L5-Transformations in the Euclidean Plane
L5 – Transformations in the
Euclidean Plane
NGEN06(TEK230) –
Algorithms in Geographical Information Systems
L5-Transformations in the Euclidean Plane
Background
In geographic analysis it is common that you have to
transform coordinates between coordinate systems.
To perform the required transformations the GIS
program provides a set of transformations
(congruent, similarity, affine, projective and
polynomial).
L5-Transformations in the Euclidean Plane
Aim
The student should understand the main empirical
transformations in the plane.
Choose the correct transformation for their applications.
The student should also understand the concept of
common point and be able to decide which common
points to use in an application (how many and the
distribution).
L5-Transformations in the Euclidean Plane
Content
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Transformations in the Euclidean Plane
Congruence (Euclidean) transformation
Similarity transformation
Affine transformation
Projective transformation
Polynomial transformation
Applying empirical transformations
Choice of empirical transformations
L5-Transformations in the Euclidean Plane
Transformations in the Euclidean Plane
Congruence transformation
Original grid
Affine transformation
Type of transformation
Projektive transformation
Maintain
Similarity transformation
Polynomial transformation
Does not maintain
Number of unknowns
Congruence (Euclidean)
Shape, size
Position
3
Similarity (Helmert)
Shape
Size, position
4
Affine
Parallelism
Shape, size, position
6
Projective
Double ratio property
Parallelism, shape, size, position
8
Polynomial
Topological relationships
Geometrical properties
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L5-Transformations in the Euclidean Plane
Transformations in the Euclidean Plane
An invariant is a property that is maintained in the
transformation.
Common points are points that are known in both
coordinate systems. Each common point will give 2
relationships (one each in x- and y- directions). In theory,
we must have at least half of the number of common
points as unknowns. In reality much more common
points should be used.
-> Choice & usage of transformations
L5-Transformations in the Euclidean Plane
Congruence (Euclidean transformation)
L5-Transformations in the Euclidean Plane
Similarity Transformation
L5-Transformations in the Euclidean Plane
Affine Transformation
L5-Transformations in the Euclidean Plane
Projective transformation
L5-Transformations in the Euclidean Plane
Polynomial Transformation
L5-Transformations in the Euclidean Plane
Common points
• Each common point gives 2 conditions.
• Theoretically it is enough with 3 common points to determine
6 unknowns in affine transformation.
• But in practice you should always use twice as many points
than is theoretically requiered.
• The common points should always be well-distributed and
circumvent the area of interest.
L5-Transformations in the Euclidean Plane
Applying empirical transformations
L5-Transformations in the Euclidean Plane
Choice of empirical transformation
• Transformation between two uniform scale coordinate systemswhere the
scale is the same in both systems -> Congruence Transformation.
• Transformation between two uniform scale coordinate systems where
the scale might differ between the systems -> Similarity Transformation.
• Transformation between two coordinate systems where at least one of
the systems might not have a uniform scale -> Affine transformation.
• Transformation between two coordinate systems where one is close to a
projection of the other -> Projective Transformation.
• Transformation between two coordinate systems where one has a bad
(or completely unknown) geometry-> Polynomial transformation.
L5-Transformations in the Euclidean Plane
Standard error of the empirical transformation