coordinatetransform

Coordinatetransform is a mathematical operation that converts coordinates from one coordinate system to another. Transformations arise when changing frames of reference, projecting spatial data, or mapping measurements between devices. They can operate in two or three dimensions and may be linear, affine, or nonlinear.

Mathematically, a coordinatetransform maps a coordinate vector x in source coordinates to a vector x' in target coordinates. Linear transforms use a matrix multiplication x' = Ax. Affine transforms add a translation, x' = Ax + b. In homogeneous coordinates, 2D uses 3x3 matrices and 3D uses 4x4 matrices to combine rotation, scale, translation, and perspective in a single matrix.

Common types include: rotation, translation, scaling, and shear (collectively affine transformations); rigid body transforms (rotation and translation only); projective transformations (homographies) used for perspective mapping; and geographic coordinate transforms that convert between coordinate reference systems, such as converting latitude/longitude to projected coordinates, often using datum and ellipsoid models.

Applications span geographic information systems, computer graphics, computer vision, and robotics. In GIS, coordinatetransforms align maps with different CRS; in graphics pipelines, model, view, and projection transforms map 3D scenes to screen coordinates; in robotics, frame transformations relate measurements in different sensor coordinates. In practice, implementations rely on libraries and standards that provide predefined transforms and interpolation for nonlinear conversions, with attention to numerical precision and inverse transformations when possible.