Spatial constraints

Eine Strasse kann beispielsweise für kleine Tiere eine räumliche Einschränkung sein. (Photo: Joël Fisler)

There is a fundamental difference between the geographical space and the metric space. The metric space is adequate for many applications of spatial analysis, such as distance calculations. In the geographical space, relations between objects are not only determined by the metric distance that can be calculated based on the given coordinate system. In order to explain the spatial objects, restrictions and conditions have to be considered. Up to now, we assumed space to be homogeneous and isotropic.
The distance was determined only by the underlying metric. Nothing has affected the calculation of distance, except the metric space. But homogeneous and Isotropic space is rare. Let’s illustrate this with the following example: The diagonal is the shortest path in a rectangle, if you want to path from a vertex to its nonconsecutive vertex. If your rectangle is a corn field, more effort is required to cross it than in the case it was a freshly cut meadow. Imagine a bull standing on the meadow, it could be reasonable to go around it. The effort made to overcome the field (friction) is called cost. The obstacle that has to be overcame (in this case the field), is called friction. If, in a given space unit, the cost, that is needed to overcome an obstacle, is known and displayed, the surface is called cost surface. Cost surfaces are used for weighting purposes.

Eisenbahnlinien trennen den Raum und können als räumliche Einschränkung aufgefasst werden. (Photo: Joël Fisler)

Für gewisse Menschen stellen auch Check-Points eine räumliche Einschränkung dar. (Photo: Joël Fisler)

Until now we calculated the distances in Manhattan or the Euclidean space. But there can be obstacles on the roads, such as railways, or the road can be composed of different road classes with different speed limits. All these attributes have to be included into the calculation, if you want to calculate fastest route between a starting point and the final destination. In the case that someone is interested to get a cheap flight ticket, e.g. to get from Zürich to Havana, he might take an indirect flight and change plain in Paris. Thus, in a monetary view, the indirect connection is shorter than the direct connection. In practice, distances are in general not symmetric and the triangle inequality doesn’t make sense. Thus, space can't be defined only by metric space.