Question 1: Recalling that , explain why the value of A is smaller for flatter curves.
Question 2: Suppose that k' is continuous on [a,b] and the vertices of the curve have parameter values s1 < s2 < ... < sn. Prove that
This is the sum of the changes of curvatures at the vertices. Using this, explain why this means the value of A will be larger for a curve with lots of sharp corners than a curve with fewer, less-sharp corners.
Question 3: What's wrong with using arc length in practice?
Question 4: What changes if we integrate with respect to a parameter other than arc length?
Head back to the concepts page to learn how computers choose curves.