III.4 - Transforming Coordinates from One Coordinate Plane to Another

Abstract

This chapter discusses transforming coordinates from one coordinate plane to another. A common problem in graphics requires converting Cartesian coordinates from one reference system to corresponding points on a different reference frame. Other applications might include registration of overlays on an existing map or drawing in which the coordinate systems of each can only be determined empirically. The chapter describes methods for conversion of local (digitized) coordinates to a common world coordinate system. To find a general transformation between 2D coordinate systems, coordinates of some corresponding points are known. A transformation will be determined that will then map any other point from one system to the other. Schematically, one wishes to convert the position of any point on A to its corresponding position on B as shown in the chapter by finding equations that convert (xi, yi) to (ζi, ni). In the simplest case, the transformation determines an origin offset between the two systems, a scale factor difference, and a relative rotation. To solve for these unknowns (three in each coordinate), at least three pairs of corresponding points are required to determine a unique transformation.