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



On 04/30/2009 06:42 AM, Anthony Lapinski wrote:
I've always wondered about this solar image thing. Using a rectangular
mirror, the image of the sun "nearby" (few meters away) is rectangular,
but "far away" (over 15 m) is circular.

OK.

I thought it had something to do
with the fact that a mirror has an infinite focal length.

Well, indirectly it does. If it had a non-infinite focal
length, it would be badly out of focus either "nearby" or
"far away" or both. As it is, it is equally-slightly out
of focus everywhere.

Not sure my (high school) students have ever heard of "convolution,"

Well, that's why they go go school, to learn about such things.

Is there another ("basic") explanation to use when trying to explain this
optical phenomenon to students?

Not that I can think of. Even if there were, the importance of
convolution is so wide and deep that it would still be the right
way to go.

The basic idea is not tricky. There are a gazillion non-basic
applications, but don't let that scare you away; stick to the
basics for now.

Briefly ignoring my own advice, let me mention some not-so-basic
applications:
*) Electronics
-- AC circuits.
-- Analog and digital filters.
-- Echo cancellation (for long-distance telephony, modems, etc.)
*) Spectroscopy.
*) MRI.
*) Optics.
*) Digital image processing
-- blurring
-- sharpening
-- edge finding
-- subsampling
-- MPEG compression.
-- etc.
*) Radar.
*) Acoustics
-- echo cancellation
-- sonar
-- MP3 compression
*) Green functions, including
-- Elementary particles and quantum field theory.
-- Seismology.
-- Mechanical engineering.
-- Civil engineering (bending of beams in bridges etc.)
-- Heat conduction.
*) etc. etc. etc.

Note that autocorrelations and cross correlations are just big convolutions.
Also note multiple connections to Fourier transforms (plain olde convolution
theorem, also Wiener-Khinchin theorem).

And that's just what pops to mind. I have on my shelf entire books on
convolution and deconvolution.

Note that modern PCs are rather good at doing convolutions. There are
special instructions and dedicated hardware on the CPU just for doing
dot products, and that includes correlations as a special case. That
should give you some idea of how important the applications are; folks
don't put dedicated hardware into every PC on the planet just on a whim.
(Beware: sometimes you have to jump through hoops to get your application
to actually use the special instructions.)

I'm not suggesting that you cover all of that in HS. I'm just trying to
hint at the importance, to convince you that time spent introducing the
idea is time well spent.

I don't know of any web sites that offer good tutorials on convolutions.
If anybody on the list knows of any, please let us know! The topic is
covered in typical "Mathematical Methods of Physics" books. I believe
Boas is as good as any, but my copy seems to have walked off so I can't
confirm this.

Also: As a starting place, and for many serious real-world applications,
the convolution is a sum not an integral. You can with a small amount of
cleverness do the sums using a spreadsheet; here's an example that you can
play with:
http://www.av8n.com/physics/convolution-intro.xls

Feel free to change the kernel (in the light green area) and the main
input (in the light blue area).

Once you see what a convolution is supposed to do, you can modify the
spreadsheet to model the sun/mirror system. It's not very tricky.