Ellipse Illusion
Copyright 2015 by Paul Niquette. All rights reserved.

What must be the most extraordinary coincidence in the universe is the fact that movements of celestial bodies follow curves that comply with the conic sections.  One might be reminded of that astronomical reality while sipping a martini.

Gravity compels every trajectory in space to fit the mathematical equations for slices of cones at various angles.  Sure enough, you have your parabola, your hyperbola, your ellipseOne enthusiast shows how to bake cone scones to prove the theory.  But -- hey, out there in space, where the hell are the cones

For stars and planets, for moons and satellites of all kinds, orbits are the thing.  Each is said to be an ellipse.  Several handy methods are available for drawing an ellipse on paper: pin-and-string, trammel, parallelogram.   However, curves followed by celestial bodies are drawn on invisible planes coerced by ellipses sliced from fictitious cones at various angles.  Which would not be especially bothersome were it not for the Ellipse Illusion...

Ellipse Illusion

Not everybody suffers from the Ellipse Illusion, but I sure do, and -- hey, I admit that it's all in my mind.  The sketch above is definitely not correct.  It depicts an oval not an ellipse.  That's the shape my personal intuition stubbornly projects for a sliced cone.
Many solvers of Measuring the Moon have studied my proposed celebration of  ironic fitness values from The Moon IllusionBe advised, the Ellipse Illusion is merely an embarrassment.  It does not afford evolutionary benefits to its victims.
The explanation for the Ellipse Illusion is simple enough [deep breath here]: The local radius of curvature for a smooth shape is defined by the radius of the osculating circle at each point such that one end of the oval drawn upon the sectioning plane seems to share the osculating circle of the cone at a point near its apex where the radius is small while the opposite end of the oval drawn upon the same sectioning plane seems to share the osculating circle of the cone at a point farther away from its apex where the radius is larger. 

Can you explain why that conic section is not an oval?

Slicing a cylinder with a plane at any old angle will always produce an ellipse, which my personal intuition does not resist.  Why don't astronomers use cylindrical sections instead of conic sections for celestial orbits?

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