Question to 6.2

Question to 6.2

av Sonja Hatzenbühler -
Antal svar: 2

Hello everyone,

we have two question concerning exercise 6.2, or more generally the calculation of the sum over C(r).

In the picture below, the left shows an example how we set the angles alpha and hence the values for the sums when considering the outer face f0 or the inner face f1. The values do not match the values (-4, +4) we would expect from the lecture. Maybe someone can help us and tell us what we did wrong.

On the right, we have a second question: In the script it is said that alpha can be 2pi. The only case we found, where that would be the case, is when we have a vertex with degree one. The question is if we imagined the waking through the edges correctly, or if this would oppose the calculations/algorithms in any way.

Som svar till Sonja Hatzenbühler

Re: Question to 6.2

av Johannes Zink -
Hello Sonja,

for computing the C(r) values, note that if we do not have any bends, the formula boils down to C(r) = alpha / (pi / 2) + 2.

In your example, the C(r) values for f_0, which are all equal, are C(r) = - (3 pi / 2) / (pi /2) + 2 = - (3 pi * 2) / (pi * 2) + 2 = - 3 + 2 = -1.
Since we have four edges, the sum of the C(r) values is - 4.

Similarly, the C(r) values for f_1, which are again all equal, are C(r) = - (pi / 2) / (pi /2) + 2 = - 1 + 2 = 1.
Since we have four edges, the sum of the C(r) values is + 4.


For your second question:
Yes, the only case to have an angle of 2 pi is if we have a vertex of degree 1. The direction for traversing the outer face (and only the outer face) is counterclockwise, hence, this is correct in your picture.

Does this answer your questions?

Best,
Johannes