# Approximating length of a curved line based on Begining and End points of line

I have two points, a known distance apart. At each of these points I have a sensor that gives me flow speed and direction.

I originally assumed the flow path between the first point and second point was a straight line connecting the two, but I now realize it is a curved path. I know I don't have enough information to precisely calculate the distance of the curved path, but I'm hoping to approximate the distance of the curved path between the two points based on the speed and direction of the 1st and 2nd points.

Something along the lines of:

The distance between point 1 and 2 is 15 meters
The bearing between point 1 and 2 is 245 degrees

The measured speed at point 1 is 25m/s at 220 degrees
The measured speed at point 2 is 10m/s at 230 degrees


Is there a way to approximate the length of the path of a particle as it passes from point 1 to point 2?

• Is it uniform decelration? The path length is already given as 15 meters. Do you want path as a function of time? Apr 18, 2015 at 22:15
• Perhaps I am asking for something that does not exist, but I am only interested in the approximate length of the most likely path a particle would take give the known vectors influencing it at points 1 and 2. No function of time is needed Apr 19, 2015 at 18:36

There is no hope of making a justifiably good estimation!

Lacking any details about the path inbetween, an adequate guess would be to approximate the curve with a cubic Bezier curve, where $P_0$ is the starting point, $P_3$ the end point, and $\vec{P_0P_1}$, $\vec{P_2P_3}$ are proportional to the measured velocities. This still leaves one degree of freedom, so maybe we should pick the scale so that the length $|P_1P_2|$ is "compatible" with $|P_0P_1|$ and $|P_2P_3|$, for example their geometric mean. In principle, this finally determines a curve the length of which can be computed. However, the details of the computation are possibly more complicated than the little available input can justify.

So alternatively, especially in a situation like your example where the curve must have a point of inflection, one could try to fit in two touching circles with radii proportional to the measured speeds. However, this does not produce a unique curve either (though the length computation should be simpler)

So finally a very crude (or: quick and dirty) method: When moving along a line at an angle of $\delta$ against the straight connection, the effective "progress" in the intended direction is shorter by a factor of ${\cos\delta}$. Now let's simply assume that about $\frac13$ of the time, we move parallel to the direct connection (at a speed between the two measured ones), $\frac13$ of the time we move at the first measured speed and at a deviation against the straight line as measured there, and the remaining $\frac13$ of the time we move at the second measured speed and at a deviation as measured there. In your example, the "progress" made per $t$ seconds would be $$\tag1\frac t3\cdot(25\cdot\cos(220^\circ -245^\circ) +\tfrac{25+10}{2}+10\cdot \cos(230^\circ-245^\circ))\approx 16.605\cdot t$$ and the actual distance traveled along the curve in the same time would be $$\tag2\frac t3\cdot(25+\tfrac{25+10}2+10) =17.5\cdot t$$ We need to pick $t$ so that $(1)$ produces the given distance of $15$ meters and then $(2)$ shows the result. This way we estimate the curve length as $$\frac{17.5}{16.605}\cdot 15\,\text{m}\approx 15.8\,\text m.$$ As said above, this method is very quick (really not much calculation involved) and dirty (approximations made that could make a physicist happy, but certainly not a mathematician) and I won't rely on it in cases where the start or end dirction deviate substantially from the straight direction (the $25^\circ$ as in your example may still seem reasonable, maybe even up to $45^\circ$) and/or where the initial and final speed differ substantially (your example has a factor of $2.5$, which may or may not be considered reasonable; certainly a factor of $10$ would make the estimate totally untrustworthy).

It seems to me that you'd need more information. I drew this in GeoGebra based on what you wrote: The particle starts at point $A$, goes through some unknown path, and ends at point $B$. The larger angle between the $y$-axes and the segment $AB$ is $245^{\circ}$.

I had initially thought that we could get an approximate answer by assuming that the particle has constant acceleration, but this is not possible, because both the starting vector and ending vector are at angles below the straight-line path from $A$ to $B$, so there must be an inflection point in the particle's path. So we can't assume constant acceleration.

The shape of the path could be so many different things that it is impossible, given just what you've told us, to approximate the length of the path. However, if you found out that the jerk was constant, or some other bit of information, it would be easier.

If you pressed me to guess the length of the path I'd throw out something like $20$ meters, just because it has to be a little bit longer than $15$ ...