In the usual parametrization (you need to choose coordinates) of the torus, the sign of the curvature is determined by the angle of the meridian. To see that is a nontrivial but also not hard calculation. Instead of trying to do it here in the comments, which will be a mess, let me rather link a nice explanation: www.rdrop.com/~half/math/torus/curvature.xhtml I hope that helps!
This was really helpful, thank you so much😌
Welcome, I am glad that you found the video helpful. That is why I do it - and also because its fun (very selfish, shameful....;-) )
In short gauss bonnet theroem = 2pi * ( V - E + F)
Hah, I like that! Thank you for sharing.
Maybe even shorter: Gauss-Bonnet = 2pi * Euler ;-)
i dont understand why the inside of the torus has negative curvature
In the usual parametrization (you need to choose coordinates) of the torus, the sign of the curvature is determined by the angle of the meridian.
To see that is a nontrivial but also not hard calculation. Instead of trying to do it here in the comments, which will be a mess, let me rather link a nice explanation:
www.rdrop.com/~half/math/torus/curvature.xhtml
I hope that helps!
The curvature is negative if there are to cross-sections that curve in opposite ways.