Calculating the inverse of a system of 4 simultaneous equations that deal with image conversion from image space to cortical space.

Question

I am completely stuck in calculating the inverse of these equations.

These equations show how to convert a image coordinate to a cortical coordinate (x,y being the image coordinate and alpha being a constant). However, I need to figure out how to calculate x,y given Xleft, Yleft, Xright, Yright. I am absolutely stuck at this because not even wolfram alpha seems to be able to do it.

Thank you!

The equations I need help with.

Answer 1

First we do

$$X_{left}^2 = (x-\alpha)^2 + y^2 \tag 1$$

Then square $X_{right}$

$$X_{right}^2 = (x+\alpha)^2 + y^2 \tag 2$$

Subtracting $(1)$ from $(2)$

$$X_{right}^2 - X_{left}^2 = 4\alpha x$$

$$ x = \frac{X_{right}^2 - X_{left}^2}{4\alpha}$$

Similarly you can get $y$ by putting $x$ in any of the above equations