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docnet

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- Homework Statement
- this map multiplies all lengths of tangent vectors by the same factor. what is the factor?

- Relevant Equations
- p^2+q^2=1

r^2+s^2=1

w^2+x^2=1

y^2+z^2=1

w=pr+qs

x=qr-ps

y=pr-qs

z=qr+ps

Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c.

I think this map multiplies tangent vectors by a factor of sqrt(2). We can think of tangent vectors as velocity while traveling along a curve on the surface. From simply plotting points, we know if travel around the pq plane or the rs plane once, we travel around both the wx and yz planes. We also know the two planes are orthogonal to each other because it's the definition of the torus under this equation. Under this map, the velocity gains an additional orthogonal component of equal magnitude, and from Pythagorean theorem the net magnitude is sqrt(1+1) = sqrt(2).

Is my intuition correct?

I've calculated the metric in part a and solved part b. but, I don't think these parts will help with c. thanks

I think this map multiplies tangent vectors by a factor of sqrt(2). We can think of tangent vectors as velocity while traveling along a curve on the surface. From simply plotting points, we know if travel around the pq plane or the rs plane once, we travel around both the wx and yz planes. We also know the two planes are orthogonal to each other because it's the definition of the torus under this equation. Under this map, the velocity gains an additional orthogonal component of equal magnitude, and from Pythagorean theorem the net magnitude is sqrt(1+1) = sqrt(2).

Is my intuition correct?

I've calculated the metric in part a and solved part b. but, I don't think these parts will help with c. thanks