Double-Cylindrical PointFocus - animation




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Published on Oct 5, 2008


The parabola has the well-known property of reflecting axis-parallel rays to a point http://tinyurl.com/6aqpxa

If we rotate the parabola around its axis, we create a parabolic disc, which has the well-known property of reflecting parallel rays (= planar wave-fronts) that are incident along the axis direction of the disc to a point. An animation that shows this process is available at http://tinyurl.com/57adll

We can avoid the "astronomical costs" associated with creating (= casting) an ordinary parabolic disc, and harness the workpower of the sun by bending two flat mirror sheets in the shape of parabolic cylinders to create an exact point focus. This is due to the

Double Cylindrical Point Focus principle:

If the focal line of the first cylinder is identical to the generating line of the parabola that is the intersection of the second cylinder with a plane perpendicular to its axis, then the incoming rays will be reflected to a perfect point.

For a proof of the DCPF principle, see http://tinyurl.com/595fsf

The DCPF principle was discovered on November 16, 1976 by Ambjörn Naeve http://tinyurl.com/5gbz8j

The video shows an animation (created with the Graphing Calculator http://www.pacifict.com ) of a planar wavefront being reshaped into a spherical wavefront by reflection in two parabolic cylinders configured according to the DCPF principle.

Besides being easier than the ordinary parabolic disc to build in large sizes (avoiding "astronomical costs"), the DCPF also has the advantage that the focal point can be placed outside of the solar influx area, where it is freely available to do work.
See http://tinyurl.com/69pusb

The DCPF also has the advantage that the number of planar approximator strips of fixed width grows LINEARLY with the overall size (since one dimension is unaffected) instead of QUADRATICALLY, as with an ordinary parabolic disc. For a comparison, see http://tinyurl.com/6xmpua
and http://tinyurl.com/5ass3j

A VR-based lecture from October 2000 can be found at http://tinyurl.com/5hby74 This lecture was created using CyberMath http://tinyurl.com/68fypm and DIVE http://www.sics.se/dive

For more films and interactive material,
see http://tinyurl.com/4prqzy
and for the story behind the DC PointFocus,
see http://kmr.nada.kth.se/wiki/Main/Poin...


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