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Deposit #98

Robert G. Chester
Intellectual Property Associates
Tumwater, WA


Good visual imagination of motion in more than one plane may be enhanced by first thinking about the following:

Equatorial orbit . . .
Equatorial animation

combined with a polar orbit:
Polar animation
Follow the red point/circle in both animations,


The movies have been updated from the .mov format to the .mp4 format.

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Quasi-Spherical Orbits and Surfaces
"Life of a Point under Dual Rotations"

This section . . . .
Click here to see the Quick Time Player Movie Animation   Be patient.

Follow the red dot as the wheel rotates with dual
Wheel animation
motions.  The orbit traced is quasi-spherical.

Click here to read the "pdf" file with equations.
Click here to see a movie of the orbit connecting
the vertices of an octahedron.

Click here to see a movie of the orbit of the red dot.

Chester created the above animations along with many others using
Graphing Calculator 3.2 by Pacific Tech and a Quick Time
Movie player.

Did you know . . .
    A Quasi-Spherical 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.

Chester, Robert G, QSO - the Mathematics and Physics of Quasi-Spherical Orbits, Intellectual Property Associates, 2009.
ISBN 978-0-9840727-0-5

See Robert Chester's YouTube streaming video.

Fuller, R. Buckminster,

Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press, 1998, pp. 927, 933.

Simpson, Andrew, Pursuit Curves for the Man in the Moone, The College Mathematics Journal, vol. 38, no. 5 (2007), pp. 330-338.

Pae, Peter, Aviation:  Taking a ride on a 'perfect flight,'  Los Angeles Times: Business, September 13, 2007, p. C1.

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.

From Hubble's copy of Viviani's book.
From Leibniz's article in Acta Eruditorum.
From J. Bernoulli's article also in Acta Eruditorum.