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...