Since it's reasonably straight, I'd say just running along the curve calculating the distance to each endpoint, and then picking the one where the difference between the two distances is minimum, should be pretty close or even exact.
However if the curve is all twisty and distorted, then you'll have to use John D'Errico's interparc http://www.mathworks.com/matlabcentral/fileexchange/?term=authorid%3A679
