Distances in analytical geometry

[silent gyro]

As we already know, there exists a formula to find the distance of a point of a line, namely $\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2+B^2}}$. Is it possible to find a formula for the distance of a point from a polynomial function, or even better, the distance of a point from any given f(x) where f(x) can be polynomial, exponential, trigonometrical, piecewise etc.? The formula doesn't have to be generalized for all f(x), there can be different formulas for sinusoidal and polynomial f(x) here. I've been stuck on this for a month now...
Also, this is just a personal doubt, not something I was asked in school or anything.

[plush cobalt]

No Lifer?

[viral agate]

This is crazy hard problem to solve in general

[silent gyro]

well

[viral agate]

But one can approach it by considering it as an optimization problem.

[silent gyro]

I was stuck at it for a month

[silent gyro]

?

[silent gyro]

and still am nowhere

I have a vague idea for quadratics, but that's basically it

[silent gyro]

It's why i put it in discussion, not like help uni or someth
I don't think an answer to this actually exists yet

[viral agate]

You know distance function to certain point right

This is basically about mininization problem of that.

[silent gyro]

like, distance between 2 points right?

I know that

and i know the line thing

[viral agate]

Line thing?

[silent gyro]

distance from point to line

I posted the formula above

[viral agate]

That is not so useful

[silent gyro]

yeah ik

It's just there

[viral agate]

Yea

[silent gyro]

it helps in creating parabolas, that's all

[viral agate]

So basically, parametrize the curve

[final sky]

This reminds me of like certain metrics in probability like the mahalobonis distance but WAYY more general

[viral agate]

And formulate distance to it as a function

Boom, Minimization

[silent gyro]

hmmm

there has to be an exact func though

sinusoidal seems the easiest

let's try that one first

[final sky]

Is this not involving alg geo??

[viral agate]

Thankfully I think algebraic geometry is not needed here

[silent gyro]

but seriously, if rizzly somehow solves this and convinces the riddle team to post this as a riddle

nah

[viral agate]

This is too computation-heavy to become a riddle, sadly

@static sonnet Can you solve this

[silent gyro]

hmmm

I have an idea for sinusoidal wave

we need an f(x) that locates a certain point

draw line

do intersection

more nonsense

tada

the period is key

alr

I'll focus on my half-yearly exams

then this

[final sky]

It seems super complication most likely unsolvable unless you have some constraints on the type of surfaces ig

[silent gyro]

oof

[static sonnet]

for sin in general i believe this produces a transcendental equation

[silent gyro]

transcendental?

[iron kelp]

I think you can do it, but using infinite polynomials

[final sky]

Transcedental equation are equations that are insanely insanely hard to solve in the best case scenario. I heard that they are usually impossible to solve with algebra and approximations using computers are the only way

Like for example: x*e^x = 1

Or sin(x+ sin(x)) = 0.3333

Or something crazy like that

Don't take what I say as factual, this is what I think it is. The first and only time I heard about these things was in a Numberphile video about the Goat problem or somrething like that

@silent gyro These kinds of very general questions are hard. I also thought of a similar question: Given any 2 2D shapes(convex or concave) with their respective areas given, how can you calculate the sum of their areas? (There are cases that they might overlap and how they overlqp change the area sum and it also depends on the exact shapes)

I think general problems like ours are just so general that only special cases are solvable in a humane sense

Just how @final sky said above

[final sky]

Oh, another example I've seen in a blackpenredpen video was: sin(x)^cos(x) = 1

[static sonnet]

well you just calculate the area of intersection

[final sky]

Well yeah but what about general shapes

Like if you were given an iregular pentagon intersecting at a weird angle with just a part of an ellipse

[static sonnet]

this can be calculated elementarily
there are formulas for area of an ellipse, area of a sector of an ellipse is just the scaled area of the corresponding sector of the circle, and the pentagon can be dealt with using triangles
intersection points can be found easily as well, just plug the line equation into the ellipse

[final sky]

Good luck

[static sonnet]

that is not a pentagon

[final sky]

Well the problem is about general shapes

[static sonnet]

and yeah my method will solve that it's just difficult

as opposed to something like sin(x)cos(2x) + cos(3x-sin(4x)) = 1.1 which probably has no way to solve

[final sky]

Actually, about this topic, how do we know that these kinds of equations are not solvable?

Like are there actual proofs showing that they are impossible?

[static sonnet]

i think so, yes

[viral agate]

Approximate the ellipse with a 100-gon, then you are golden

[static sonnet]

maybe you can do that if you want the perimeter of the shape

[viral agate]

Wdym "if you want the perimeter of the shape"? @static sonnet

[static sonnet]

no approximation needed for area i mean

[viral agate]

Oh you can compute that area directly?

Thinking about it, I guess it is just some integration

[static sonnet]

this is what i think

[viral agate]

Woah

[silent gyro]

wow

transcendental

yeah um

i give up

[iron kelp]

It can always be solved, but the calculation varies depending on the shape and the measurements must be given.

[final sky]

I see

[final sky]

I think I heard about Pick's theorem or something like that

Guess that only works when the vertices of points lie only on integer coordonates points

[static sonnet]

that's polygons

so yeah not really working here

[proven halo]

we can sort the functions out and work on it

me and @final sky will help

[final sky]

Well

[silent gyro]

hmmm

[final sky]

I say create parametric form of the function

And then optimize the distance using distance formula

[silent gyro]

I don't have any idea where to start with this though

I think derivatives will be required though

[final sky]

What i am saying is a rathwr generalized technique

[final sky]

Depends

[silent gyro]

hmmm

[final sky]

That would be our last resort if inequalities dont work

[silent gyro]

I think a sinusoidal wave would be easier to start with

[silent gyro]

oh

inequalities?

how would those help here?

[final sky]

am-gm

[silent gyro]

oh

[final sky]

Cauchy

[silent gyro]

the typical inequality

[final sky]

Help us to predict range directly

[silent gyro]

ah

[final sky]

Without having us to calculate at what point its minimum

Basically

1. paramatrize the function
2. write generalized distance formula
3. optimize it using derivatives/inequalities

Thats what you need to go

[silent gyro]

parametrize the function

how would we go ahead with that though...

sorry if I'm acting stupid here

[final sky]

Well even i dont really on that part to be honest

Parametric equations get you the locus of a point on the curve using a single variable

Thus they are very helpful here

[silent gyro]

hmmm

[static sonnet]

what are you guys talking about

[final sky]

But you would need to learn how to actually paramatrize equations

[silent gyro]

something I don't know

oof

[silent gyro]

the topic of discussion

Or

Shortest distance from a point to any f(x)

[final sky]

Shortest distance of a point to any curve

[silent gyro]

any f(x), but sure

[final sky]

Any fx

This would always work

Unless @static sonnet can tell something

[silent gyro]

sinusoidal wouldn't count as a curve right?

It could be called an infinite polynomial

but

idk

[static sonnet]

lets try finding the distance from (1,1) to cos(x)
we would like the line through 1,1 that hits cos(x) perpendicular to it
so a(x-1) + 1 = cos(x) where -sin(x) = -1/a
x = arccsc(a)
we want a(arccsc(a)-1) + 1 = cos(arccsc(a))
and now there is an issue
maybe we can replace cos(arccsc(a)) with sqrt(a^2-1)/a but still going to have a problem

[silent gyro]

well

derivatives

hmmm

-sin(x) at x should be perpendicular to the line with points cos(x) and 1,1

wow

[final sky]

A tangent whose normal passes through the point

[static sonnet]

cos(x) has a very cool set of normal lines iirc

[silent gyro]

$\frac{1}{\sin{x}} = \frac{|1-\cos{x}|}{|1-x|}$

[plush cobalt]

Martian Attack Helicopter

[silent gyro]

uh

that looks wrong

[final sky]

Fuck all that logic and stuff

((1 - cos(x))^2 + (1-x)^2)^(1/2)

Minimize the abivw

And its done

Prolly

[silent gyro]

$|1-x| = \sin{x}|1-\cos{x}|$

[plush cobalt]

Martian Attack Helicopter

[final sky]

@static sonnet am i right ?

[silent gyro]

$((1 - cos(x))^2 + (1-x)^2)^{1/2}$

[plush cobalt]

Martian Attack Helicopter

[static sonnet]

here they are

[silent gyro]

damn

that looks insane

[final sky]

,w minimize ((1 - cos(x))^2 + (1-x)^2)^(1/2)

@silent gyro done

[static sonnet]

no

[silent gyro]

hmmm

[static sonnet]

it didn't find an exact solution which is my point

[final sky]

why not

[final sky]

We have x now

[static sonnet]

you have x approximately

[plush cobalt]

Results provided by WolframAlpha

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Requested by Krypton#5577

[silent gyro]

$1 + x^2 - 2x = \sin^{2}{x}(1 + \cos^{2}{x} - 2\cos{x})$

[plush cobalt]

Martian Attack Helicopter

[final sky]

@static sonnet minimise it by hand then

Numbers wont be good

[silent gyro]

,w minimize 1+x^2-2x=sin^2(x)*(1+cos^2(x)-2cos(x))

[final sky]

I assure you that

[plush cobalt]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by nolifer3527

[silent gyro]

wow

[final sky]

,w calculate (0.789781,cos(0.789781)

[plush cobalt]

Results provided by WolframAlpha

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Requested by Krypton#5577

[static sonnet]

eyy look at that it can get exact forms oh wait no

[silent gyro]

wow

what did i create

[static sonnet]

what do you mean

[silent gyro]

what have I done

but

The perpendicular thing

I never thought about that

wait

the differential of an f(x) returns a func that returns the gradient of the original func at a certain point

[static sonnet]

the derivative, yes

[silent gyro]

yep

and we want this function to return someth perpendicular

ok

if a line has slope m

a line perpendicular to it has slope -1/m

for let's say, cos(x)

and point, 1,1

[final sky]

Its not always true

[silent gyro]

wait what

[final sky]

Look at that

You need to select shortest one now

Perpendicular isnt the only condition

[static sonnet]

the shortest one will always be a perpendicular one if f(x) satisfies some sort of smoothness condition, though

[silent gyro]

hmmm

yeah what rizzly said

I'm operating on that

[static sonnet]

(also, krypton, there is a unique one that goes through (1,1))

[final sky]

.

[final sky]

Only 1 ?

[silent gyro]

If there's multiple roots

I'll take the one closest to 1,1

[static sonnet]

there would be more if it was higher up in the region of double density

[final sky]

I thought so but you posted that graph bruh

[static sonnet]

look here's 1,1

only 1 line going through it

[final sky]

Like (0,190 ?

I think

[static sonnet]

yeah that probably has multiple

i don't remember if i was successful in determining the exact area in which there is only 1 solution

[final sky]

does every point have atleast 1 line

[final sky]

I will try it

Or we can try it

And verify later between us

@static sonnet deal ?

[static sonnet]

yes

visually

it looks like they do

[final sky]

yeah it does

[static sonnet]

well of course

every point has a closest one

[final sky]

I was trying to think of edge cases

Point on the curve ones

Would they have a normal to its own curve except their own ?

@static sonnet wanna fully calculate this ?

Not rn

But like whenever each of us have time

[static sonnet]

if you draw the normals as x moves from π to 3π/2 you get one segment of the hyperbola-looking curves

then minimizing the y value it becomes transcendental so probably no

might be a parametric but i don't think even that can happen

[final sky]

I will find this

Let you know if i get anything

Anywats

Good night @static sonnet

[final sky]

Hmm

This reminds of Voronoi diagrams