On Sat, Aug 29, 2020 at 9:12 PM Anuj Verma <an...@iitbhilai.ac.in> wrote:
> Hello All, > > I have written the final report for my project. > > You can check it out here: > https://gitlab.com/-/snippets/2007070 > Thank you Anuj. I really enjoyed observing your work this summer, even though I didn't get to contribute more review. I promised you a proof and here it is: > Lemma: if the closest point on curve [0,1] is to the endpoint at t=1 and the cubic equation has no real root at t=1, the cubic equation must have at least one real root at some t > 1. > Similarly, if the closest point on curve [0,1] is to the endpoint at t=0 and the cubic equation has no real root at t=0, the cubic equation must have at least one real root at some t < 0. > > As such, you just need to compute all real roots, clamp them to [0,1] and remove duplicates. Here's the proof for the first case: Consider the derivative of the distance, called "the function" from here on. It's a continuous function. At t=1 the function is a negative number because of the assumptions. When t tends towards +infinity, the function approaches +infinity. As such, there exist t > 1 where the function is zero. Cheers, b
