# The direction between two close noisy points is mostly noise

> The angle of the vector between two noisy points has an error of about √2·σ/d radians while the points are far apart, and becomes a uniformly random direction as they meet. With noise that is correlated over time, it wanders smoothly and looks like a real signal.

- Author: Ajish Pradeep (https://ajishpradeep.com/about/)
- Published: 2026-10-07
- Area: numerics
- Canonical: https://ajishpradeep.com/blog/the-direction-between-two-close-noisy-points-is-mostly-noise/
- Cite as: Pradeep, A. (2026). The direction between two close noisy points is mostly noise. https://ajishpradeep.com/blog/the-direction-between-two-close-noisy-points-is-mostly-noise/

Many quantities in data analysis are directions computed from two estimated points: the orientation of an edge between two detected corners, the heading between two position estimates, the angle of a line fitted through two measurements. Each point carries some error, and the direction inherits it. I wanted to know how fast that error grows as the points get closer, and what it looks like when it has taken over, because a direction always comes back as a perfectly valid angle, whatever went into it.

## On paper

Take two points measured with independent Gaussian noise of standard deviation $\sigma$ in each coordinate, a true distance $d$ apart along the x-axis. The difference vector is $v = (d + \epsilon_x,\ \epsilon_y)$, where each $\epsilon$ is the difference of two independent errors and so has standard deviation $\sqrt{2}\,\sigma$. Its angle is $\theta = \operatorname{atan2}(\epsilon_y,\, d + \epsilon_x)$. For small errors, a first-order expansion gives

$$\theta \approx \frac{\epsilon_y}{d}, \qquad \sigma_\theta \approx \frac{\sqrt{2}\,\sigma}{d}$$

so the angular error is inversely proportional to the separation in units of the noise. Halve the distance and the error doubles.

At $d = 0$ the expansion has nothing left to expand around. The vector is pure noise, and since two-dimensional Gaussian noise has no preferred direction, the angle is uniformly distributed over the circle. A uniform angle between $-\pi$ and $\pi$ has a standard deviation of $\pi/\sqrt{3}$, about 104°, and a median absolute error of 90°. In between, the formula must stop holding somewhere, since it predicts errors larger than a uniform angle can have.

## Checking it

I drew 200,000 pairs of points for each separation, with $\sigma = 1$.

*(An interactive figure is on the web page; every number it shows is stated in the text.)*

| Separation $d/\sigma$ | Angle error, simulated | Formula $\sqrt{2}\sigma/d$ | Median absolute error | Share more than 90° wrong |
|---:|---:|---:|---:|---:|
| 40 | 2.0° | 2.0° | 1.4° | 0% |
| 20 | 4.1° | 4.1° | 2.7° | 0% |
| 10 | 8.2° | 8.1° | 5.5° | 0% |
| 5 | 17.1° | 16.2° | 11.0° | 0% |
| 2 | 49.7° | 40.5° | 27.5° | 7.7% |
| 1 | 75.4° | 81.0° | 47.8° | 24% |
| 0.5 | 89.9° | 162° | 66.2° | 36% |
| 0 | 103.8° | undefined | 89.7° | 50% |

The formula is exact far apart and still close at five noise lengths. Below that it fails in two directions. At $d = 2\sigma$ the real error is larger than predicted, because the expansion ignores the tails. At $d \le \sigma$ the formula overshoots, because the real error is capped by the circle. At $d = \sigma$ a quarter of the computed directions point more than 90° away from the truth, that is, roughly backwards. At zero separation the simulation matches the uniform distribution: a standard deviation of 103.8° and half of all angles wrong by more than 90°.

So a useful rule is that a direction from two points is meaningful while they are several noise lengths apart, degraded at one or two, and arbitrary below one.

## When the noise is correlated over time

Point estimates made over time often have correlated errors, for example after smoothing or tracking, where each estimate leans on the previous one. I modelled that with errors that follow an autoregressive process with coefficient 0.95, so each error keeps 95% of the last one, and compared the resulting angle series with one whose errors are independent at each step.

*(An interactive figure is on the web page; every number it shows is stated in the text.)*

| Separation $d/\sigma$ | Angle spread | Lag-1 autocorrelation | Mean change per step | Same, if the errors were independent |
|---:|---:|---:|---:|---:|
| 20 | 4.0° | 0.95 | 1.0° | 4.6° per step, autocorrelation 0.00 |
| 5 | 17.2° | 0.94 | 4.3° | 19.0° per step, autocorrelation −0.01 |
| 1 | 74.1° | 0.77 | 21.7° | 77.5° per step, autocorrelation 0.00 |

The spread of the angle is the same as before; the correlation doesn't make the noise smaller. It changes what the noise looks like. With independent errors, a noisy direction jumps around from step to step and looks like noise. With correlated errors it changes four times more slowly and wanders smoothly, with a lag-1 autocorrelation of 0.95, which is what a slowly turning real signal looks like. A plot of it would not raise any alarm. Even at $d = \sigma$, where the direction is mostly arbitrary, it still drifts with an autocorrelation of 0.77.

That is the more dangerous case of the two. Jitter invites smoothing and suspicion. Drift invites interpretation.

## What I would check before trusting such a direction

- the separation of the two points in units of their measurement noise, where anything under about five noise lengths deserves a stated error bar and anything near one deserves no interpretation;
- the angular error predicted by $\sqrt{2}\sigma/d$, reported next to the angle;
- whether a smooth trend in a direction persists when the points are far apart, or only appears when they are close.

Differences of noisy samples amplify noise in time just as they do in space, and a loss built on them pays for it, in [derivative terms in a loss amplify noise](https://ajishpradeep.com/blog/derivative-terms-in-a-loss-amplify-noise-by-one-over-h-squared/). A direction can track a signal's shape and still be wrong in level, which is the distinction in [correlation is not agreement](https://ajishpradeep.com/blog/correlation-is-not-agreement/).

## What I still don't know

I assumed equal, isotropic noise on both points. If the noise is larger along one axis, the angular error depends on the direction itself, and the uniform limit becomes a skewed distribution. Directional statistics, which Mardia and Jupp (1999) treat in depth, has the tools for that case, and averaging or testing angles properly needs them, since an ordinary mean of angles breaks at the wrap-around. I also haven't looked at directions computed from more than two points, where a fitted line can use all of them and the error falls with the number of points as well as with their spread.

## References

- Directional Statistics — Kanti V. Mardia and Peter E. Jupp, Wiley 1999 — https://doi.org/10.1002/9780470316979