Computational Euclid

TL;DR

This paper models Euclidean geometry axioms using object-oriented programming, creating C++ classes that accurately simulate geometric relations and tests, balancing computational efficiency with the limitations of floating-point arithmetic.

Contribution

It introduces a novel object-oriented framework translating Euclidean axioms into C++ classes, enabling efficient geometric computations with semi-decision algorithms.

Findings

Successfully models Euclidean axioms as object classes

Implements semi-decision algorithms for geometric tests

Balances computational efficiency with computability limitations

Abstract

We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++ classes that efficiently supports the construction of complex geometric configurations. Although all computations are performed in floating-point arithmetic, they correctly implement as semi-decision algorithms the tests for equality of points, a point being on a line or in a plane, a line being in a plane, parallelness of lines, of a line and a plane, and of planes. That is, in accordance to the fundamental limitations to computability requiring that only negative outcomes are given with certainty, while positive outcomes only imply possibility of these conditions being true.