A Modern Introduction to Fuzzy Mathematics

11Fuzzy Geometry

The theorems and propositions of classical Euclidean geometry which are based on the corresponding axiomatic corpus, study idealized shapes and figures made by crisp points and having exact dimensions on the plane or in three‐dimensional space. However, natural reality appears different. Already on 27 January 1921, in his lecture on “Geometry and Experience” at a public session of the Prussian Academy of Sciences [114, 115], Albert Einstein, made the famous statement: “As far as the propositions of mathematics refer to reality they are not certain, and as far as they are certain they do not refer to reality”. Einstein's aim was to underline the realistic possibility of adopting non‐Euclidean geometry as the true geometry of the universe. In the context of fuzzy mathematics, one follows a different route in geometry and its relation to reality. Indeed, when one studies or measures real objects, she soon realizes that the very process of measurement will often render expectations and theoretically exact predictions as improbable. Points, lines, lengths, areas, shapes, and other geometrical concepts and notions become rather uncertain. It is here where, since the late 1970s, a steadily increasing interest, coming mostly from areas such as pattern recognition and image processing and analysis where the use of fuzzy mathematics has already been established, for a new subject that has become known as geometry based on vagueness or fuzzy geometry, has come into play. ...