r/desmos

yes very awesome and completely non-self-contradictory math i know

I was thinking about ways to get the length of a Bezier curve segment.

There's the rigorous way, of breaking it up into smaller "linear" segments and then adding them up. It's tedious and not computationally cheap.

Then there's the wing it approach. I read somewhere that you could just measure the length of the net (line segments connecting the control points) and then just 1/3rd it to get a good estimate of the actual curve length.

This seemed like a plausibly valid approach, but I decided to test it out.

I built this

If you start the ticket, it randomly generates Bezier curves within the red bounding box, measures the Net Length, and the actual Curve Length and then plots them as (x, y) data points.

With enough data taken from bounding boxes of different sizes, if you perform a linear regression on it, the slope of the graph should tell you what a good "wing it ratio" is. I wanted to see if the 1/3rd estimate holds up.

It does not.

For starters, this scatter plot spreads like a cone with an upper bound slope of 1, and a lower bound slope of 0.27 (I don't yet fully understand why that is the smallest a Bezier curve can be in comparison to its net)

This is just not a good data set for a linear regression model. I was hoping for a scatter plot more or less consolidated around a line.

A cone means that there is equally valid data in a range of slope values around your "wing it ratio" .

And even if you power through with the assertion that some estimate is better than no estimate, and perform the regression, the best slopes I'm getting that carve through and bisect the data have a slope of around 1/2 and not 1/3.

You could argue that most Bezier curves people draw look like C shapes and this randomized curve generation process is not representative of curves you'd want to estimate in practice.

What do you think? And do you have any insight about why cubic Bezier curves can't be shorter than 27% of their net length?

Don't judge me for missing out on the entirety of Indonesia (I have beef with Indonesia because of geoguessr, it's my #1 opp).

1,896 points, 7 hours to make this

Graph link: https://www.desmos.com/calculator/o8ix1zxrdw

https://www.desmos.com/calculator/dsnicu9eme

Here's a song about imaginary numbers, set to the tune of "Imagine" by John Lennon, with accompanying animations made in Desmos:

https://www.youtube.com/watch?v=pOagqB8CuLE

Made for MoMath's Open Set math songs event: https://momath.org/openset/

The main Desmos graphs used:
Complex grid inversion: https://www.desmos.com/calculator/vfr0kwikp3

Mandelbrot with moving trajectory (will take a few seconds to render)
https://www.desmos.com/calculator/haablhcnxk

Domain-colouring complex roots of unity: https://www.desmos.com/3d/qckybwlbya

I hope some day you'll join us: https://www.desmos.com/calculator/7gsnqxr4up

With Fourier to guide us: https://www.desmos.com/calculator/bdnl8vzgz2

I hope you enjoy it!

Raj

official rules: https://youtu.be/hMqYC4QUvbU

Approximation of 1 using an approximation of the golden ratio, but each element in the golden ratio is an approximation, in which all of the element's in those approximations are approximations, in which all of those elements in the approximations are approximations, in which all of them are written out using 1's only after being converted from primes to 2's only, and then from 2's only to no numbers only, and then from no numbers only to 1's only.

https://www.desmos.com/calculator/v2kmitxr6f