[HN Gopher] Sunset Geometry (2016)
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Sunset Geometry (2016)
Author : johndcook
Score : 118 points
Date : 2021-10-14 14:29 UTC (8 hours ago)
(HTM) web link (www.shapeoperator.com)
(TXT) w3m dump (www.shapeoperator.com)
| [deleted]
| [deleted]
| wyldfire wrote:
| > This is 20% larger than the true value, 3960 mi
|
| Whoa, that is not insignificant error. What contributes to an
| error of this size?
| ffhhj wrote:
| The moon.
| jvanderbot wrote:
| I expect the curvature of the earth has much less effect than
| refraction and mirage effects in causing this phenomenon.
| thatcherc wrote:
| My bet would be refraction of the sunlight through the
| atmosphere. At sunrise and sunset light that reaches you on the
| ground is taking just about the longest path through the
| atmosphere that it can, and gradient of the atmosphere's
| density is definitely going to add some bending to the light
| coming in from space.
| sixbrx wrote:
| Yeah, it's significant enough that my astro mount has a "King
| tracking rate" to account for this effect. Named for Edward
| King.
|
| https://en.wikipedia.org/wiki/Edward_Skinner_King
| 123pie123 wrote:
| I was going to say something similar
|
| "we can see the Sun even when it is _geometrically_ just
| below the horizon, at both sunrise and sunset. This is
| because of the refraction of the light from the Sun by the
| Earth 's atmosphere--the Earth's atmosphere bends the path of
| the light so that we see the Sun in a position slightly
| different from where it really is."
|
| see http://curious.astro.cornell.edu/our-solar-system/52-our-
| sol...
| rrss wrote:
| Vanderbei's linked articles have some discussion of sources of
| error. The main one seems to be waves making the reflection
| appear longer.
| Keyframe wrote:
| I don't think refraction would impact the error as much as a
| focal length of the camera. Probably both however.
| Tabular-Iceberg wrote:
| What's the effect of the focal length?
| Keyframe wrote:
| https://fstoppers.com/architecture/how-lens-compression-
| and-...
| dahart wrote:
| That article specifically clarifies that moving the
| camera is what causes changes to relative sizes, not the
| focal length. But in the experiment described, the camera
| doesn't move a significant difference with respect to the
| sun, the camera can't really get anywhere close enough to
| the sun or the reflection on the horizon to make any
| difference, certainly not 20%.
| KineticLensman wrote:
| > What's the effect of the focal length?
|
| Not sure what the GP intended, but one possible effect is
| that increasing the focal length ('zooming in') would
| change the relative size of two objects (one closer, one
| further away) depending on their relative distance to the
| camera (I think the closer object might appear relatively
| larger).
|
| But if the sun and the horizon are the intended objects,
| they are both effectively at infinity, from the camera's
| perspective, so I wouldn't expect focal length to change
| anything in this case.
| terramex wrote:
| Changing focal length of a lens ('zooming in') does not
| change relative size of two objects, it is a common
| misconception. What changes relative size of objects are
| your legs when you move from one spot to another to get
| roughly the same framing with different focal lengths. If
| you stand in one spot and zoom or change prime lenses
| relative object sizes stay the same.
| KineticLensman wrote:
| Fair point - I should have tried this with an actual
| camera. But I suspect my line of reasoning still stands,
| that the focal length doesn't affect the phenomena being
| discussed here
| julienchastang wrote:
| Somewhat related: How many of you have ever seen the green flash
| [0] ? I never have unfortunately even though I've looked for it
| many times.
|
| [0] https://en.wikipedia.org/wiki/Green_flash
| abecedarius wrote:
| I believe I saw it once, at the beach looking out on the ocean.
| It was very brief and a long time ago.
| jedimastert wrote:
| Sebastian Lague made a fantastic video about
| approximating/simulating Rayleigh scattering in real-time (using
| Unity) to simulate sky color and sun sets. Really interesting
| stuff
|
| https://www.youtube.com/watch?v=DxfEbulyFcY
| darthoctopus wrote:
| > I have never seen someone try to use Pauli matrices to solve a
| trigonometry problem, but it can certainly be done.
|
| The Pauli matrices in this context are isomorphic to quaternions,
| which certainly have been used to solve geometric problems in 3
| dimensions (although not necessarily by physicists), as has been
| discussed many many times here on HN. The property of describing
| spin-1/2 particles (i.e. generating SU(2)) is precisely the same
| property that makes the quaternions amenable for use in reasoning
| about 3D rotations!
| jacobolus wrote:
| > _describing spin-1 /2 particles (i.e. generating SU(2))_
|
| This has the relationship backwards. Particles have this spin
| group because they can rotate in 3-dimensional space, not
| because they have some mysterious association with the complex
| plane.
|
| That we can represent rotations using 2x2 unitary matrices of
| unit determinant is just a coincidence. There are a bunch of
| such coincidental isomorphism between groups. The explanation
| is comparable to the "strong law of small numbers"
| https://en.wikipedia.org/wiki/Strong_law_of_small_numbers
|
| (The unitary matrices come about when we take points on the
| 2-sphere and stereographically project them onto the plane,
| representing points in the plane by complex numbers. Then
| concentric rotations of 3-space of correspond to particular
| Mobius transformations of the complex plane, which can be
| represented as the special unitary matrices. This works out
| cleanly when projecting onto a 2-dimensional plane, but
| representing rotations gets trickier in higher dimensions.)
|
| * * *
|
| But you are missing Jason's point. There are clearly many
| possible ways of representing the same relationships. The
| underlying relationships don't change if you e.g. use classical
| spherical trigonometry. The question he is trying to ask
| instead is: which representation is most conceptually clear and
| intuitive to work with (after some experience), has the nicest
| and easiest to manipulate notation, etc.
|
| There are valid reasons to use a stereographic representation
| of points on the sphere. (For instance, it involves 2
| coordinates instead of 3.) And from there, representing
| spherical rotations as Mobius transformations is convenient and
| effective. But _conceptually_ , if trying to solve arbitrary
| problems with pen and paper, representing points as vectors and
| rotations as scalar+bivector "quaternions" is a lot more
| natural. Especially if you have a problem where some parts are
| not confined to the sphere.
|
| For more on representing spherical geometry stereographically,
| cf. https://observablehq.com/@jrus/planisphere - there are some
| tools even here where we get leverage out of treating
| stereographically projected points as 2D vectors rather than as
| complex numbers, and separating the concepts of scalar, vector,
| and bivecor.
| thatcherc wrote:
| The relationship between 3D rotations, spin-1/2 particles, and
| unit quaternions appears on one of the best-named Wikipedia
| pages: Exceptional Isomorphisms! [0]
|
| [0] -
| https://en.wikipedia.org/wiki/Exceptional_isomorphism#Spin_g...
| lqet wrote:
| > If the earth was flat, photographs of the sun setting over
| water would look like this:
|
| I am curious: has this argument, historally, ever been used
| against the idea that the world is flat?
| dudus wrote:
| The idea that the world is flat is not something that exists
| due to lack of arguments.
| TruthWillHurt wrote:
| Too much math..
| okmathlessone wrote:
| 'Flat-earth', i wrote this to give you a smile, a couple of
| years ago i'd seen a wristwatch -i thought that is must have
| been made vor classic painters, not using a photoshot or
| better their own mind, but a sketch drawn outside (in the
| wild), its display showed a model of the time-dependent
| shaddow cast, using two overlaying layers which build the
| shaddow. It took a while and reading till i fiddling out how
| the time-setting to get a correct shaddow-cast will be done
| (hypothetical) by using a list on the internet. Hint: 'What
| may be possible done with building a watch' ^^
| abecedarius wrote:
| Great question. Skimming through the Spherical Earth page on
| Wikipedia I don't see it, though it could conceivably be among
| the phenomena listed by e.g. Strabo and not enumerated there.
| twic wrote:
| Does geometric algebra provide an alternative to pseudovectors
| for representing things like angular velocity?
|
| The fact that you have to flip the sign of pseudovectors
| sometimes feels like a hint that they aren't the right
| representation, somehow.
| ajkjk wrote:
| Yes (to repeat the other reply here more confidently).
| Pseudovectors are (n-1)-graded vectors in exterior algebra, ie
| bivectors in 3d and trivectors in 4d. No sign flip is required
| required if you write them this way. Algebraically, P(x^y) =
| P(x)^P(y) (basically by definition), where P is a parity
| transformation.
| ogogmad wrote:
| I think angular velocities are precisely the bivectors (which I
| take to be the grade-2 elements of an exterior algebra or
| Geometric Algebra). The change of basis is indeed different
| than for ordinary vectors. The exponential map (the obvious
| generalisation of e^x) then takes bivectors to the "rotors",
| which represent rotation operators.
|
| In terms of Lie theory, the rotors are a Lie group and the
| bivectors are the corresponding Lie algebra.
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