How to find function for a parabola that goes though 3 points and is rotated 90° clockwise?

Question

Short background: I am building a laser engraving system that can engrave on wood. The power Is always at its maximum, what changes is the speed of the laser on the surface. slow movement produces dark burns, fast movements light burns. Currently the speed is proportional and linear to the brightness. But once a spot gets lightly burned, it absorbs the light faster, and burns very quickly. Therfore I need my speed to vary according to the function shown below:

i know the values of 2 points and the vertex.

P1 = (0,300)

P2 = (255,1000)

Pv = P1 - 1 = (-1,299)

The values 300 and 1000 can change based on the material, Pv is alway P1 - 1. The other values stay the same. How can I calculate the speed for any given brightness?

Answer 1

The general equation for such a parabola is $$(y-a)^2 = bx+c $$

Plug in the three points in this equation to get three equations in three variables, which you can solve.

For example in your specific case, the three equations are $$(300-a)^2=c \\ (1000-a)^2 =255b+c \\ (299-a)^2 =-b+c$$ which give $$a= \frac{151451}{78}, b =-\frac{98140}{89}, c=\frac{9613998601}{31684} $$

Answer 2

I would suggest the following:

1. Flip the diagram along the diagonal. In other words declare the vertical axis to be $x$ and the horizontal axis to be $y$.
2. Given any three points without common $x$-value you can find the quadratic polynomial through these points by solving $y=ax^2+bx+c$ for $a,b,c$
3. Solve for $x$. Note that as $x$ appears as a square this will include solving for a root, hence making a decision of whether taking $+\sqrt{...}$ or $-\sqrt{...}$. Computing both solutions $x_0$ and $x_1$ for $x$, take their maximum ie. $x=\max (x_0,x_1)$