A New Look at the So-Called Trammel of Archimedes

The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity

Real circles tangent to 3 conics

In this paper we study circles tangent to conics. We show there are generically 184 complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit example of 3 conics with 136 real circles tangent to them. We conjecture that 136 is the maximal number of real circles. Furthermore, we implement a hill-climbing algorithm to find instances of conics with many real circles, and we introduce a machine learning model that, given three real conics, predicts the number of circles tangent to these three conics

On Heatshield Shapes for Mars Entry Capsules

The 70deg sphere-cone - the standard geometry for all US Mars entry missions - is thoroughly examined via flow field simulations at a select few peak heating points along candidate flight trajectories. Emphasis is placed on turbulent heating based on the Baldwin- Lomax turbulence model. It is shown that increased leeward turbulent heating for a 70 sphere-cone flying at angle of attack is primarily due to the discontinuity in curvature between the spherical nose cap and the conical frustum - the attachment of the sonic line at this sphere-cone junction leads to a supersonic edge Mach number over the leeward acreage. In an attempt to mitigate this problem of elevated turbulent heating, alternate geometries, without any curvature discontinuities in the acreage, are developed. Two approaches, one based on nonlinear optimization with constraints, and one based on the use of non-uniform rational B-splines, are considered. All configurations examined remain axisymmetric. The aerothermal performance of alternate geometries is shown to be superior to that of the 70 sphere-cone

Lightweight conical components for rotational parabolic domes: geometric definition, structural behaviour, optimisation and digital fabrication

Although initially intended for academic purposes, the research shown in this paper was drawn towards the development of hollow lightweight conical com-ponents to materialise rotational parabolic domes. The starting point is a projec-tive interpretation of an Archimedean property of rotational paraboloid planar sections. This is used to discretise the parabolic surface with a set of tangent ellipses obtained via planar circle-packing algorithms. The ellipses are then mate-rialised with components composed of three truncated conical surfaces, which may be composed of several laminar materials. The geometry and economy of the material, the good structural behaviour, the simple solution for fabrication and assembly, and the tests on a full-scale prototype prove this component to be an efficient self-supporting system for wide-span structures against the use of solid boundary rings, not only for rotational parabolic domes, but also for a possible translation to other types of surface

General theory of three-dimensional radiance measurements with optical microprobes

Measurements of the radiance distribution and fluence rate within turbid samples with fiber-optic radiance microprobes contain a large variable instrumental error caused by the nonuniform directional sensitivity of the microprobes. A general theory of three-dimensional radiance measurements is presented that provides correction for this error by using the independently obtained function of the angular sensitivity of the microprobes. © 1997 Optical Society of America