This is a real-world problem that I can tackle by brute force but I'd like to know if there is a faster way to do it. I suspect there is.....
I have a dataset $d$ where $d(0)$ $d(1)$ $d(2)$ are an ordered set of data. Each of them contains an $x$ and $y$ coordinate, and the distance precalculated from the previous point. So typical data might be: $$\begin{align} &d(0): x=0,\ y=0,\ d=0\\ &d(1): x=3,\ y=0,\ d=3\\ &d(2): x=2,\ y=1,\ d=1.414\\ &\quad\vdots \end{align}$$
The real-world data has tens or hundreds of thousands of points, and the distance between points is small $O(10^{-5})$, but the principle is the same. The important figure is the total length of the path.
We then apply a simple linear skew +stretch to the plane, and to all the points: $x\rightarrow x - Ay$, $y \rightarrow By$, and need to calculate the new total length.
I need to create a lookup table of this for about $100,000$ combinations of $A$ and $B$, so that for any given $(A, B)$ in the set, I can find the length of the curve when transformed.
Clearly it's not that demanding computationally; also clearly the scale factor isn't constant for all curves and skews, it depends on the exact curve shape. I could iterate through each $(A, B)$, and for each, skew the $10$-$100$k points and recalculate the total new distance, but that seems inefficient. Its also about $10^{12}$ point recalculations and Pythagorases (is that a valid plural?).
So I wonder, is there any generalised faster way?