Because of relativistic effects, quickly moving objects appear to shift in color: quickly approaching objects seem bluer, and quickly departing objects seem redder. Astronomers use this phenomenon to determine the speed of distant objects. First, they measure the frequency of the radiation that reaches them. Next, they infer the frequency of radiation that was originally emitted by the object. (This surprising feat is possible because each chemical element has a recognizable light signature that's not distorted by the doppler effect.) By comparing the emitted frequency with the observed frequency, astronomers can calculate the speed of distant objects.

You can use this nomogram to perform the same calculation:

- Find the
*observed frequency*of the object on the lower horizontal axis. - Find the
*emitted frequency*of the object on the upper horizontal axis. - Draw the line passing through the two points. That line will cross the vertical axis in exactly one place— that's the velocity of the object relative to the observer.

With the visible spectrum overlaid, you can also answer questions such
as *“How fast would you have to travel in order for
red objects to appear green?”* (Answer: you'd have to
travel towards the red object at about 0.35 times the speed of
light.)

As you approach the speed of light, time dilates and length contracts along your direction of travel. The relation is remarkably simple to depict using the unit circle.

This is because the effects of relativity are trigonometric in nature: for each point on the unit circle, the y-coordinate corresponds to a velocity, the x-coordinate coordinate corresponds to the length contraction factor at that velocity, and the length of the secant line (the line from the origin to the point on the circle, stopping at x=1) is the time dilation at that velocity.

Officially, this relation doesn't even require a nomogram, since the variables are all monotonic functions of each other and could all three be plotted on a single curve. But on the other hand, this nomogram is “metrically accurate” — every quantity is a geometric length in the image.

Here, the orange line indicates how you can use the nomogram: at half the speed of light, time dilates by a factor of about 1.2, and length contracts by about a factor of 0.87.

The relations depicted are $$\ell^\prime/\ell= \sqrt{1-\beta^2}$$ $$t^\prime / t = \frac{1}{\sqrt{1-\beta^2}}$$ Hence we can define an angle \(\theta\) for which $$\begin{eqnarray*} \beta &=& \sin{\theta}\\ \ell^\prime/\ell &=& \cos{\theta}\\ t^\prime / t &=& \sec{\theta}.\\ \end{eqnarray*}$$

According to special relativity, observers who are moving at different speeds will disagree about measurements of length and duration. This nomogram enables you to measure those differences.

Here, the horizontal axis represents your velocity relative to another observer, vertical displacement represents your choice of a length or time measurement in your reference frame, the blue curve represents the other measurement (i.e. time or length, respectively) in your reference frame, and the grey curve represents how it appears to another observer.