APOD: A Laser Strike at the Galactic Center (2019 Jan 06)

Comments and questions about the APOD on the main view screen.
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MarkBour
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Re: APOD: A Laser Strike at the Galactic Center (2019 Jan 06)

Post by MarkBour » Sun Jan 13, 2019 6:21 am

neufer wrote:
Fri Jan 11, 2019 6:20 pm
https://en.wikipedia.org/wiki/Non-reversing_mirror wrote: <<A non-reversing mirror (sometimes referred to as a flip mirror) is a mirror that presents its subject as it would be seen from the mirror.>>
When I first saw this, the term "non-reversing mirror" made perfect sense for what such a mirror does. Because I know that we tend to view a mirror reflection as a sort of "reversal", based on the way we think about it all. But upon further reflection (sorry) it seems that regular mirrors are actually the non-reversing ones, and it would be more appropriate to call these double mirrors "reversing mirrors". I guess this is kind of a subtle issue in human psychology, and that most people would disagree with me here. If you rotate these mirrors 90 degrees and see yourself standing upside down in the image, however, then I think the majority of folks would be in agreement with me. But I'm sure the terminology is too entrenched, to try to overturn it, anyway. I see that these doubly-reflecting arrangements are also called "flip mirrors". I could go for that term as the one I much prefer.
Mark Goldfain

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neufer
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Re: APOD: A Laser Strike at the Galactic Center (2019 Jan 06)

Post by neufer » Sun Jan 13, 2019 9:40 pm

MarkBour wrote:
Sun Jan 13, 2019 6:21 am

When I first saw this, the term "non-reversing mirror" made perfect sense for what such a mirror does. Because I know that we tend to view a mirror reflection as a sort of "reversal", based on the way we think about it all. But upon further reflection (sorry) it seems that regular mirrors are actually the non-reversing ones, and it would be more appropriate to call these double mirrors "reversing mirrors". I guess this is kind of a subtle issue in human psychology, and that most people would disagree with me here. If you rotate these mirrors 90 degrees and see yourself standing upside down in the image, however, then I think the majority of folks would be in agreement with me. But I'm sure the terminology is too entrenched, to try to overturn it, anyway. I see that these doubly-reflecting arrangements are also called "flip mirrors". I could go for that term as the one I much prefer.
https://www.etymonline.com/word/reverse#etymonline_v_29895 wrote:
reverse (adj.) c. 1300, from Old French revers "reverse, cross, opposite" (13c.), from Latin reversus, past participle of revertere "turn back, turn about, come back, return" (see revert). Reverse angle in film-making is from 1934. Reverse discrimination is attested from 1962, American English.

reverse (v.) early 14c. (transitive), "change, alter;" early 15c. (intransitive), "go backward," from Old French reverser "reverse, turn around; roll, turn up" (12c.), from Late Latin reversare "turn about, turn back," frequentative of Latin revertere.

reverse (n.) mid-14c., "opposite or contrary" (of something), from reverse (adj.) or from Old French Related: revers "the opposite, reverse." Meaning "a defeat, a change of fortune" is from 1520s; meaning "back side of a coin" is from 1620s. Of gear-shifts in motor cars, from 1875. As a type of sports play (originally rugby) it is recorded from 1921.
Putting your car in reverse is different from turning around
(even if revertere means "turn back, turn about, come back, return").

The operation of a 1D mirror is essentially the "square root"
of the operation of a 2D mirror (= a 180º rotation).

30 years ago I coauthored a paper on "square root" F(0.5, z) operations:
--------------------------------------------------------------------------
"Functional Powers near a Fixed Point"
by Lawrence J. Crone and Arthur C. Neuendorffer,
_Journal of Mathematical Analysis and Applications_,
Vol.132, No.2 June 1988.

Abstract: It is proved that if a function F(z) is analytic in a neighborhood of a fixed point z0, and if 0 < ¦F′(z0)¦ < 1, then there exists a family of related functions F(p, z), each defined in a neighborhood of z0, which act as functional powers of F(z). In particular, F(0, z) = z, F(1, z) = F(z), and F(p, F(q, z)) = F(p + q, z). It is further demonstrated that the family of functions F(p, z) is identical with the set of nonconstant analytic functions with fixed point z0 which commute with F(z).
-------------------------------------------------------------
Art Neuendorffer

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MarkBour
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Re: APOD: A Laser Strike at the Galactic Center (2019 Jan 06)

Post by MarkBour » Wed Jan 16, 2019 12:48 am

neufer wrote:
Sun Jan 13, 2019 9:40 pm
The operation of a 1D mirror is essentially the "square root"
of the operation of a 2D mirror (= a 180º rotation).

30 years ago I coauthored a paper on "square root" F(0.5, z) operations:
--------------------------------------------------------------------------
"Functional Powers near a Fixed Point"
by Lawrence J. Crone and Arthur C. Neuendorffer,
_Journal of Mathematical Analysis and Applications_,
Vol.132, No.2 June 1988.

Abstract: It is proved that if a function F(z) is analytic in a neighborhood of a fixed point z0, and if 0 < ¦F′(z0)¦ < 1, then there exists a family of related functions F(p, z), each defined in a neighborhood of z0, which act as functional powers of F(z). In particular, F(0, z) = z, F(1, z) = F(z), and F(p, F(q, z)) = F(p + q, z). It is further demonstrated that the family of functions F(p, z) is identical with the set of nonconstant analytic functions with fixed point z0 which commute with F(z).
-------------------------------------------------------------
Not quite sure I follow your 1D/2D terminology, but I gather you're just labeling the number of times the light changes direction. Then, sure, one can view the twice-reflected transformation as the product of two single-reflection transforms. (Although when used in its usual configuration, I'd say a single-plane mirror's transform, when squared, would result in the identity transform.)

Interesting paper abstract. Are these functions of the complex plane (locally analytic there)?

Was 1988 really 30 years ago, already? Yeeesh!
Mark Goldfain

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neufer
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Re: APOD: A Laser Strike at the Galactic Center (2019 Jan 06)

Post by neufer » Wed Jan 16, 2019 2:38 pm

MarkBour wrote:
Wed Jan 16, 2019 12:48 am
neufer wrote:
Sun Jan 13, 2019 9:40 pm
The operation of a 1D mirror is essentially the "square root"
of the operation of a 2D mirror (= a 180º rotation).

30 years ago I coauthored a paper on "square root" F(0.5, z) operations:
--------------------------------------------------------------------------
"Functional Powers near a Fixed Point"
by Lawrence J. Crone and Arthur C. Neuendorffer,
_Journal of Mathematical Analysis and Applications_,
Vol.132, No.2 June 1988.

Abstract: It is proved that if a function F(z) is analytic in a neighborhood of a fixed point z0, and if 0 < ¦F′(z0)¦ < 1, then there exists a family of related functions F(p, z), each defined in a neighborhood of z0, which act as functional powers of F(z). In particular, F(0, z) = z, F(1, z) = F(z), and F(p, F(q, z)) = F(p + q, z). It is further demonstrated that the family of functions F(p, z) is identical with the set of nonconstant analytic functions with fixed point z0 which commute with F(z).
-------------------------------------------------------------
Not quite sure I follow your 1D/2D terminology, but I gather you're just labeling the number of times the light changes direction. Then, sure, one can view the twice-reflected transformation as the product of two single-reflection transforms. (Although when used in its usual configuration, I'd say a single-plane mirror's transform, when squared, would result in the identity transform.)
Barber shops used to always have opposite walls of parallel mirrors where one could observe an infinite series of one's image... half of which would include the back of one's head such that "a single-plane mirror's transform, when squared, would result in the identity transform."

:arrow: I'm thinking rather of "corner reflector type" mirrors such as telescopic diagonal mirrors which involve a 90º rotation plus a left/right (or up/down) parity shift.
MarkBour wrote:
Wed Jan 16, 2019 12:48 am

Interesting paper abstract. Are these functions of the complex plane (locally analytic there)?
Yes. A simple linear example NOT included (because ¦F′(z0= 1) might be a multiplication of minus 1 where the square root operation would be a multiplication of i (where the fixed "point" is the imaginary axis).
Art Neuendorffer