L6 – Transformations in the Euclidean Plane

NGEN06(TEK230) –

Algorithms in Geographical Information Systems

by: Irene Rangel, updated by Sadegh Jamali (source: Lecture notes in GIS, Lars Harrie)

L6- Transformations

Background

In geographic analysis, it is common that you have to transform coordinates between coordinate systems.

The most GIS programs provide a set of transformations (congruent, similarity, affine, projective and polynomial).

To choose the right transformation it’s important to know their differences from a geometrical viewpoint.

L6- Transformations

A note of warning!

All transformations in this lecture are of type Empirical Transformations.

Empirical transformations are used when the true relationships between the coordinate systems are unknown.

L6- Transformations

Aim

… to understand the most common empirical transformations in plane to be able to choose a suitable one.

… to understand the concept of common point and be able to decide which common points to use in an application (how many and the distribution).

L6- Transformations

Content

1. Transformations in the Euclidean Plane

2. Congruence (Euclidean) transformation

3. Similarity transformation

4. Affine transformation

5. Projective transformation

6. Polynomial transformation

7. Applying empirical transformations

8. Choice of empirical transformations

L6- Transformations

Transformations in the Euclidean Plane

Original grid Congruence transformation Similarity transformation

Affine transformation Projektive transformation Polynomial transformation

How to classify the transformations? Invariant

Invariant: a property that is maintained in the transformation

L6- Transformations

Transformations in the Euclidean Plane

Original grid Congruence transformation Similarity transformation

Affine transformation Projektive transformation Polynomial transformation

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 -

Type of transformation Invariant (maintain) Does not maintain No. of unknown parameters

L6- Transformations

Transformations in the Euclidean Plane

To be able to use the transformations, unknown parameters have to be determined.

Common points: known points in both coordinate systems

Each common point provides 2 relationships (x- and y- directions).

In theory: no. of common points = (½). no. of unknowns. In practice: no. of common points > (½). no. of unknowns.

Common pointsHow?

How many common points?

L6- Transformations

Congruence (Euclidean) transformation

It models translation and rotation.

L6- Transformations

Congruence (Euclidean) transformation model

Note: to determine the 3 transformation parameters , and α) at least 2 common points are required.

or in matrix form:

10

L6- Transformations

Similarity transformation

It models translation, rotation, and uniform scaling.

L6- Transformations

Similarity transformation model

Note: to determine the 4 transformation parameters , , m, and α ) at least 2 common points are required.

Or in matrix form:

L6- Transformations

Affine transformation

It models translation, rotation, non-uniform scaling in different directions, and shear.

non-uniform scaling shear

L6- Transformations

Affine transformation model

Note: to determine the 6 transformation parameters , , , , α , and ) at least 3 common points are required.

L6- Transformations

Projective transformation

Figure: Projective transformation. The planes do not have to be parallel. 15

L6- Transformations

Projective transformation modelThe double ratio property

Note: to determine the 8 transformation parameters (a1 , a2 , a3 , b1 , b2 , b3 , d1 , d2) at least 4 common points are required.

16

L6- Transformations

Polynomial transformation model

Polynomial transformation (degree = n)

The total number of unknown parameters are 6, 12, and 20 for transformation of degree 1, 2, and 3 respectively. 17

L6- Transformations

Polynomial transformation model

n = 1 no. of parameters = 6n = 2 no. of parameters = 12n = 3 no. of parameters = 20n = 4 no. of parameters = ?n = 5 no. of parameters = ?

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠=(𝑛+1)×(𝑛+2)

L6- Transformations

Applying empirical transformations

Example: how to estimate affine parameters?

Original affine transformation

Rewritten affine transformation

where

Or

L6- Transformations

Step 1

Equations system for solving the unknown parameters using 3 common points

𝑎0 ,𝑎1 ,𝑎2

𝑏0 ,𝑏1 ,𝑏2

, , , , α ,

Note: In case of having more than 3 common points, Least Squares technique is used.

L6- Transformations

Step 2

Common points

• Each common point gives 2 relationships.• 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

as is theoretically required.• The common points should always be evenly-distributed and

circumvent the area of interest.

L6- Transformations

Standard error of the empirical transformation

The estimated standard error (transformation quality at the common points)

L6- Transformations

Choice of empirical transformation

In general, it depends on type of

geometric distortions between the two coordinate

systems

L6- Transformations

Choice of empirical transformation

• Transformation between two uniform scale coordinate systems where the scale is the same in both systems

---> Congruence Transformation.

• Application: transformation between two geodetic reference systems

expressed in the same map projection and with equal scale between the systems (e.g. the same measuring techniques and instruments).

L6- Transformations

Choice of empirical transformation

• Transformation between two uniform scale coordinate systems where the scale might differ between the systems

---> Similarity Transformation.

• Application: transformation between two geodetic reference

systems expressed in the same map projection but with different scales between the systems (e.g. NOT the same measuring techniques and instruments)

L6- Transformations

Choice of empirical transformation

• Transformation between two coordinate systems where at least one of the systems might not have a uniform scale

----> Affine transformation.

• Application: transformation used for digitizing a paper map or a

photograph that might have different scales in the two main directions (due to e.g. non-uniform shrinkage of paper).

L6- Transformations

Choice of empirical transformation

• Transformation between two coordinate systems where one is close to a projection of the other ---> Projective Transformation.

• Application: suitable for rectification of aerial images that are not

taken along the plumb line. Rectification is a technique used in photogrammetry to obtain distortion-free images.

L6- Transformations

Choice of empirical transformation

• Transformation between two coordinate systems where one has a bad (or completely unknown) geometry

----> Polynomial transformation.

• Application: - geocoding of remote sensing images that are normally a mosaic of several minor parts.- digitizing historical maps or other maps with an

unknown map projection.

L6- Transformations