title: A new unity for angular measurements in strabismus.
abstract: The practical advantages of quantifying an angle by a ratio of linear lengths instead of arcs of circles has led to the definition of the prism-diopter, a conventional unity for numbering prisms and measuring strabismic deviations. However, a major inconvenience of using prism-diopter unities to express angular measurements is the non-linearity of the scale, which reaches an infinite value for the angle of 90º, then becomes negative, with decreasing magnitudes for increasing angles between 90º and 180º. As a consequence, arithmetical operations and comparisons of angles measured by such unities present errors of very great magnitudes. In order to retain the advantages of defining an angle by straight line dimensions but to diminish the severe inconveniences of this method, a new definition of the prism-diopter is proposed. Here, instead of defining the prism-diopter by the asymmetrical condition, the conception of this new unity is based on a geometrically symmetrical condition; that of the relationship of an isosceles triangle (where the leg is perpendicular to the bisector of the angle and the bisector itself ). The condition of symmetry for the definition of the new unity represents a conceptual advance because it reproduces the already well accepted, conventional criteria for quantifying the value of a prism, that of its minimum deviation. Furthermore, it corresponds to the most commonly observed clinical conditions of binocular balance. The absolute differences between the unitary values of the prism-diopter and that of the new unity are negligible (0.0025%), but the scale of values expressed by the new unity is closer to the ideal scale of angular measurements. (With the new unity, the infinite value is only reached for an angle of 180º and the errors due to arithmetical operations are much smaller.) Numerical examples showing the advantages of using the new unity of angular measurements instead of the prism-diopter are presented. A mathematical generalization of the modifying concept (partition of an angle) with which the new unity is based is also provided.
authors.affiliation: Department of Ophthalmology, Faculdade de Medicina de Ribeirão Preto, Universidade de São Paulo, Ribeirão Preto, SP, Brazil.
authors.forename: Harley E A
authors.initials: HE
authors.lastname: Bicas