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## Procrustes solution

### Algebraic Geodesy and Geoinformatics (2010-01-01): 111-135 , January 01, 2010

This chapter presents the minimization approach known as *“Procrustes”*
Procrustes partial solution
which falls within the multidimensional scaling techniques discussed in Sect. 9-22. Procrustes analysis is the technique of matching one configuration into another in-order to produce a measure of match. In adjustment terms, the partial Procrustes problem is formulated as the least squaresleast squares
problem of transforming a given matrix
$$ {\mathbf A} $$
into another matrix
$$ {\mathbf B} $$
by an orthogonal transformation
transformations orthogonal
matrix
$$ {\mathbf T} $$
such that the sum of squares of the residual matrix
$$ {\mathbf E} = {\mathbf A} - {\mathbf {BT}} $$
is minimum. This technique has been widely applied in shape
shape analysis
and factor analysisfactor analysis
. It has also been used for multidimensional rotation and also in scaling of different matrix configurations. In geodesy and geoinformatics, data analysis often require *scaling*, *rotation* and *translation* operations of different matrix configurations. Photogrammetrists, for example, have to determine the orientation of the camera during aerial photogrammetry and transform photo coordinates into ground coordinates. This is achieved by employing scaling, translation and rotation operations. These operations are also applicable to remote sensing and Geographical Information System (GIS) where map coordinates
coordinates maps’
have to be transformed to those of the digitizing table. In case of robotics, the orientation of the robotic arm has to be determined, while for machine and computer visions, the orientation of the Charge-Coupled Device (CCD)
CCD cameras
cameras has to be established. In practice, positioning with satellites, particularly the Global Navigation Satellite Systems (GNSS) such us GPS
satellites GPS
and GLONASS
satellites GLONASS
has been on rise. The anticipated GALILEO
satellites GALILEO
satellites will further increase the use of satellites in positioning. This has necessitated the transformation of coordinates from the Global Positioning System (WGS 84) into local geodetic systems and vice versa.

## Algebraic Geodesy and Geoinformatics

### Algebraic Geodesy and Geoinformatics (2010-01-01) , January 01, 2010

## From Riemann manifolds to Euclidean manifolds

### Map Projections (2006-01-01): 97-112 , January 01, 2006

## Cartesian to ellipsoidal mapping

### Algebraic Geodesy and Geoinformatics (2010-01-01): 155-171 , January 01, 2010

In establishing a proper reference frame of geodetic point positioning, namely by the Global Positioning System (GPS) - the Global Problem Solver - we are in need to establish a proper model for the
*Topography*
Topography
of the Earth, the Moon, the Sun or planets.By the theory of equilibrium figures, we are informed that an ellipsoid, two-axes or three-axes is an excellent approximation of the *Topography*. For planets similar to the Earth the biaxial ellipsoid, also called “*ellipsoid-of-revolution*”
ellipsoid-of-revolution
is the best approximation.

## “Sphere to cone”: polar aspect

### Map Projections (2006-01-01): 379-393 , January 01, 2006

## “Sphere to cylinder”: polar aspect

### Map Projections (2006-01-01): 273-284 , January 01, 2006

## “Sphere to tangential plane”: polar (normal) aspect

### Map Projections (2006-01-01): 161-207 , January 01, 2006

## Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others

### Map Projections (2014-01-01): 609-671 , January 01, 2014

Up to now, we treated various mappings of the *ellipsoid* and the *sphere,* for instance of type conformal, equidistant, or equal areal *or* perspective and geodetic.