1. Not finding help here? Sign up for a free 30min 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!

Euclidean Geometry Question

  1. Oct 29, 2005 #1
    Suppose you have a smooth parametrically defined volume V givin by the following equation.

    f(x,y,z,w)= r(u,s,v) = x(u,v,s)i + y(u,v,s)j +z(u,v,s)k + w(u,v,s)l

    Consider the vectors ru=dr/du, where dr/du is the partial derivitive of r with respect to the parameter u. Similarly, rv = dr/dv, rs=dr/ds are the partial derivitives of r with respect to the parameters v and s, respectively.

    I presume that ru(u0,v0,s0), rv(u0, v0, s0), and rs(u0,v0,s0) form a three dimensional parallelpiped that represents the rate of change of the three dimensional surface volume in four dimensional Euclidean space.

    The angle theta between ru and rv is arccosine((ru*rv)/(|ru||rv|))

    Similarly, the angle phi between ru and rs is arccosine((ru*rs)/(|ru||rs|).

    The height h1 of the parallelogram p1 formed by ru and rv is the magnitude of the projection of rv onto the perpindicular of p1 and is equal to
    |rv|sin(theta). The area A1 of this parallelogram is
    |ru|*h1 = |ru||rv|sin(theta).

    The height h2 of the parallelogram p2 formed by ru and rs is the magnitude of the projection of rs onto the perpindicular of p2, and is equal to |rs|sin(phi)
    The area of A2 of this parallelogram is |ru|*h2 = |ru||rs|sin(phi).

    The volume of the parallelpiped V = the area of either of the parallelograms times the height of the other parallelgram.

    A2*(h1) = A1*(h2) =|ru||rv||rs||sin(phi)*sin(theta)|

    Based on this, I conjecture the following:

    If a smooth parametrically defined volume V is givin by the following equation:

    r(u,s,v) = x(u,v,s)i + y(u,v,s)j +z(u,v,s)k + w(u,v,s)l

    Where (u,s,v) are elements of E, and

    V is covered just once as (u,v,s) varies throughout the parameter domain E, then the "Surface Volume" is

    the tripple integral over E of =|ru||rv||rs||sin(phi)*sin(theta)|dV

    Where ||sin(phi)*sin(theta)|is a constant.

    Does this sound accurate?


  2. jcsd
  3. Oct 30, 2005 #2
    Where ||sin(phi)*sin(theta)|is a constant.

    Oops! ||sin(phi)*sin(theta)|is not a constant!

    it's |arccosine((ru*rv)/(|ru||rv|))*arccosine((ru*rs)/(|ru||rs|)|.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Euclidean Geometry Question