Projection stereographic and second fundamental form

In summary, the conversation is about finding the second fundamental form of r:R2 →R3 using a shortcut involving the unit normal n. The shortcut involves thinking geometrically about what vectors are orthogonal to the surface of a sphere. It is also mentioned that this shortcut does not apply to an ellipsoid with a normal vector of (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)).
  • #1
Simone Furcas
10
0
Let r:R2 →R3 be given by the formula
rac{2u}{u^2%20+v^2%20+1},%20\frac{2v}{u^2%20+v^2%20+1},%20\frac{-1+u^2%20+v^2}{u^2%20+v^2%20+1}).gif
Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n).

I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained impossibile result... The solve is too long to write down here... I use I used
gif.latex?N=\frac{ruXrv}{||ruXrv||}.gif
 
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  • #2
Think geometrically: what vectors are orthogonal to the surface of the sphere?
 
  • #3
I was thinking it was very difficult... Now I think to be a bit silly! :) thx
 
  • #4
Is it the same with an ellipsoid (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)) ? N is (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)) ?
 
  • #5
No.
 

Related to Projection stereographic and second fundamental form

1. What is projection stereographic?

Projection stereographic is a method of mapping points on a curved surface, such as a sphere, onto a flat plane. It involves projecting points from the surface onto the plane using a specific mathematical formula.

2. How is projection stereographic used in mathematics?

In mathematics, projection stereographic is commonly used in the study of geometry and topology. It allows for easier visualization and calculation of geometric properties on curved surfaces, such as the curvature and area.

3. What is the second fundamental form?

The second fundamental form is a mathematical concept used in differential geometry to describe the curvature of a surface. It contains information about the change in the surface's normal vector in different directions.

4. How is the second fundamental form related to projection stereographic?

The second fundamental form is related to projection stereographic in that it is used to calculate the curvature of a surface in the projection. By analyzing the change in the normal vector, the second fundamental form can help determine the shape and properties of the projected surface.

5. What are some real-world applications of projection stereographic and the second fundamental form?

Projection stereographic and the second fundamental form have many applications in various fields, including cartography, computer graphics, and the study of curved objects in physics and engineering. They are also used in the development of maps and projections, as well as in the analysis of surfaces in medical imaging and geology.

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