Re: Ray inside a cone.
- From: "Narasimham" <mathma18@xxxxxxxxxxx>
- Date: 17 Nov 2006 04:24:34 -0800
Narasimham wrote:
pierre.bornsztein@xxxxxxxxx wrote:
''A ray of light is issued from a point interior to a given cone. All
the internal surface of this cone can reflect the light.
Prove that after a finite number of reflections, the ray will never
meet the cone again.''
For a cone r = tan(al)*z let us say a ray originates on cone at P1
(r1,th1) and strikes it at P2 (r2,th2) and again after second
reflection strikes at P3 (r3,th3). Let us express P3 as a function of
P1 and P2.That way we could get result for cone and also later handle
any surface of revolution to find emergent ray caustics etc.
I think this can be extended from physics/maths of reflections by
Fermat's least distance principle as light "knows" the shortest path.
During reflections a ray of light takes minimum time between successive
points of incidence of a cone mirror by Fermat 's principle, and path
length between successive points of incidence is a minimum for given
speed of light. When interval between successive points is
infinitesimally reduced, the line projected on cone surface becomes a
geodesic; plane of reflection becomes the plane of osculation. We need
only to consider the motion of the projected point on the geodesic of a
cone that closely follows the light path. For a finite length
inter-incidence points, the light rays are chords spanning between
vertices of a twisted polygon whose sides inscribe discontinuous
geodesic arcs on a cone without torsion when considering only two arcs,
but with torsion when a third arc/chord is added.
When drawn on surfaces of revolution, geodesics are torsion-less in
three exceptional cases 1) meridional 2) circumferential 3) when on a
sphere, in general they have a geodesic torsion.
Geodesics on a cone semi-vertical angle al, polar angle th from minimum
radius rmin have the parameterization:
rmin*(cos(th), sin(th), cot(al) )/cos(sin(al)*th). The geodesics go
to the nearest radius rmin even if the ray is initially directed
towards the cone vertex to eventually leave the cone with angle
reducing to zero when th -> pi/(2 sin(al)).
The following is no direct response to the question, but for
understanding a generalization. For non-spherical ellipsoid mirrors,
the rays are chords based on endless returning geodesics paths of the
ellipsoids. For cones and hyperboloids, the small rays/geodesics do not
return.
Fun with science,agreed?
Narasimham
.
- References:
- Ray inside a cone.
- From: pierre . bornsztein
- Re: Ray inside a cone.
- From: Narasimham
- Ray inside a cone.
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