Solving Parametric Equations for a Torus: Normal & Surface Areas

[fabsuk]

Could someone please give me a clue how to solve these parametric equations or a starting position.

torus specified by these equations

x=(R+rcosΦ)cosθ
y=(R+rcosΦ)sinθ
z=rsinΦ

calculate the normal to the torus N(θ,Φ) and entire surface area

p.s anyone recommend a book or a webresource that discusses this in detail as I am very confused.

 

[CarlB]

The formula for surface area in your book is:

$$\int dx\; dy$$

where the integral is over the whole surface.

You need to convert this into an integral over theta and phi. This might require some work.

To get the surface normal, you might take the cross product between two vectors that are tangent to the surface. To get vectors tangent to the surface, take a look at the deriviative of x(theta,phi) where "x" stands for the three vector (x,y,z), with respect to theta and phi. Make a drawing and you may likely see what is going on.

Carl