Math Concepts

Used to find optimal translations, rotation and scaling. Example used is aligning the big dipper and little dipper. This works amazingly well. Very easy python example, golang required gonum.

Manhattan Distance

he distance between two points is the sum of the absolute differences of their Cartesian coordinates -- https://en.wikipedia.org/wiki/Taxicab_geometry

Used to quantify distance between points. In 3d it would be:

pointA = [1105,-1205,1229]
pointB = [-92,-2380,-20]
|(1105 + -92)| + |(-1205 + -2380)| + |(1229 + -20)| = 3621

Lagrange Interpolation

The Lagrange interpolation formula is a way to find a polynomial, called Lagrange polynomial, that takes on certain values at arbitrary points. Lagrange’s interpolation is an Nth degree polynomial approximation to f(x). Let us understand Lagrange interpolation formula using solved examples in the upcoming sections.

Given n distinct real values $x_1, x_2,...,x_n$ and n real values $y_1, y_2,...,y_n$ (not necessarily distinct), there is a unique polynomial P with real coefficients satisfying $P \left(x_i\right) = Y_i$ for $i \in {1, 2, …, n}$, such that $deg\left(P\right) < n$. Lagrange interpolation formula for different order of polynomials is given as,

$f\left(x\right)=f\left(x_0\right) + \left(x - x_0\right)\left(\frac{f\left(x_0\right)-f\left(x_1\right)}{x_0-x_1}\right)$

simplified:

$f\left(x\right) = \frac{\left(x - x_1\right)}{\left(x_0 - x_1\right)}f_0 + \frac{\left(x - x_0\right)}{\left(x_1 - x_0\right)}f_1$

-- https://www.cuemath.com/lagrange-interpolation-formula/