I am currently working on an emergency license for math. I am trying to pass the Math Praxis 0061 to receive my certification. Need some help on some of the questions.
Thanks,
karies4083
Hello, I have graduated from math -uhm- maybe 20 years ago and now, after some turmoil in ICT field, I have scrubbing a job from here and there (part time teaching and re-warming the research career.)
My current problem is this: Given a unit hypersphere S_3 (in R^4) and four normalized vectors n1,n2,n3,n4 (ni<>nj), how to calculate the -uhm- space angle defined by the vectors (subvolume of the sphere)?
I am familiar of the normal unit sphere S_2 which have a space angle Phi defined by 3 vectors n1,n2,n3 as: Phi= a12+a23+a31 -pi where a_{ij} = cos^{-1}(ni . nj). (E.g. the familiar rectangular corner slice of the sphere which has the area Phi = 3 pi/2 -pi = 4pi / 8 .)
Is there a nice equation for S_3 case? Preferably one using a_{ij} angles. Or a recursive one? Actually my hunger goes up to S_11 (12 vectors in R^12), so I need references if the problem is
not trivial. The intended use is non-commercial, so I am not trying to milk you for my profit.
Problem (almost) solved. I stumbled upon Heinz Hopf, Selected Chapters of Geometry, 1940, which answers my problem with the first incoming breeze. (Let's see if the computation is too expensive next... And I had a mistake in the definition of cos -terms, have to snip away the components along nk from ni and nj... )