Invariance property of coordinate transformation

Is coordinate transformation invariant with respect to coordinate systems? Intuitively one assumes so, but the answer is more complex as in some cases it is invariant and in some cases it is not. Invariant coordinate transformation is a transformation where the quadratic form of the weighted residuals (v^TPv) is preserved in any coordinate system in which the transformation parameters are calculated. Consequently, invariant coordinate transformation allows converting the transformation parameters that were calculated in one coordinate system to another coordinate system correctly, without the need to calculate the parameters again in the new coordinate system. This paper discusses the invariance property of coordinate transformation and shows the complexity of geodetic coordinate transformations, where some are invariant and others are variant. In variant transformations, the transformation parameters and the quality of the transformation depend on the coordinate system and its definition, an important situation to be aware of.