How to linearize the distance formula?

Question

So I have an equation that contains the distance formula squared. However, I am interested in linearizing this equation.

My equation is: Constant/distance squared

My distance is between a fixed point and a variable point. So x1 and y1 are known but x2 and y2 are variables. Any idea how to linearize this? I thought of expanding the distance and then linearizing the squares.

Answer 1

From the Taylor development,

$$\sqrt{(x_2+\delta x-x_1)^2+(y_2+\delta y-y_1)^2}\approx \\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}+\frac{\delta x(x_2-x_1)+\delta y(y_2-y_1)}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}.$$

which is linear in $\delta x, \delta y$, the variations around some "central" $x_2,y_2$.

Notice that this gives you the algebraic distance to the plane tangent on the sphere centered at $p_1$, and passing through $p_2$.

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You can reason similarly for the case of $\dfrac c{d^2}$.