How do you parametrize paths with (1-λ) in a twin paradox problem?

[Kuma]

Homework Statement

Hi, I'm just seeking an explanation of this solution for question 4 on the following document.

https://docs.google.com/viewer?a=v&...Sg3wEk&sig=AHIEtbSrlOPM8RXfN8I7gBoTxzXwVzwddg


I'm just wondering how did they just pick out the vectors A, B and C arbitrarily? And I'm more confused about how they parametrized the paths c1, c2 and c3. Why choose (1-λ)?

 

[v2536]

What you have to calculate is the length of paths which are all geometric object. So no matter how to parametrize them (pick out vector in this case) the integral will give you the same result (the length).
And if you use the fact that given two points A and B you can find the parametrization of a line from A to B by c(l) = A + (B-A)*l (l=0 to 1 and B-A give you direction) then you can rewrite this as c(l) = (1-l)*A + l*B and this is where (1-l) came in.