1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Spherical cap

  1. Jan 9, 2016 #1
    I am trying to understand N-spherical cap area formula (surface area of blue part), but it seems to give wrong answers.
    lossless-page1-220px-Spherical_cap_diagram.tiff.png

    for 1 dimensional cap obviously ## \frac{l_{cap}}{l_{sphere}}=\frac{l_{arc}}{l_{circle}}=\frac{r*θ}{r*2π}=\frac{θ}{2π} ##

    But according to wikipedia formula ##\frac{l_{cap}}{l_{sphere}}=\frac{I(\frac{(2r-h)h}{r^2},\frac{n}{2},\frac{1}{2})}{2}##
    https://en.wikipedia.org/wiki/Spherical_cap#Hyperspherical_cap

    since ##h=r(1-cos(θ))##
    ## I(a,b,c)=\int_0^a (dx*x^{b-1}*(1-x)^{c-1})## (incomplete beta function)
    and ## n=1##

    ##\frac{l_{cap}}{l_{sphere}}=\int_0^{sin^2(θ)} (\frac{dx}{\sqrt{x-x^2}})/2##

    But
    ##\int_0^{sin^2(θ)} (\frac{dx}{\sqrt{x-x^2}})≠\frac{θ}{2π}##
    Where is mistake?
     
  2. jcsd
  3. Jan 9, 2016 #2

    Svein

    User Avatar
    Science Advisor

    Look at your figure. You have an angle θ on both sides of h. Therefore, the ratio is θ/π. Sanity check: If θ=π/2, the blue part is half the circle - which agrees with my revised expression.
     
  4. Jan 9, 2016 #3
    You are right ,but still
    ##\int_0^{sin^2(θ)} (\frac{dx}{\sqrt{x-x^2}})/2≠\frac{θ}{π}##
     
  5. Jan 10, 2016 #4
    So can anybody derive 1- or 2-dimensional spherical cap formula from N-spherical cap formula?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Spherical cap
  1. Spherical Trigonometry (Replies: 2)

  2. Spherical Caps (Replies: 1)

  3. Spherical coordinates (Replies: 26)

Loading...