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April 26

duality vs. conjugacy

I noticed that Isbell conjugacy and Isbell duality have the same meaning. So, I would like to know the difference in meaning between duality and conjugacy in mathematics. Also, I found Category:Duality theories, but what is the field of mathematics called Duality theory? However, since Baez (2022) said that the Isbell conjugacy is an adjoints rather than a duality of the category, so I changed the category to Category:Adjoint functors. Thank you, SilverMatsu (talk) 03:54, 26 April 2024 (UTC)[reply]

The original duality occurs in projective geometry, see Duality (projective geometry). At some point people noticed that the axioms of the projective plane where the same (or equivalent) if you swapped the undefined terms "point" and "line". So any theorem in projective geometry can be transformed to a dual theorem by changing the roles of points and lines. The new theorem may simply be a restatement of the original theorem as in Desargues's theorem, but sometimes it's not as in Pappus's hexagon theorem. The result is that you often get two theorems for the price of one proof. You can define a dual category for a given category by reversing the arrows, but category theory was invented long after projective geometry so that's not the original meaning. You can also define the dual curve of a plane curve, the dual space of a vector space, the dual polytope of a polytope, etc. As far as I know there is no all-encompassing "theory of duality", just the custom of using "dual" to describe when mathematical objects seem to occur in pairs in some way. Calling something a dual usually implies that the dual of the dual is in some way identifiable with the original object, but this is not always required. For example the dual of a dual vector space is not identifiable with the original vector space unless it's finite dimensional. Duality does not always exist, for example there doesn't seem to be a useful concept for the dual of a finite group, though you can define one for abelian groups. And sometimes there is a duality that's not called that, for example cohomology can be viewed as the dual of homology. I don't think there is a formal distinction between a "dual" and a "conjugate", but usually a conjugate is the the result of applying an automorphism of order two. For example a complex conjugate is the result of applying the automorphism a+bi → a-bi. Again, this is more of a naming custom than a formal mathematical concept, and there is (apparently) some overlap. I'd say a "conjugate" is usually used when the two objects live inside the same structure, and "dual" is used when you're talking about two different structures. For example the dual of a plane curve lives in the dual of the plane in which the original curve lives. Category theory blurs the distinction between an object and a structure so I can see how the distinction is rather meaningless there. --RDBury (talk) 07:12, 26 April 2024 (UTC)[reply]

Thank you for teaching me so kindly. I'm going to re-read some of the references, keeping in mind what you've taught me. --SilverMatsu (talk) 16:02, 27 April 2024 (UTC)[reply]

Oblate spheroid

I thought of Googling orthographic projection ellipsoid and found these, did I interpret everything right?:

ν=a/(1-e²*(sinΦ)²)^1/2 (why not square root?) x=(ν+h)*cosΦ*cosλ y=(ν+h)*cosΦ*sinλ z=(ν*(1-e²)+h)*sinΦ This seems to be a simple spherical to Cartesian converter with latitudes (Φ) "massaged" so it's not slightly wrong (eccentricity²=0.00669437999014 so not much massaging). Then they convert that to topocentric Cartesian with a matrix I can't solve (now I know why galactic Cartesian's UVW!) but it seems like they also say surface points are U=ν*cosφ*sin(λ-λ_O) V=ν*(sinφ*cosφ_O-cosφ*sinφ_O*cos(λ-λ_O))+e²*(ν_O*sinφ_O-ν*sinφ)cosφ_O where _O means "of the topocentric origin". Did I get that right? If so then I can set an initial guess point at or about 0.25 circumference from the W-axis, use the formulae to find its U and V in "W-axis place"-centered coordinates and the test point is of course √(U²+V²) meters from the W-axis and the part of the ellipsoid with the most meters without being too far from the W-axis-test point plane is the limb of the Earth from infinite distance. The worst-case scenario for how spindly a pie the test points have to be in would be looking at ~the 45th parallel limb with the W-axis in the equator plane. The geocenter depth increases roughly quarter mile from 45.5N to 44.5N so ~56 meters poleward shortens the limb to W-axis line segment by 8 inches which is how much Earth curves in a mile. Azimuth accuracy needed increases "exponentially" with limb coordinate accuracy desired though so 10 miles accuracy would be a lot more than 10x easier than 1 mile. Sagittarian Milky Way (talk) 06:24, 26 April 2024 (UTC)[reply]

(Why not square root? Because the site may not have the facility to extend the radical bar over the whole expression.) —Tamfang (talk) 17:19, 7 May 2024 (UTC)[reply]

It's a pdf https://www.hydrometronics.com/downloads/Ellipsoidal%20Orthographic%20Projection.pdf since it has images couldn't they put a radical bar instead of 1/2th power? Anyway did I get it right, I never got to the part of school where they taught matrixes but maybe the stuff below is equivalent and not screwed up by my 100% truancy from late high school math conventions education. Sagittarian Milky Way (talk) 23:19, 7 May 2024 (UTC)[reply]

The text on these slides was obviously created using a word processing tool, and (apart from matters of taste) the choice of presentation may be affected by limitations of that tool. It all looks complicated, but actually the orthographic projection is as simple as it gets: map the 3D point

(x,y,z)

to the 2D point

(x,y),

"forgetting" one of the three spatial coordinates. For a sphere, one can use the transformation of spherical coordinates

(r,\theta ,\phi )

to Cartesian coordinates

(x,y,z)

(see the section Spherical coordinate system § Cartesian coordinates). A rotational ellipsoid requires an additional scaling of one of these coordinates. If the desired view of a 3D object is from an angle, rotate it in 3D space before projecting it. The general case is merely the chain of these by themselves standard, fairly simple transformations, of which spatial rotation is the most difficult. (It definitely helps to understand the notion of multiplying a 3×3 matrix and a 3D vector to get a new 3D vector; see Rotation matrix § In three dimensions.) A more natural perspective projection is only moderately more complex: one needs in general to also translate the object in 3D space, and the final mapping is given by

(x,y,z)\mapsto (x/z,y/z).

--Lambiam 09:19, 8 May 2024 (UTC)[reply]

I long wondered what that was, then wondered how they could simplify 2-D lists, then wondered if it was multiplying everything by everything like a 2-dimensional version of first-outside-inside-last. When I read matrix (math) I'll probably slap forehead at how few seconds it takes to teach and how simple it is compared to how hard it is to understand the hieroglyphics completely without a dictionary or Rosetta Stone. Similar to when I treated the options, futures, future options and sports betting odds in the newspaper as puzzles for 5-6 years and though I gave up and Googled at age ~18 even ten or twenty more years might've not been enough despite each of these conventions being easily teachable in seconds (i.e. USA has its own odds convention for non-horse sports, why was one team always -105 or less (minus many thousands if they're very likely to win) the other team was +100 to +many thousands (always somewhat less positive than the other was negative) but sometimes it's -115 -105 or PK which obviously meant pick 'em/50% chance. Completely stumped me what that means, if it was consistently how much a win profits per $100 or how much to win $100 I would've figured it out but it was just the bet per $100 bet without the $ sign, minus="you bet this not him". Very anticlimactic, like Googling how magicians drop a cup being filled by a cup without it falling (ROT13: Pyrneguernq) Sagittarian Milky Way (talk) 16:28, 8 May 2024 (UTC)[reply]

Ah so that's what dot product is, draw a matrix aligned with the 2 matrices, multiply corresponding cells of equal outwardness, add all the shit in the product cell (which would be a lot for those matrices far bigger than 3x3). I was wondering what cells multiplied with what (everything times everything?) and how you smooshed 3x3 to 1x3. And adding real number matrices to each other and multiplying them by real numbers is way simpler than it looks! (though unneeded here). So I learned more than if I data entried into a matrix solver. Sagittarian Milky Way (talk) 19:45, 8 May 2024 (UTC)[reply]