Vector space of generalized radii of ellipses for quantitative analysis of blur patches and other referred apertures in the astigmatic eye.

Ellipses are features of several structures in astigmatic eyes; they include retinal blur patches. How can one calculate change or averages or perform other quantitative analyses on such elliptical structures? The matrix A in the equation r ^T Ar=1, commonly used to represent an ellipse, is positive definite; such matrices do not define vector spaces. They are unsuitable, therefore, for quantitative analysis of ellipses. This paper defines a generalized radius R of an ellipse that is positive or negative definite for locally diverging or converging rays, respectively, indefinite between line foci, and singular at foci. Generalized radii of ellipses constitute a vector space and are suitable for quantitative analysis of elliptical ocular structures.