Graphing implicit equations

[reef briar]

I'm making a graphing calculator with JavaScript and I'm able to graph simple functions in the form y=f(x). However, I can't figure out how I'd graph implicit equations(eg. y^2+x^3=3x+3x^2-2). I figure you can get a bunch of points belonging to the function by iterating over values of y and solving for x using root estimation methods, but I have no good way of figuring out how to connect the points. Are there any good ways to do this?

Graphing implicit equations

[remote monolith]

Just change your code instead of going over every x value and plotting a point based on the value of the function there,
have it go over every x and y, and plot a point if the equation is true

[reef briar]

Unfortunately doesn't work

The thing is I'm dealing with pixels

And that means you can't have infinite precision

You have to include pixels that technically aren't part of the graph to make the whole thing continuous

Consider a•sin(x), where a is some large number

[remote monolith]

Yes no graphing calculator has that

[reef briar]

Yea

So I have to connect the points together

If not I could just get a bunch of disconnected points

[reef briar]

Also there's a possibility that it misses some points, for the range of x given

[remote monolith]

Yes

plotting a function also misses points on the x axis

[reef briar]

Yea so for function plotting in the form y=f(x),I just iterate over the pixel x coordinates, translate them to graph x coordinates and find the corresponding y

Which does the job quite well

Added bonus of no nested for loops

For implicits I'm a bit stumped though

An example of what would happen in some cases if we're iterated over every pixel, but didn't connect the points, with some vertical line

[reef briar]

@weary rover very goated conversation

What do you think I could do

[weary rover]

differentials

[reef briar]

I use the derivative to find out which points to connect?

[reef briar]

@pulsar flame would I be able to graph a line segment of length √2 on a screen

[pulsar flame]

No. The pixels deceive you.

[reef briar]

Alright mb

[reef briar]

How would I graph implicit equations on a screen

[reef briar]

Go on

[pulsar flame]

The Constructivist Perspective: Pixels

A constructivist wants every step in the graphing process to be given by a finite, explicit procedure. This means you need:

• An algorithmm that takes as input an implicit equation (for example, F(x, y) = 0) and a finite region of the plane to examine.

• Finite resolution “pixels” or intervals, aannd

• A method to decide - using an algorithm - that a piece of the graph exists in each pixel with controlled error.

In other words, you want to “construct” the graph by showing, in each finite subinterval, how the function behaves.

[reef briar]

Yea fair

So yea im using interval bisection

Then iterating over y for the range of values of y to solve for x such that F(x,y)=0

So that gives me a bunch of coordinates

[reef briar]

But they won't be necessarily be continuous

For functions I just estimated by drawing vertical lines to connect the pixels to those beside it

But with implicit equations I don't know how I'd link them

[pulsar flame]

Exactly. It’s like every pixel is a tiny, finite universe where a bit of the graph appears - except the cosmos refuses to hand out irrationals and real numbers on a silver platter. As well as continuous curves. We’re stuck with rationals, pixels, and a perpetual “this is as close as you get” vibe. https://media.tenor.com/FNvvUTbZoU8AAAAC/yes-thanos.gif

[reef briar]

It's not that deep 😭

I just want to know how to connect these points

[reef briar]

So how do we at least estimate the shape

[pulsar flame]

https://media.tenor.com/UDSW1iPhpUgAAAAC/goku-bye.gif