Q: Extract coords from distance-defined parabola?

Questions
1

Dear Desmos folks,

I recently stumbled across the cool Desmos feature that it can draw curves implicitly defined by distance relationships, ancient-Greek-style, e.g. this for a parabola:
distance( (x,y), (0,1) ) = distance( (x,y), (x,0) )
I learned about it here, by the way:

My question is: can I extract the coordinates of specific points from such a curve? For example, here:

The green line successfully uses a distance formula to draw a line at the y-value of the point equidistant between the focus and the draggable point on the directrix, but I can’t figure out how to actually extract that y-value from Desmos.

It would of course be possible to calculate that y-value analytically, but that would be ugly and would go against the whole spirit of the distance-defined curve. Also, it seems pointless, because Desmos clearly has the value in it somewhere, as it uses it to draw the green line at the correct height!

Any help greatly appreciated,

Raj

2

Unfortunately, there are quite a few occasions where Desmos “knows” certain values - and is happy to display them on a trace - but it is difficult to surface them to use elsewhere. Particularly roots of a polynomial or points of intersection. It would be wonderful to have, even in beta form, access to whatever is clearly known…

In this specific case, you can get what you want by changing y to any unused variable (other than x) and changing the = to a ~. Essentially it’s a sort-of hack of the regression calculator, but in this case it surfaces an estimation of the value of “y” (whatever you’ve changed it to):
image

Be aware that this requires a lot more computation - it slows interactions with the graph down quite noticeably on this basic Chromebook I’m using presently - but it at least surfaces the “y” for you to use elsewhere.

If you’ve not played with this before, you can ‘hack’ certain single solutions out of the regression calculator using things like f(b)~0 or whatever - but it’s not an exact science, and if you’re looking for multiple solutions it is not going to give those.

3

Thanks, that regression trick is a good one, and I didn’t know about it. Thanks! I fully agree with you that it’s not a very resource-efficient way of extracting info that Desmos has already internally calculated!

Raj