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Gavin R. Putland

Publications and source records attributed to Gavin R. Putland.

5 recordsLinked to original sources
arXiv2026

The unreasonable effectiveness of the cathetus rule in ancient and modern optics

The "cathetus rule" in optics alleges that the image of an object-point, formed by reflection or refraction at a surface, lies on the perpendicular ("cathetus") from the object-point to or through the surface. The first known statement of the rule, attributed to Euclid, was for a plane or spherical mirror. The rule was extended to refraction by Ptolemy.... Kepler was universally credited with the first disproof-and-salvage of the cathetus rule until 2018, when Benedetti's priority was exposed by Goulding. Kepler notwithstanding, the rule was reaffirmed by Tacquet for plane and spherical mirrors, except for the case in which the rays converge toward a point behind the eye; this became known as the "Barrovian case" because it troubled Barrow, in spite of his modern concept of an image. Barrow demolished the cathetus rule for the tangential image except in the paraxial limit, and Newton salvaged it for the sagittal image. The rule then seems to fade from history. But the rule is equivalent to the assumption that the image is stigmatic and the cathetus well defined. This narrow assumption is approximately true in the first-order (paraxial, "Gaussian") analysis of lenses and mirrors; and unacknowledged applications of the ancient rule can indeed be discerned in modern expositions of that subject. Moreover, the validity of the rule for the sagittal image fills a critical gap in meridional ray-tracing through spherical surfaces: by tracing the chief ray from an off-axis object-point, then applying the cathetus rule to the successive surfaces, one can locate successive sagittal image-points on the chief ray (produced rectilinearly through surfaces as necessary), and hence assess astigmatism to leading order, without tracing any rays outside the meridional plane.

physics.hist-ph↗
arXiv2026

Obituary for Augustin Fresnel

Annotated English translation of Duleau's "Notice sur A. Fresnel" in Revue encyclopédique, vol.39, pp.558-67 (September 1828), and of the shorter obituary for Fresnel in id., vol.37, pp.316-7.

physics.optics↗
arXiv2025

Generalized Gregorian quadrature, including end-corrected weights for the midpoint rule

A class of numerical quadrature rules is derived, with equally-spaced nodes, and unit weights except at a few points at each end of the series, for which "corrections" (not using any further information about the integrand) are added to the unit weights. If the correction sequences overlap, the effects are additive. A fundamental parameter ("alpha") in the derivation is the distance from the endpoint of the range of integration to the first node, measured inward in step-lengths. Setting alpha to 1/2 yields a set of corrected composite midpoint rules. Setting alpha=0 yields Gregory's closed Newton-Cotes-like rules, including (for sufficient overlap) the standard closed Newton-Cotes rules (trapezoidal rule, "1/3 Simpson rule", "3/8 Simpson rule", "Boole's rule", etc.). Setting alpha=1 yields open N-C-like rules, again including the standard ones. A negative alpha means that the integrand is sampled outside the range of integration; suitably chosen negative values yield centered finite-difference end-corrections for the trapezoidal rule and the midpoint rule. One can even have different values of alpha at the two ends, yielding, inter alia, Adams-Bashforth and Adams-Moulton weights. Thus the title could have been "Unified derivation of equispaced quadrature rules".

math.HO↗
arXiv2025

Exact formulation of Huygens' principle in terms of generalized spatiotemporal-dipole secondary sources

A "spatiotemporal dipole" wave source, as defined by D.A.B. Miller (1991), differs from an ordinary ("spatial") dipole source in that the inverted monopole is delayed relative to the uninverted monopole, the delay being equal to the propagation time from one monopole to the other. A "generalized" spatiotemporal dipole (GSTD), as defined here, is generalized in two ways: first, the delay may be smaller in absolute value (but not larger) than the propagation time, so that the radiated waves cancel at a certain angle from the axis of the dipole; second, one monopole may be attenuated relative to the other, so that the cancellation is exact at a finite distance - on a circle coaxial with the dipole. I show that the Kirchhoff integral theorem, for a single monopole primary source, gives the same wave function as a certain distribution of GSTD secondary sources on the surface of integration. In the GSTDs, the "generalized" delay allows the surface of integration to be general (not necessarily a primary wavefront), whereas the attenuation allows an exact match of the wave function even in the near field of the primary source. At each point on the surface of integration, the circle of cancellation of the GSTD secondary source passes through the primary source, which therefore receives no backward secondary waves, while the direction of specular reflection of the primary wave passes through the same circle, giving a geometrical-optical explanation of the suppression of backward secondary waves at any field point.

physics.optics↗