I have learned the paper,"Shape similarity retrieval under affine transforms". Paper is here
I tried to use the formula (4) in the paper to calculate the curvature of a straight line.But i get the result that the curvature of every point is not zero.It shows that the curvature of the first point and the last point are 5.7 and 7.6.The curvature of others points are the same,1.6. I don't konw why i get this result.Can somebody help me?Thanks very much!
Depends on your defining metric. All certain circles in hyperbolic geometry are deemed to be straight due to a certain defining or modeling metric function.
In a simpler example if you define
$$ \int \dfrac{\sqrt{ r^2 + {(r d \theta)}^2}}{r^2} $$
as your arc length , then any circle through the origin is quite straight.