Home #1 Mathematics What Revolution Surface is the following ? ...
What Revolution Surface is the following ? ... PDF Print E-mail
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Thursday, 04 November 2010 15:56

Greetings,

    The study of Differential Geometry was introduced like "standard subjects" after Professor Einstein realize his "Relativity Theory".

    Permuting to comment that I consider several assertions made by Einstein Theory complete wrong like the speed of Light constant, and I consider variable. As well the Lorentz equations wrong or valid not for the cases explained by Professor Einstein ... I still consider the study of Differential Geometry fundamental.

    What Revolution surface is:

     This surface is the Catenoid like the surface of revolution the catenary, which we find in any place like in a parking ... place.

    Mathematically speaking it is graphically the following curve,

     which equation is:

     Now, that we now what is the curve, I want to explain some notions about Differential Geometry ... in quick, easy and speedy mode.

     All these notions are available at: Manfredo Carmo - Differential Geometry of Curves and Surfaces.

(Click to download and read)

     Probably one of the most important and fundamental definitions of Differential Geometry is the First fundamental form.

     It is nothing but the expression of how the surface S inherits the natural inner product in R3.

     You can read Section 2-5 in the book. (page 92 in the printed edition and page 101 in the PDF file)

     Then, you will get, three coefficients, E, F and G that are the expression of that inner product.

     Therefore, if

                           Ip(α'(0)) = E (u')2+2Fu'v'+G(v')2

     we will have,

            E (u0,v0)=<xu,xu>p

            F(u0,v0) = <xu,xv>p

            G(u0,v0) =<xv,xv>p

      This will lead E, F and G for any Surface, as well a formula for the area of that surface that will be, Ip=|w|2.

      I, Giovanni will give three examples about these numbers and the last will be the catenoid.

      Example 1. Please consider, the Right cylinder x2+y2=1, which parametrization is,

                               x(u,v)=(cos u, sin u, v), U = {(u,v) ε R2; 0<u<2π, -∞<v<∞}

Cylinder

     Please note that "any" surface of revolution includes a parametrization like (cos u, sin u, v).

     Now, to calculate the "First fundamental Form", or the functions, E, F and G we will note that,

               xu= (-sin x, cos u, 0) and xv = (0,0,1)

     Therefore,

                     E = sin2 u + cos2 u = 1, F =0, G=1.

     Please remember that Linear Algebra lead us to adopt also a Matrix edition of the First Fundamental Form,

     Example 2. If the Helix ... has the following figure,

     What will be the Surface of Revolution, it equation and the value of E, F and G?

     The surface of Revolution will be the "Helicoid". The equation is a little bit different ...

  x(u,v)=(v cos u, v sin u, av), U = {(u,v) ε R2; 0<u<2π, -∞<v<∞}

      The Surface of Revolution will be like,

     It is almost immediate that,

        E(u,v) = v2+a2 ... F(u,v)=0, G(u,v)=1

     Example 3. What are the value of E, F and G for a Surface of Revolution? ... In particular for the catenary?

     This example is available at page 221 of the printed book, previously displayed.

     Like commented previously (without proof) a surface of Revolution has the following equation,

x(u,v)=(f(v) cos u, f(v) sin u, g(v)), U = {(u,v) ε R2; 0<u<2π, a≤v≤b, f(v)>0}

     Now, the coefficients of the First Fundamental Form of S, are:

E(u,v) =(f(v))2 ... F(u,v)=0, G(u,v)=(f'(v)2+g'(v))2

     In the case of the catenary we have,

x = a cosh v, z = a v       -∞<v<∞

     and therefore,

    x(u,v)=(a cosh v . cos u, a cosh v . sin u, av), U = {(u,v) ε R2; 0<u<2π, a≤v≤b}

     and therefore,

E = a2 cosh2 v, F = 0, G= a2(1+sinh2 v) = a2 cosh2 v.

Thanks,

Giovanni A. Orlando.

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Last Updated on Thursday, 04 November 2010 16:00