14:45 〜 15:00
[G01-3-06] Variant and Invariant Properties of Coordinate Transformation
An invariant coordinate transformation is a transformation where the quadratic form of the weighted residuals is preserved regardless of the coordinate system in which the transformation parameters are calculated. This implies that the quality of the transformation is invariant with respect to the coordinate system.
An 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. It allows for a proper transformation of the same point in different coordinate systems.
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.
We discovered that the most common transformation in geodesy, surveying and related professions, the 7 parameters transformation, is invariant. However, with the widespread use of GNSS measurements, transformations with more than 7 parameters become more common and some of them are variant. The most general affine transformation with 12 parameters is invariant, but all transformations with a number of parameters greater than 7 and less than 12 are variant. Thus, the transformation is variant with respect to the coordinate system.
An 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. It allows for a proper transformation of the same point in different coordinate systems.
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.
We discovered that the most common transformation in geodesy, surveying and related professions, the 7 parameters transformation, is invariant. However, with the widespread use of GNSS measurements, transformations with more than 7 parameters become more common and some of them are variant. The most general affine transformation with 12 parameters is invariant, but all transformations with a number of parameters greater than 7 and less than 12 are variant. Thus, the transformation is variant with respect to the coordinate system.