I wish to find the asymptote between two cubic Bezier curve segments as shown in the following diagram.



The end of the P segment will always coincide with the start of the Q segment, and the angle P0-Q0-Q3 will always be greater than the angle P0-Q0-Q1; and both of these angles will be between 90 and 180 degrees (I believe this ensures a unique solution).

I am specifically interested in the end-points where the asymptote intersects the two curves (the ends of the orange line in the diagram) and, if possible, I would prefer a simple formula solution, rather than an algorithmic one (such as walking along the curves).

I am in no particular rush as I will be traveling over the next few weeks, but would really appreciate any guidance provided.

Just noticed this post so doing some necrobumping here... aren't you actually looking for a common tangent?

The problem doesn't seem so trivial. There is a research paper on the subject:

http://www.cs.nyu.edu/parida/res/public/tan95.ps.gz

I don't have a ready answer only visualizing.

Maybe those that are expert in circle packing can figure this one out. As for figuring out coordinates from such arbitrary points escapes me now. My brain's now too old. lol

I think I need to know this too when I wanted to do text path to stretch out in a different direction. I didn't know how to draw a line/path that would make it like this.

I'm amazed that 8 years ago lylejk exclaimed "My brain's now too old. lol"
Lyle if you're reading this your doing OK. Your posts are still coherent and you still hold down 3 jobs. Good on you

Only 2 jobs, Tas, but, have to pull 2 14 hour shifts this weekend. lolololol

Glad G'MIC made up for my brain issues. lol

ChatGPT gave derivative answers to calculate slope of bezier curve point given t but when asked if it can come up with straight formulas, it said that you'd have to use math analytical something. So I gave up querying it.

This thread is 11 years old and even ChatGPT can't answer it?

Lyle I hope your 14 hour shifts have long breaks LoL.

I suspect there will be few calls so there will be a lot of dead time, Tas. lol

Hope you all have a great weekend; about to soon force myself to sleep (way too early for me, but need to try; lol).

It answered it best it can which is giving formulas to sort of walk the paths. but no directly formula as that might not be possible.

I think the problem is even a bit ill-defined. SaulGoode's question is about single Bezier splines, and we usually consider paths... But even then two splines can have 4 tangents:



And even with the condition that they have a common anchor, 1) there are obvious cases with no tangents, and 2) there are cases with two tangents.

Edit: there are even cases with three tangents:

wow, so many different cases, i don't even know how to solve the 1 case.