Re: [Phys-l] Sun's image due to reflection by not a pinhole.

[ludwik kowalski <kowalskil@mail.montclair.edu>]

On May 1, 2009, at 12:42 PM, John Denker wrote:

> On 04/30/2009 06:42 AM, Anthony Lapinski wrote:
>
> > Not sure my (high school) students have ever heard of "convolution,"
>
> Gaudeamus igitur !
>  (That's Latin for "party on".)
>
> Think how much easier it is when the students have never heard the
> word.  Contrast that with, say, conservation of energy, where the
> students arrive knowing the vernacular meaning of energy and the
> vernacular meaning of conservation, both of which must be unlearned
> before they can learn the physics meanings.
>
> Besides, the word isn't important.  The idea is important.  And
> the idea is simple.
>
> > Seems to be an "advanced" calculus topic/idea, well beyond most
> > introductory high school physics courses.
>
> Absolutely not.
>
> ============================
>
> You can demonstrate the whole idea in less time than it takes to
> tell about it.
>
> Start with a board with three pinholes in it.  Let the arrangement
> of pinholes be non-symmetric enough to be recognizable.  This
> board will be called the _mask_ and will serve as the lens in a
> slightly peculiar pinhole camera obscura.
>
> Also arrange for a bright scene to serve as the _object_.  Form an
> image on a screen in the usual way.
>
> Case 1:  This the intermediate case, where each pinhole produces a
> separate subimage that does not overlap with the other subimages.
> Each subimage will be a clear representation of the object.
>
> Case 2:  In this case, let the object be very small (but still bright)
> so that each subimage becomes small and structureless.  In this case,
> the image as a whole is a one-to-one representation of the mask.
>
> Case 3:  This is the opposite extreme from case 2.  In this case, let
> the object be large and/or let the spacing between pinholes be small.
> Then there will be nearly 100% overlap between the subimages, so that
> in effect there is only one image.  It is a single representation of
> the object, just slightly blurred.
>
> In all three cases, the image as a whole is the convolution of a  
> function
> representing the object with a function representing the mask.  The  
> mask
> can be considered the kernel of the convolution.
>
> This thread started by calling attention to the two extreme cases  
> (case 2
> and case 3).  I suggest that the intermediate case (case 1) is  
> easier to
> understand.  There's nothing tricky about it.  No calculus,  No math  
> of
> any kind.  Just look at the three subimages.

1) Suppose the object is a flat square (each side is 2 cm) in the z=0  
plane. The center of the square, whose sides are parallel to axes, is  
at the origin. Each "pixel" (small area) emits equal amount of light  
(100 units) isotropically.

Write the "function representing the object." Why should it be a  
polynomial?

2) Suppose the mask consists of a black paper located in the plane  
z=10 cm, with three circular holes. The radium of each circle is 1 cm.  
Centers of circles are specified as follows:

circle 1 (x=0, y=0); circle 2 (x=5 cm, y=5 cm); circle 3 (x=5 cm, y=-5  
cm)

Write the "function representing the mask." Why should it be a  
polynomial?

3) Suppose the resulting light spot is observed on a screen situated  
in the plane z=30 cm

Show how to convolute the two functions in order to predict the  
distribution of light on the screen (how many units of light per  
square cm at a given x and y).

P.S.
This is probably far fro being trivial. I have no idea how to do this.  
Specific examples of that kind would probably be helpful.

- - - - - - - - - - - - - -
Ludwik Kowalski, a retired physics teacher and an amateur journalist.  
Updated links  to publications and reviews are at:

http://csam.montclair.edu/~kowalski/cf/   http://csam.montclair.edu/~kowalski/my_opeds.html 
   http://csam.montclair.edu/~kowalski/revcom.html

Also an ESSAY ON ECONOMICS at: http://csam.montclair.edu/~kowalski/economy/essay9.html