Subject [math-fun] (More) area on a sphere. From Bill Gosper <billgosper@gmail.com>[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_search=932771c375d2fc6311380a21c7bacda8&_caps=pdf%3D1%2Cflash%3D1%2Ctif%3D0#add> Sender math-fun <math-fun-bounces@mailman.xmission.com>[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_search=932771c375d2fc6311380a21c7bacda8&_caps=pdf%3D1%2Cflash%3D1%2Ctif%3D0#add> To math-fun@mailman.xmission.com[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_search=932771c375d2fc6311380a21c7bacda8&_caps=pdf%3D1%2Cflash%3D1%2Ctif%3D0#add> Reply-To math-fun <math-fun@mailman.xmission.com>[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_search=932771c375d2fc6311380a21c7bacda8&_caps=pdf%3D1%2Cflash%3D1%2Ctif%3D0#add> Date 2015-03-10 13:08 I love playing with NeilB's (& Gene's) solid angle integral. In terms of latitude and longitude, the area enclosed by a series of arcs, (lat(t),long(t)), 0<t<1 is just the sum (over the arcs) of spharea[lat_, long_] := long[1] - long[0] - Integrate[D[long[t], t]*Sin[lat[t]], {t, 0, 1}] (times r²). E.g., for the four sides of a "rectangle", (lat(t), long(t) = {{south*(1 - #) + north*# &, east &}, {north &, east*(1 - #) + west*# &}, {north*(1 - #) + south*# &, west &}, {south &, west*(1 - #) + east*# &}}; In[4]:= Factor[Total[spharea @@ # & /@ %]] Out[4]= (east - west) (Sin[north] - Sin[south]) consistent with the "hatbox" (cylindrical projection) equal area theorem. The projection of an a×b (semiaxes) by (height) z ellipse onto the unit sphere has area 2 (π- (2 b z EllipticPi[1 - b^2/a^2, (a^2 - b^2)/(a^2 + z^2)])/(a Sqrt[a^2 + z^2])) (done in two halves), presumably a crypto-symmetric function of a and b. (Does the area formula for the general a,b,c ellipsoid have six-way crypto-symmetry?) ------ Indeed, Macsyma and Mma get six different but numerically equivalent expressions. Neither can prove the equality of any two, nor even expand them in comparable series. ---------------- Checking, replace z by 1 and a,b by h a, h b, small h, In[37]:= Series[% /. {a -> a*h, b -> b*h}, {h, 0, 2}] Out[37]= SeriesData[h, 0, { 2 Pi - 2 a b^(-1) (a^(-2) b^2)^Rational[1, 2] Pi, 0, a b Pi}, 0, 3, 1] In[38]:= Simplify[Normal[%], a > b > 0] Out[38]= a b h² π I.e., the area of an h a × h b ellipse. -------- Similar were my series solutions (recounted here) to finding the semiaxes of an ellipsoid given its principal girths. ------------- I think I once mentioned here NeilB's proud ownership of a plastic-laminated cheat-sheet -------- gosper.org/cheetsheat.pdf (scanned with Neil's IPhone!) --------- of handy-dandy geometry formulas, purporting to give the area of an elliptical cone in elementary functions. Probably planted by the KGB or Red Chinese during the Cold War. ------------- But lurking in Canada. I misrecalled: it was an elliptical cylinder, not an elliptical cone. Note also the a b c sin theta volume for the general parallelepiped, given only one of the angles between adjacent edges. This is known as the a b c conjecture. Finally, the reverse of the card helpfully protects you from looking like an idiot by reminding you how to spell "frustrum". --rwg _______________________________________________