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I have a problem where I need to find the equation :

$y=ax^2+bx+c$

for the quadratic curve that pass at a certain distance from 4 points on a grid.

Visual description of problem

At each 4 point I know that the curve is at a certain distance but I don't know any points that are on the curve.

I though of maybe doing a system of 4 equations in the form $(x-Px)^2+(ax^2+bx+c-Py)^2=distance^2$

But I'm a bit stuck on how to solve for a,b,c

Am I going in the right direction ? And if not, what other way is there to solve that problem.

Thank you !

Benjamin_Trebuchet
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  • You can write that system, but 1) the $x$ is not the same in the four equations, and 2) that just gives the condition that point $(x,y)$ on the parabola is at distance $d$ from a fixed point $(P_x,P_y)$, but what you need is for $d$ to be the minimal distance. Once you add those conditions, you'll have $8$ equations with $7$ unknowns ($a$,$b$,$c$,$x_{1,2,3,4}$) which is overdetermined and won't have a solution in general. – dxiv Dec 07 '21 at 23:56
  • yeah I've been searching this afternoon and I've also come to the conclusion that i was mistaken to think x was the same. If I choose only 3 points instead of 4 can I have a proper solvable system ? Also how did you construct the proper 8 equations for the minimal distance ? Thank you ! – Benjamin_Trebuchet Dec 08 '21 at 00:12
  • Yes, with $3$ points you would have a solvable system in general, though exact calculations are heavy. Finding the distance between a point and a parabola involves solving a cubic, see for example Find a point on a parabola that's closest to another point. – dxiv Dec 08 '21 at 00:43

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