Conic sections embedded in a torus must be circles of special types: (i) proole circles, (ii) cross-sectional circles , and (iii) Yvone-Villarceau circles. Based on this classiication, we present eecient and robust geometric algorithms that detect and compute all degenerate conic sections (circles) in torus/plane and torus/natural-quadric intersections. 2 Introduction Simple surfaces (such as p...

The locus of points where two non-concentric circles c1 and c2 are seen under equal angles is the equioptic circle e. The equioptic circles of the excircles of a triangle ∆ have a common radical axis r. Therefore the excircles of a triangle share up to two real points, i.e., the equioptic points of ∆ from which the circles can be seen under equal angles. The line r carries a lot of known triang...

We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras. For example, we show how Armstrong’s description ([1, 2]) of knot colourings and knot groups fit into t...

For a point T and a circle δ, if two congruent circles of radius r touching at T also touch δ at points different from T , we say T generates circles of radius r with δ, and the two circles are said to be generated by T with δ. If the generated circles are Archimedean, we say T generates Archimedean circles with δ. Frank Power seems to be the earliest discoverer of this kind Archimedean circles...

An inhomogeneous cosmological magnetic field creates vortical perturbations that survive Silk damping on much smaller scales than compressional modes. This ensures that there is no sharp cut-off in anisotropy on arc-minute scales. As we had pointed out earlier, tangled magnetic fields, if they exist, will then be a potentially important contributor to small-angular scale CMBR anisotropies. Seve...