
QuasiSpherical Orbits and Surfaces "Life of a Point under Dual Rotations"
A QuasiSpherical Orbit results from the simultaneous rotation of a point
about two or more axes with a common center. Initially, an
individual might use a
mental model of the dual motions of our Earth. Visualize the spin
the Earth on a daily basis of 24 hours while simultaneously moving
about the sun in a 365 day cycle. Though easy to understand, this
mental model is not completely accurate. While the Earth rotates
simultaneously on at least two axes, the axes of the rotations do not
intersect; there is no common center of rotation. A "QSO"
must have a common center of mass for all rotations. Moreover,
there is nothing that says the rotation has to be circular, or even on
a unit sphere. A heliocentric model is merely a simplication to
help the viewer's
initial imagination.
Chester uses a unicycle to model rotation around the fork of two planes while the wheel itself is also in motion on the exterior about the axis. He has produced an exhaustive study of the "life of a point" in both planar and space curves. Obviously, the mathematics of relative speed and position is often quite complex. View the attached "pdf" file for the unicycle as a sampler.

Historical Sketch . . . . Galileo’s spectacular findings influenced scholarly communities well beyond those of astronomy and religion. In particular, Vincenzio Viviani (1622 – 1703), also of Florence, posed eight problems in his Aenigma geometricum (1692), challenging mathematicians regarding surfaces on a dynamic hemisphere. Those doing analysis were asked to investigate the new Galilean “Architecture of Geometry.” One proposition became known as “Viviani’s window.” But in posing the problem, Viviani used the expression “quadrable Florentine sail.” As he noted, a surface removed from a sphere may resemble a sail. Europe’s leading scholarly journal, Acta Eruditorum, was the forum of exchange. Those publishing response articles included none other than Leibniz (1691, 1693), [Fig. 5] J. Bernoulli (1692) [Fig. 9 and Fig 10] and l’Hospital (1694). In addition, Viviani’s book was reviewed in 1694. His propositions and figures were repeated. The Huntington Library of San Marino, CA has graciously permitted us to view these late 17th century illustrations. Moreover, the images taken from Viviani’s book are from Edwin Hubble’s own copy.
