# Thread: patch of a torus

1. ## patch of a torus

Let $\displaystyle C$ be the circle in the x-z plane with radius $\displaystyle r>0$ and center (R,0,0). A torus of revolution is obtained by revolving $\displaystyle C$ about the z-axis. Show that the patch is given by $\displaystyle x(u,v)=((R+r cos u)cos v, (R+r cos u)sin v, r sin u)$.

Ahmmm, a patch is just a mapping x:$\displaystyle R^{2} to R^{3}$ that defines the whole curve.

2. Originally Posted by wiz_girl
Let $\displaystyle C$ be the circle in the x-z plane with radius $\displaystyle r>0$ and center (R,0,0). A torus of revolution is obtained by revolving $\displaystyle C$ about the z-axis. Show that the patch is given by $\displaystyle x(u,v)=((R+r cos u)cos v, (R+r cos u)sin v, r sin u)$.

Ahmmm, a patch is just a mapping x:$\displaystyle R^{2} to R^{3}$ that defines the whole curve.
I've attached a drawing how to calculate the coordinates of the points of the surface of a torus.

1. The black circle in the x-y-plane is the path of point R. In the picture of the torus I've choosen R=5

2. The red circle has the radius $\displaystyle \rho = R+r \cdot \cos(u)$. I've taken r = 3.

With $\displaystyle \rho$ and the angle v you can calculate the x and y-coordinate of the points.

3. The z-coordinate of all points only depends on r and the angle u.