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## Re: [ft-devel] GSoC 2020 - Integrating Distance Field Generation in Free

 From: Behdad Esfahbod Subject: Re: [ft-devel] GSoC 2020 - Integrating Distance Field Generation in FreeType Date: Mon, 7 Sep 2020 21:00:00 -0600

On Sat, Aug 29, 2020 at 9:12 PM Anuj Verma <anujv@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