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

 From: Anuj Verma Subject: Re: [ft-devel] GSoC 2020 - Integrating Distance Field Generation in FreeType Date: Tue, 8 Sep 2020 09:30:13 +0530

> Thank you Anuj. I really enjoyed observing your work this summer, even though I didn't get to contribute more review.

Thank you for checking it out and for your suggestions.

>> 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.

Thanks for proof. I finally understand what is happening in the code that I have written and can update the comments.

Best,
Anuj