systems. Equation 1 transforms the coordinates of any point from the x-y-z system to the X-Y-Z system, and Equation 2 transforms the coordinates from the X-Y-Z system to the x-y-z system.

Coordinate transformations between Cartesian systems can often be interpreted as certain geometrical changes, typically, the translations or the shifts of theO[Page 487]origin, rotation, warping, and scale change, through which one of the coordinate systems is made to overlap exactly with the second system. For example, for the two coordinate systems shown in Figure 1, the translations of the origin of one of the systems are c and f, respectively, in the X and Y directions, and the rotation angle for the x-y axes to become parallel with the X-Y axes is α.

Figure 1 Points in Two Different Coordinate Systems

Coordinate Transformation Models

Different coordinate transformation models are suited to different kinds of problems. Each model has its own special properties. Some of the most commonly used models for transforming coordinates between Cartesian or near-Cartesian coordinate systems are summarized below.

2D Conformal Transformation

Conformal transformation is one of the most commonly used coordinate transformation models. The model is also known as similarity, or Helmert transformation. The general 2D conformal transformation model is as follows:

where (see Figure 1)

c, f translations of the origin of the x-y system (the old system) along the X-and Y-axes, respectively, or the coordinates of the origin of the x-y system in the X-Y system (the new system);

α rotation angle of the x-y axes to the X-Y axes. α is defined positive when the rotation is clockwise and is zero if the axes are parallel; and

s scale factor between the two systems. When the unit length of the new system is shorter than that of the old system, s is larger than 1, and vice versa.

Equation 3 transforms the coordinates of any point from the x-y system to the X-Y system. Coefficients c, f,α, and, s are the transformation parameters. The parameters define the relationship between the two coordinate systems.

Equation 3 is often written in the following form

where a, b, c, and f above represent another form of the four transformation parameters in the coordinate transformation model. Since in either Equation 3 or 4, four parameters define the relationship between the two reference systems completely, the model is a fourparameter transformation model.

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