Geometry for Navigation

This year I’ve spent a lot of time trying to understand various forms of navigation, primarily due to my interest in space exploration and colonization of other planets.

The questions I’ve been asking myself is how do you figure out where you are in space if you are piloting a spacecraft and if you land on another planet how would you navigate?

A lot of what we take for granted with respect to navigation, cannot be done on other planets. Venus, Mars and the Moon don’t have a magnetic field and they certainly don’t have GPS satellites. Nor do they have a good north star like Polaris.

The problem with the books I’ve been reading on navigation is that the of course mainly concern themselves with the Earth, and quickly get into very many earth specific details, making it difficult to grasp the fundamental concepts of navigation.

So that is what I intend to do here. Basically explain the concepts, one small step at a time without assuming too much previous knowledge.

Trigonometry Crash Course

So let us start with the most basic geometry. If we got a right triangle (one angle is 90°) and we know either:

  1. The length of two sides
  2. A length and an angle

Then we can calculate an unknown side or angle. Using the triangle as an example we got these relationships.

a² = b² + c²
sin(α) = b/a
cos(α) = c/a
tan(α) = b/c

Similar for other angles

sin(β) = c/a
cos(β) = b/a
tan(α) = c/b

The trigonometric functions sin, cos and tan are defined in terms of a unit circle. If we imagine a circle with radius r equal to 1, placed at the center of the coordinate system, then the (x, y) coordinates of this point is defined as

x = cos(θ)
y = sin(θ)

Which is another way of saying the the width and height of the triangle inscribed in a circle with radius r is.

w = r*cos(θ)
h = r*sin(θ)
tan(α) = c/b

Navigation on a Flat World

Lets avoid complicating things, and consider the cases when the earth is flat and the sun is relatively close. A point directly below a heavenly object (a star, moon, sun or planet), such as is called a Geographical Position (GP). In this example A is GP. If we got a person at some position B, who doesn’t know where he is, he tries to figure it out by measuring the angle θ to the sun at his position.

Angles could be measured using a clinometer, theodolite or sextant. Whether you are at sea or land is going to decide what is best suited. Finding the angle however this doesn’t precisely define your position. As you can see

If he knows the distance h from the earth to the sun, he can figure out his distance to A. Our lost person then knows that he must be on some point along a circle with a radius r

r = h/tan(α)

Far Away Objects

A common issue when dealing with navigation is that we try to find angles to objects extremely far away, such as stars or the sun.

Assume we are standing at point marked in blue looking at an object far away marked in yellow, such as the sun. What you can see is that as the distance grows the angle starts approaching 90°.

In such a scenario, measuring angles isn’t going to help you determine where you are because you are going to measure 90° at any location.

As in this example where A, B and E all give 90°. In this case it actually helps us that the earth has curvature.

Navigation on a Curve World

In the animation below we assume a curved world, with a sun so far away that the sun lights hit the surface with the same parallel lines.

However since the surface curves, the angle the sun rays will make with the horizon (tangent) will vary). So a person looking at the sun with a sextant and measure the angle between the horizon the sun, will determine that he/she must be one of two places on an entirely flat world.

This is because a given angle can be made be measure in two places. If the world is a cylinder which continues into the paper, then the person would determine that they are at any place along the two lines.

In this example you can see sun rays coming in with red color making an angle with the horizon represented as a black line. Our explorer would then be along the blue line, alternatively a blue line on the opposite side of the cylinder.

Spherical World

Of course our actual world, the earth is spherical. If we measure an angle to a distant object, we are not on some point along a line but rather along a circle, as you can see below.

If we consider the red lines sun rays, then the blue dot is the Geographical Position (GP) of the sun on the earth surface. This is the point where the sun would be observed directly above the person. So if we measure the angle against the horizon, we would read 90° with the sextant. In this example we are making the angles with the nadir, which is an imagine line running from your current positioned through the center of the earth. A weight on a string would point in this direction. Basically it is the direction of the gravity. This is also an alternative reference point to the horizon.

As mentioned in my coverage of navigation instruments, a sextant on an airplane works on this principle, by using a spirit level, since you can typically not observe a horizon from an airplane.

Trilateration

Of course we would not be happy simply knowing which circle were are on. We want to know a single point. To accomplish that we use triangulation. Or more specifically when we are dealing with circles we call it trilateration.

The process involves observing multiple landmarks and measuring the distance to them. Measuring the distance can be done with trigonometry.

Say we are located at point A looking at a lighthouse at B with the top of the lighthouse at point C. Given that we got a map with details about each lighthouse such as its height h, then we can find the distance d with.:

d = h*tan(α)

Determining the distances to two landmarks would then allow us to draw two overlapping circles. Our position must be in one of the intersection points marked in orange. In practice it is two risky to rely on accurate measurements of just two landmarks. Instead one will make multiple measurements.

Assuming we measure the distance to 3 landmarks A, C, E, we will get three overlapping circles. Given that neither measurement is entirely accurate, the intersection will not be a single point but three G, H, I, which can be used to construct a triangle.

Our position will then be found somewhere within this triangle. For increased accuracy it is normal to use more circles, potentially creating a more complex polygon.

Triangulation

When we find our position by measuring angles rather than distances (radius of a circle) we call that triangulation. We can e.g. look at a lighthouse using a bearing compass to determine how many degrees relative to magnetic north it is. A slightly more complicated case is to find the angle between two known objects. That would be required if e.g. we don’t have a magnetic north, or we don’t have a compass. Other planets such as Mars of Venus don’t have magnetic norths.

Again such angles could be measure by a number of instruments, both a sextant and a theodolite e.g. We are used to using a sextant to measure angles between the horizon and a celestial object, but it could be put on its side and used to measure angle between objects on the horizon.

But first lets look at the simple case, of measuring angles relative to north. We got our map, and we mark of A, B, C, which are the features on the map we are measuring angles to. This could be light houses, mountain tops or some other feature we can see from our position and identify on our map.

For each of these we measure angles relative to north marked by 3 vectors u, v and w on our north compass rose on our navigation map. We got to get lines with these angles going through points A, B and C, to do our triangulation.

First we get lines going north-south through these points. You can accomplish in several ways. One approach is to construct a perpendicular line through a point twice as I described earlier.

You could also use a parallel ruler, which is common for anybody navigating on the ocean using paper maps.

You can use the same method to create parallel lines with the vectors u, v and w running through the points A, B and C

This gives us three intersection points in orange labeled K, L and M. Your location has to be within the beige triangle formed by these intersection points.

Next time we’ll look at how we can combine this knowledge to perform navigation by the sun, moon and stars. This is what we call celestial navigation.