4 ms·

Fresnel Integral

taneq 4y ago
Why, though? Just because it’s pretty? Edit: their application to smooth path generation is pretty interesting: https://en.m.wikipedia.org/wiki/Track_transition_curve https://en.m.wikipedia.org/wiki/Track_transition_curve
- sidlls 4y ago
  They have applications all over physics.
  - taneq 4y ago
    Yep, cool. It's much more fun to discuss something if you have some kind of starting point for discussion, though. There seems to be a number of interesting things about these - which ones were OP thinking of? Otherwise it's just a random wiki link.
    - exmadscientist 4y ago
      Strongly agreed. Why are these more interesing than the Airy functions? What's cool about them today? Posting random Wikipedia links is about as interesting as posting random dictionary entries. I'd much rather see a short intro or at least comment on the article rather than just a link.
      - Agentlien 4y ago
        I disagree about random Wikipedia link. While I sometimes wish for more context, most times when a link to a Wikipedia page hits the front page I read the article and learn something cool and unexpected.
        kragen 4y ago
        When I scanned the Oxford English Dictionary, that happened to me often with random dictionary entries. I probably spent as much time reading random dictionary entries as I spent actually scanning the damn thing.
  - Yajirobe 4y ago
- mcnamaratw 4y ago
  I think the Wikipedia article shows that the original importance of the Fresnel integral is in optics. It's also a precursor to the Feynman path integral way of thinking. The plot may or may not be pretty, but that's not why it matters in engineering and physics.
raphlinus 4y ago
Also the mathematical basis of the Euler spiral, also known as Cornu spiral and clothoid, which is a beautiful curve and a very good basis for curvature-continuous splines. I wrote about that extensively in my thesis.
- kragen 4y ago
  That was the first thing I thought of when I visited the Wikipedia page and saw the link to [[Euler spiral]]!
- breakds 4y ago
  The thesis is a master piece, highly recommended!
  - eecc 4y ago
    Link?
    - nyankosensei 4y ago
      https://www.levien.com/phd/thesis.pdf https://www.levien.com/phd/thesis.pdf
- meltyness 4y ago
  Just as the radius of a circle determines the radial acceleration needed to maintain a circular path, in general, a given path's curvature may be related to this net force, a result from applying correct principles and an intuition regarding the Frenet-Serret formulas.
- drsopp 4y ago
  I made an introduction video in norwegian on these 8 years ago. https://m.youtube.com/watch?v=-WFVJh-dMgY https://m.youtube.com/watch?v=-WFVJh-dMgY
peter_d_sherman 4y ago
Greetings all! >[tanek]: Why, though? Just because it’s pretty? Isn't that reason enough? <g> Usually in Math/Science/Physics -- if something is pretty -- then usually it has a lot of purposes/functions around it that are employed by Nature in various ways... One example of this might be Phi, AKA The Golden Ratio: https://www.goldennumber.net/ https://www.goldennumber.net/ Both beautiful and purposeful! But back to Fresnel Integrals! >[sidlls]: They have applications all over physics. Indeed they do! >[exmadscientist]: Strongly agreed. Why are these more interesing than the Airy functions? Well, I don't know! <g> I do know that prior to your comment, I didn't know what Airy functions were! So, here's a link for the folks back home, just tuning in!: https://en.wikipedia.org/wiki/Airy_function https://en.wikipedia.org/wiki/Airy_function Note the similar features (at least in part!) of the graph of Fresnel Integrals (https://en.wikipedia.org/wiki/File:Fresnel_Integrals_(Unnormalised).svg https://en.wikipedia.org/wiki/File:Fresnel_Integrals_(Unnorm...) and Airy Functions (https://en.wikipedia.org/wiki/File:Airy_Functions.svg https://en.wikipedia.org/wiki/File:Airy_Functions.svg). It's sort of like, if you took one graph and flipped it over 180 degrees, and cut out the part where the two functions start converging or start diverging, and ignore the units of measure, then the two graphs almost look similar... So maybe they're related in some way... maybe there's something there... Airy functions -- seem like they would be very interesting to study alongside Fresnel Integrals! So, thank you for suggesting them! The rest of my comments are for everyone in general -- so back to the Fresnel Integrals! OK, so here's what my intuition tells me... We've all seen a hologram right? OK, so a hologram is basically a 2D space -- masquerading as a 3D space. First question (discovered easily by inverting/reversing the above idea): Is it possible to have a 3D space -- which is actually a 2D space, in disguise? ? Sort of like a "reverse-hologram" -- for lack of better terminology... In other words, is it possible, for an actual, real, 3D space in reality -- to have similar characteristics to a flat, 2D hologram? ? Well, I for one don't know! <g> But what I do know is that IF such a thing is possible, IF it is (again, this is highly speculative!) -- then IF so -- then on the 2D holographic plane -- ALL of the 3D information -- must exist! So now the question is (again, IF such a thing is possible, which is highly speculative!) -- How/where is that INFORMATION -- which is in the 3D view of reality -- encoded into the 2D plane, which is a basically hologram of that view of reality? The answer for that question, is simply: "I don't know!!!" But Fresnel Integrals (and Airy Functions!) -- seem like they might have something to offer here -- some way of encoding extra (depth!) information -- into what is basically a 2D frame! Now IF that's true (and I should point out that we're beyond speculative at this point -- we're in "super-crackpot-they've-gone-to-plaid" land now, just for reference! <g>), then Fresnel Integrals do not just generically have applications all over physics -- but they're also intimately, intimately related to such things as the double-slit experiment, gravity ("A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration." -- well, this is information, and this information encoded in a 2D image would determine gravity (gravity = acceleration)), and of course, diffraction -- which is sort of like the measure of how fuzzy distant objects in a 2D image are, depending upon where the camera lens is focused... So, that's why Fresnel Integrals (and Airy Functions!) -- might be interesting for Future Physicists (and people who are not physicists -- but just like "pretty" things! <g>. I should point out that I am in the latter camp! <g>) Anyway, since you asked! <g>
eecc 4y ago
Oh this is something 3brown1blue should do
- peter_d_sherman 4y ago
  Agreed entirely!

Why, though? Just because it’s pretty? Edit: their application to smooth path generation is pretty interesting: https://en.m.wikipedia.org/wiki/Track_transition_curve https://en.m.wikipedia.org/wiki/Track_transition_curve

sidlls 4y ago
They have applications all over physics.
- taneq 4y ago
  Yep, cool. It's much more fun to discuss something if you have some kind of starting point for discussion, though. There seems to be a number of interesting things about these - which ones were OP thinking of? Otherwise it's just a random wiki link.
  - exmadscientist 4y ago
    Strongly agreed. Why are these more interesing than the Airy functions? What's cool about them today? Posting random Wikipedia links is about as interesting as posting random dictionary entries. I'd much rather see a short intro or at least comment on the article rather than just a link.
    - Agentlien 4y ago
      I disagree about random Wikipedia link. While I sometimes wish for more context, most times when a link to a Wikipedia page hits the front page I read the article and learn something cool and unexpected.
      - kragen 4y ago
        When I scanned the Oxford English Dictionary, that happened to me often with random dictionary entries. I probably spent as much time reading random dictionary entries as I spent actually scanning the damn thing.
- Yajirobe 4y ago
mcnamaratw 4y ago
I think the Wikipedia article shows that the original importance of the Fresnel integral is in optics. It's also a precursor to the Feynman path integral way of thinking. The plot may or may not be pretty, but that's not why it matters in engineering and physics.

raphlinus 4y ago

Also the mathematical basis of the Euler spiral, also known as Cornu spiral and clothoid, which is a beautiful curve and a very good basis for curvature-continuous splines. I wrote about that extensively in my thesis.

kragen 4y ago
That was the first thing I thought of when I visited the Wikipedia page and saw the link to [[Euler spiral]]!
breakds 4y ago
The thesis is a master piece, highly recommended!
- eecc 4y ago
  Link?
  - nyankosensei 4y ago
    https://www.levien.com/phd/thesis.pdf https://www.levien.com/phd/thesis.pdf
meltyness 4y ago
Just as the radius of a circle determines the radial acceleration needed to maintain a circular path, in general, a given path's curvature may be related to this net force, a result from applying correct principles and an intuition regarding the Frenet-Serret formulas.
drsopp 4y ago
I made an introduction video in norwegian on these 8 years ago. https://m.youtube.com/watch?v=-WFVJh-dMgY https://m.youtube.com/watch?v=-WFVJh-dMgY

peter_d_sherman 4y ago

Greetings all! >[tanek]: Why, though? Just because it’s pretty? Isn't that reason enough? <g> Usually in Math/Science/Physics -- if something is pretty -- then usually it has a lot of purposes/functions around it that are employed by Nature in various ways... One example of this might be Phi, AKA The Golden Ratio: https://www.goldennumber.net/ https://www.goldennumber.net/ Both beautiful and purposeful! But back to Fresnel Integrals! >[sidlls]: They have applications all over physics. Indeed they do! >[exmadscientist]: Strongly agreed. Why are these more interesing than the Airy functions? Well, I don't know! <g> I do know that prior to your comment, I didn't know what Airy functions were! So, here's a link for the folks back home, just tuning in!: https://en.wikipedia.org/wiki/Airy_function https://en.wikipedia.org/wiki/Airy_function Note the similar features (at least in part!) of the graph of Fresnel Integrals (https://en.wikipedia.org/wiki/File:Fresnel_Integrals_(Unnormalised).svg https://en.wikipedia.org/wiki/File:Fresnel_Integrals_(Unnorm...) and Airy Functions (https://en.wikipedia.org/wiki/File:Airy_Functions.svg https://en.wikipedia.org/wiki/File:Airy_Functions.svg). It's sort of like, if you took one graph and flipped it over 180 degrees, and cut out the part where the two functions start converging or start diverging, and ignore the units of measure, then the two graphs almost look similar... So maybe they're related in some way... maybe there's something there... Airy functions -- seem like they would be very interesting to study alongside Fresnel Integrals! So, thank you for suggesting them! The rest of my comments are for everyone in general -- so back to the Fresnel Integrals! OK, so here's what my intuition tells me... We've all seen a hologram right? OK, so a hologram is basically a 2D space -- masquerading as a 3D space. First question (discovered easily by inverting/reversing the above idea): Is it possible to have a 3D space -- which is actually a 2D space, in disguise? ? Sort of like a "reverse-hologram" -- for lack of better terminology... In other words, is it possible, for an actual, real, 3D space in reality -- to have similar characteristics to a flat, 2D hologram? ? Well, I for one don't know! <g> But what I do know is that IF such a thing is possible, IF it is (again, this is highly speculative!) -- then IF so -- then on the 2D holographic plane -- ALL of the 3D information -- must exist! So now the question is (again, IF such a thing is possible, which is highly speculative!) -- How/where is that INFORMATION -- which is in the 3D view of reality -- encoded into the 2D plane, which is a basically hologram of that view of reality? The answer for that question, is simply: "I don't know!!!" But Fresnel Integrals (and Airy Functions!) -- seem like they might have something to offer here -- some way of encoding extra (depth!) information -- into what is basically a 2D frame! Now IF that's true (and I should point out that we're beyond speculative at this point -- we're in "super-crackpot-they've-gone-to-plaid" land now, just for reference! <g>), then Fresnel Integrals do not just generically have applications all over physics -- but they're also intimately, intimately related to such things as the double-slit experiment, gravity ("A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration." -- well, this is information, and this information encoded in a 2D image would determine gravity (gravity = acceleration)), and of course, diffraction -- which is sort of like the measure of how fuzzy distant objects in a 2D image are, depending upon where the camera lens is focused... So, that's why Fresnel Integrals (and Airy Functions!) -- might be interesting for Future Physicists (and people who are not physicists -- but just like "pretty" things! <g>. I should point out that I am in the latter camp! <g>) Anyway, since you asked! <g>

eecc 4y ago

Oh this is something 3brown1blue should do

peter_d_sherman 4y ago
Agreed entirely!