Computing tangent spaces of implicitly defined manifolds

In summary, the conversation discusses finding the tangent space at a specific point to an implicitly defined manifold. The speaker mentions using an explicit parametrization and computing the Jacobian matrix or the normal vector to the manifold, but is unsure of any other methods. The other person mentions that the gradient of f is perpendicular to the tangent space and provides the equation gradf(x).v = 0 as the normal vector.
  • #1
sin123
14
0
Hi there,

Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix, sometimes I can compute the normal vector to the manifold (when c is just a real number), but that's where I am running out of ideas. I am hoping that there might be some sort of implicit differentiation trick that I have not figured out yet.
 
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  • #2
sin123 said:
Hi there,

Is there an "easy" way to find a tangent space at a specific point to an implicitly defined manifold? I am thinking of a manifold defined by all points x in R^k satisfying f(x) = c for some c in R^m. Sometimes I can find an explicit parametrization and compute the Jacobian matrix, sometimes I can compute the normal vector to the manifold (when c is just a real number), but that's where I am running out of ideas. I am hoping that there might be some sort of implicit differentiation trick that I have not figured out yet.

i am not sure if I am telling you something that you already know but the gradient of f is perpendicular to the tangent space of f(x) = c. So the equation for it is gradf(x).v = 0
 

1. What is the purpose of computing tangent spaces of implicitly defined manifolds?

The purpose of computing tangent spaces of implicitly defined manifolds is to understand the local geometry of the manifold. Tangent spaces provide information about the direction and rate of change of the manifold at a specific point, which is useful in applications such as optimization and simulation.

2. How are tangent spaces of implicitly defined manifolds calculated?

Tangent spaces of implicitly defined manifolds are typically calculated using the implicit function theorem, which states that the tangent space at a point on the manifold is spanned by the gradients of the defining equations at that point.

3. What are the challenges in computing tangent spaces of implicitly defined manifolds?

One of the main challenges in computing tangent spaces of implicitly defined manifolds is dealing with high-dimensional and complex manifolds. This can lead to computational difficulties and require efficient and robust algorithms for accurate calculations.

4. Can tangent spaces of implicitly defined manifolds be approximated?

Yes, tangent spaces of implicitly defined manifolds can be approximated using numerical methods such as finite differences or tangent space estimation algorithms. These approximations can be useful in cases where the exact tangent space cannot be calculated analytically.

5. What are the applications of computing tangent spaces of implicitly defined manifolds?

There are many applications of computing tangent spaces of implicitly defined manifolds, such as in computer graphics, robotics, and computer-aided design. They are also used in scientific and engineering fields, such as in physics and biomechanics, to model and analyze complex systems.

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