Torus parametrization and inverse

In summary, a torus is a 3-dimensional geometric shape with a curved surface and a hole in the middle, often described as a donut or inner tube. Parametrization is the process of representing a geometric object using variables, and a torus can be parametrized using two parameters, u and v. Inverse parametrization involves finding the parameters corresponding to a given point on the torus surface, which is useful in torus modeling and computer graphics.
  • #1
Jess_l
1
0
I've been looking at the torus parametrization
\begin{equation}
\phi(u,v)=((r\cos u+a)\cos v, (r\cos u +a)\sin v, r\sin u)
\end{equation}
with \begin{equation}a>0, r\in(0,a)\end{equation}. I want to invert this map to get a chart map for the torus.
Can anyone give me a hand with this?
Thanks!
 
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  • #2
well, u seems to equal arcsin(z/r). how's that for a start?
 

1. What is a torus?

A torus is a geometric shape that resembles a donut or inner tube. It is a 3-dimensional object with a curved surface and a hole in the middle. It can also be described as a surface of revolution generated by revolving a circle in 3-dimensional space.

2. What is parametrization?

Parametrization is a process of describing a geometric object or shape in terms of parameters or variables. These parameters can be used to represent the location, size, and orientation of the object in mathematical equations. In the case of a torus, parametrization involves using equations to represent its curved surface and inner hole.

3. How is a torus parametrized?

A torus can be parametrized using two parameters, typically denoted as u and v. These parameters correspond to the angles used to generate the torus by revolving a circle. The parametric equations for a torus can vary, but they often involve trigonometric functions such as sine and cosine.

4. What is inverse parametrization?

Inverse parametrization is the process of finding the parameters that correspond to a specific point on a geometric object. In the context of a torus, inverse parametrization involves determining the values of u and v that correspond to a given point on its surface. This can be useful for calculations and visualizing the torus.

5. How is inverse parametrization used in torus modeling?

Inverse parametrization is an essential tool in torus modeling. It allows for precise placement of points on the torus surface, which is crucial for creating accurate models. Inverse parametrization is also used in computer graphics and animation to generate 3-dimensional torus objects with smooth surfaces.

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