Torus Parametrization Hint (Problem 5 from the
Lesson 6 HW)
Here is a hint to Problem 5 in the 6.th lesson Problem set:
We keep the angle theta as one of the parameters and let r the distance of a
point on the torus to the z-axis. This distance is r=2+cos(phi) if phi
is the angle you see on the animated figure below to the left. Note that phi
has no relations with the angle phi in spherical coordinates. The blue
segment you see has the length r.
You can read off from the same (left) picture
also that z=sin(phi).
To finish the parametrization problem, you
have to translate back from cylindrical coordinates
(r,theta,z)=(2+cos(phi),theta,sin(phi)) to Cartesian coordinates (x,y,z).
Write down your result in the form r(theta,phi)= (x(theta,phi),y(theta,phi),
z(theta,phi)).
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| Changing the angle phi. In this picture the vertical axes is
the z-axes. This picture obtained by cutting through the doughnut. For
example along the xz-plane. |
Changing the angle theta. In this picture the axes are
the x and y axes. You look onto the doughnut from above. |
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