# Upkeep - Examples congruency test

SECOND DRAFT

## Definitions

Congruent: situation where two polygons (parcel representations) have the same shape and dimension as a result, all their corresponding sides (perimeter segments) have the same length, the angle between any two corresponding segments has the same value.

In general, congruency indicates that one object can be transformed onto the other by a combination of translation and rotations (but not resized). In LPIS context, both representations need to be in the national CRS, so rotations are not expected and only translation are relevant.

LPIS polygons are congruent if they have the same number of sides, all the corresponding sides have the same length and orientation. It follows that all corresponding interior angles have the same opening (in the theory).

Shape: geometrical description of the part of the (2-dimensional) space occupied by an object, as determined by its external boundary.

Dimension: distance between two opposite sides of a feature.

Angle: measure of rotation by the diagonal axis.

Side: segment of a shape between two vertices.

Axis-aligned bounding box (AABB): generalized polygon representation of the parcel, with the coordinate axis aligned to the longest axis of the parcel. It can be constructed using of four specific vertex points:

• the 2 vertices on the perimeter that have the longest possible line segment between them (i.e. the define the long diagonal).
• the vertices removed at the longest distance from either side of the long diagonal. The sum of the respective (not necessarily collinear) distances offers the total length of the perpendicular axis.

It is the box, aligned with the local coordinate axes, with the area that fully contains the polygon.

## Illustration of the congruency test to detect manifest change

Figure 4d.1 presents the RP representation (= the polygon) retrieved from the LPIS and the new representation (=the image) on the orthoimage of a later date.

figure 4d.1: Existing parcel representation (left) and new parcel representation (right)

Figure 4d.2 shows the construction phases of the AABB derived from the existing RP representation. Key elements are the determination of the longest diagonal [AC] and the two perpendicual axis [DD'][BB'] where the dimension is maximal.

figure 4d.2: the axis-aligned bounding box construction (the dashed rectangle)

Figure 1.3 compares the original vector and associated AABB with the corresponding cardinal points on the orthoimage representation.

figure 4d.3: LPIS parcel and AABB (left), Cardinal points of vector and orthoimage representation (right)

Figure 4d.4 shows on the left the two aligneed minimum bounding boxes superimposed on the the orthoimagery. The shift between the two AABBs is obvious. In the right image, the LPIS AABB is translated so that the origins of the AABB coincide. Only the small rotation remains visible.

figure 4d.4: original AABBs superimposed (left), translated LPIS AABB and image AABB superimposed (right)

The AABBs derived above yield the following parameters.

 AABB coordinate Start point End point parameter observed value decimal 1 Easting(Y) 327575,217 327998,139 length 574,2841 1 Northing(X) 705215,785 705604,294 angle 47°25'42.505 47,428474 2 Easting(Y) 327578,387 328003,573 length 573,2119 2 Northing(X) 705220,766 705605,199 angle 47°52'53.794 47,88161

As a result:

• the observed difference in length is 1,0722 meter, less than the 2.5 m threshold
• the observed difference in angle is 0,453136 degrees, less than the 1 degree threshold

No manifest change has been detected, the conclusion must be that the reference parcel has not changed.

## Illustration of the congruency procedure to determine coordinate shift

Figure 4d.5 presents the RP representation (= the polygon) retrieved from the LPIS and the new representation (=the orthoimage) at a later date.

figure 4d.5: Existing parcel representation (left) and new parcel representation (right)

Figure 4d.6 compares the original vector and associated AABB with the corresponding cardinal points on the orthoimage representation.

figure 4d.6: LPIS parcel and AABB (left), Cardinal points of vector and orthoimage representation (right)

Figure 4d.7 shows on the left the two axis aligned bounding boxes superimposed on the orthoimagery. The shift between the two AABBs is obvious. In the right image, the LPIS AABB is translated so that the origins of the AABB coincide. Only a very small rotation remains visible.

figure 4d.7: original AABBs superimposed (left), translated LPIS AABB and image AABB superimposed (right)

The AABBs derived above yield the following parameters.

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