The Unfolding

Surah 74 AlMuddathir (الْمُدَّثِّرُ) (The Enveloped/Enfolded/Wrapped Up One)

(74:1) O You Enfolded One (Enveloped/Wrapped Up In Your Mantle)
(74:2) Arise and Warn! (Ascend/Mount Up To Deliver The Message)


You, My Dear, Can give any Message You would like at this point. Since I've been hung out to dry for over 14 years by my reckoning, the only message I feel the need to deliver is in answer to the question, "Why are the planetary orbits elliptical?"

Plane curves (lines, circles, parabolas, sine curves, etc.) that are wrapped onto cylinders of varying radii exhibit a sinusoudal influence, and can be characterised in terms of equations whose solutions are simple 2-Dimensional geometric transformations(rotation, translation, etc.), without resorting to calculus.
A cylinder is a developmental surface it can be 'flattened' or 'unwrapped' onto a plane without distortion--Mathematically expressed as having Zero Guassian Curvature Κ , (intrinsic curvature is Zero). Gaussian curvature is defined as the product of the principal curvatures κ1 and κ2.




Κ = κ1κ2
And since cylinders can be transformed flat without compression or expansion (distortion), unwrapping a curve that lies on the surface of a cylinder preserves the distance between points on that curve. In other words, any arc lengths on their profile unfolds/unwraps/uncurls onto a line segment of the same length. The diagram above shows the simple case where an arc on a right cylinder's circular profile is unwrapped by rolling the cylinder along its side; the end point on the circle projects onto a point t, but unwraps onto a point x. And since the sinusoidal influence is determined by the relationship of the arc length to the linear projection, unwrapping curves that lie along the cylinder surface gives rise to their periodic nature.









By extension, space curves unwrapped from the lateral surface of a right circular cone produce periodic plane curves. (An interesting anolog found in nature of curves unwrapped from conics is the lunar obit).



A line segment (the shortest distance between two points) wrapped onto a right circular cylinder is also a geodesic arc (shortest path) on the cylinder, no matter how tightly it is rolled (decreasing r) and the profile of a geodesic arc on a right circular cylinder is a function of arcsine. * A geodesic on a circular cylinder is always part of a circular Helix.



The profile of such a wrapped line is given by:





p(t)=cr arcsin (t/r) (where; c= constant, r=radius, t= projected point)
In case of a right circular cylinder, eccentricity is 0, where such a line in the xyz plane is given by z=t lies at a 45 degree angle to the generators (a=pi/2), the tightly wrapped line would stack into circular ringlets should the cylinder collapse orthogonally (like a Slinky would when there is no tension). The profile of such a line in this particular case is given by:









p(t)= ct^2





The stacked ringlets (circles) it forms are given by:





z^2 + t^2 = r^2
There are 3 ways to distort these circles into ellipses, by changing the viewing angle either via rotating the cylinder or changing the position of the outside Observer or by deforming the cylinder itself (squeezing it) so that the eccentricity is no longer 0. Changing the viewing angle is simply a matter of shifting perspective (--); physically deforming the right cylinder such that its eccentricity is no longer 0 requires effort (Force).









So, this begs the question, when considering planetary orbits, is it the perspective that makes the orbits appear to be elliptical or something more intrinsic?





From a purely geometric consideration, and realizing that circles are in fact merely ellipses with eccentricity=0, it is strange that the more generic elliptical orbits are the common case and not the perfect circular orbit.





In a 2-body system where a much heavier object is at the center, a circular orbit would be the expected outcome, or the ideal case, in any event. Neptune's largest moon, Triton, comes closest to this ideal case in the Solar system, with eccentricity=0.000016. {That of the Earth's orbit around the Sun is 0.01675 (a far cry from the exaggerated elliptical orbit often depicted in the literature)--based on a cursory look at the limited curricular activity on the web, it's no longer a surprise that the Germans landed first--in my living room--& very good helpers they are, too, even if they do tend to overshop the tomatoes.





The event alluded to happened over a dozen years ago and has nothing to do with my state of mind which was as sound then despite the intractable headache as it is now despite the fact that a casual trip to the local library to pick up a book accompanied by my child has some people conspicuously talking to You about 'stopping her, now!'





Is 'stop her' code for 'shoot her'? (^_~)





And it's a good thing You didn't because it isn't like I don't give You ample opportunities to do just that without the children present; and it would be in exceedingly poor taste if something like that were to happen right in front of their eyes given how much of a predictable, isolated, easy mark I make myself the rest of the time when they are not around.} "<o _<o"





Anyway, back to the present.





That the geodesic is the shortest distance between two points on the surface of a cylinder, regardless how tightly it is rolled up, is interesting from the perspective of an outside observer in that an object or person following this route would appear to be going around in circles (elliptical path), but from that traveller's or object's point of view they think they are travelling in a straight line.





An interesting variation on this theme is if the object or person were to follow a weaving path in a sinusoidal fashion in a plane that is then rolled up into a right cylinder. The sinuous path can represent Time as well as Displacement. One familiar representation of just such a path is the apparent Time Difference due to the Earth's orbit about the Sun (The Time Equation).










A knack for stating the obvious is one of my strong suits. Orbits "unfolded" in such a manner are indistinguishable from displacement due to simple harmonic motion, as found in springs that obey Hooke's Law or the motion of a Pendulum.





In terms of the Geometry, an Orbit is essentially a Straight Line 'Wrapped' around a Developmental Surface (like that of a Cylinder), and its Profile is essentially a Circular (Elliptical) Path.





By extension: Time (t) is a Projection of Orbit onto a Tangential Viewing Plane. Each Point in Time is also a Position (x) in the Orbital path. As that Path (displacement) 'unfolds/unwraps' (theta) onto the Viewing Plane, the Time correlates to a particlular Place (x) on the Orbit's Profile given by:





x= (Time*Theta)/(Sin (Theta))





or x= Time*θ Csc[θ] = ; and





alternatively;











and for the Real Time, and Place and Angular Displacement (Azimuth)











Interstingly has only 1 Real Integer Root at t=0;

Does this mean that In Reality Nothing Really Ever Happens?





(Projecting a Point and/or Unwrapping a Curve from a Developmental Surface is like Stepping Down from a Higher Dimension. When the need to refresh What this is all about arises, we can always go back to How Emily treats Bob, and Where Angels come from--always leaving the residual question, Why?)

"They forgot Y I told U!"
The Argyle Sweater -Confusion at the Alphabet Gang Hideout
Appeared in the March 1, 2014 Los Angeles Times Comics Section

*Unwrapping Curves from Cylinders and Cones, Apostol and Mnatsakanian, 1997, The Mathematical Association of America; pp 392-415