Toepfer Selection Criterion

The Töpfer Selection Criterion (also called Selection Law or Radical Law) is used to determine the number of objects of the derived map in comparison to the source map. The choice has to be done manually. It is a method to generalize a map and to perform object selection in a controlled way. It is based on the idea to maintain relative object density. In most cases the Töpfer Selection Criterion is applied to topographic maps with large or medium scale.

The following formula describes the Töpfer Law:
\[ n_D=n_S\sqrt{\frac{m_S}{m_D}} \]

\( n_D/n_S \): number of objects on the derived/source map
\( m_D/m_S \): scale denominator of the derived/source map

For small-scaled maps it is important to consider additional influencing factors:
  • Objects that should be highlighted for a certain map use type
  • Coefficient of character key and meaning:
    • \(>1\): objects with special meaning
    • \(<1\): objects with poor meaning
Very important is also to adapt the size of the objects to the remaining scale.

Here is a simple Example:

\(~\)Source Map\(~~~~~~~\)\(~\)Derived Map\(~~~~~~~\)
\(~\)Number of Objects\(~~~~~~~\)7736
\(~\)Scale Denominator\(~~~~~~~\)10 00015 000

\( ~~~~~~~~~~~~~~~~~~~~~~~~~n_F=77\sqrt{\frac{10~000}{15~000}} \approx \) 36


The following map shows Coffee-Shops in the area of Munich. By selecting different scales the amount of visible icons is generalized/decimated by the means of Töpfer Selection Criterion.

Coffee-Shops in Munich