To perform the calculation, the algorithm requires three coordinates for two points, for a total of six numbers. The point where you are standing is called Point A. The target point to be observed is Point B. For both points you need:
- Latitude: how many degrees north or south of the Earth’s equator the point is. Positive values are north of the equator, and negative values are south of the equator.
- Longitude: how many degrees east (positive) or west (negative) the point is from the prime meridian in Greenwich, England.
- Elevation: the point’s height above (positive) or below (negative) mean sea level, expressed in meters.
This algorithm also works if your target location is a geostationary satellite. All geostationary satellites orbit directly above the Earth’s equator (latitude = 0 degrees) at an elevation of 35,786,000 meters. Because the latitude and elevation are already known, the only thing missing is the satellite’s longitude. My online calculator has a checkbox for geostationary satellites to make this case easier. When that checkbox is enabled, you only need to enter the satellite’s longitude. People have used this feature to help aim satellite dishes.
Given the two sets of coordinates, the algorithm calculates the following quantities:
- Azimuth: The compass direction of Point B as seen from Point A. This is a number from 0 to 360 degrees. The value 0 indicates north, 90 is east, 180 is south, and 270 is west.
- Altitude: This is the angle above (positive) or below (negative) the horizon of Point B as seen by an observer at Point A. This is not to be confused with the elevation input defined above.
- Distance: The straight-line distance (not the distance around the Earth) from Point A to Point B, expressed in kilometers.
How It Works
The first thing the algorithm does is convert both points, Point A and Point B, to Cartesian (x, y, z) coordinates. This is complicated by the fact that the Earth is not a perfect sphere. Due to rotation, the Earth bulges at the equator, forming a shape known as an oblate spheroid. The distance between the poles is 6357 km, but the distance of a diameter passing through the equator is 6378 km. Consider this diagram, in which the bulging is exaggerated for the purposes of illustration:
Here, X is a point on the Earth’s surface, N is the north pole, S is the south pole, and C is the center of the Earth. The equator is represented by the horizontal line passing through C. Imagine you are standing at the point X. The dotted line tangent to the Earth there indicates your local horizon.
Perpendicular to your horizon is another dotted line that passes into the Earth. Where that dotted line touches the plane of the Earth’s equator, you see an angle marked φ (Greek letter phi). This angle is called the geodetic latitude. This is the kind of latitude reported by GPS.
Indeed, geodetic latitude is the classic kind of latitude used since ancient times. Roughly speaking, it indicates a latitude measured by astronomy: it is the angle of the North Celestial Pole (a point in the sky near the pole star Polaris) above your local horizon. If you are standing on the Earth’s north pole, Polaris will be straight up, or 90 degrees above the horizon. Likewise, if you are standing on the equator, Polaris will be on the horizon, or 0 degrees above the horizon. Thus geodetic latitude can be measured directly by any sailor with a clear nighttime sky.
However, to calculate the Cartesian coordinates of X with respect to the center C of the Earth, the calculator needs geocentric latitude, indicated by θ (Greek letter theta) in the diagram. This is the angle between the equator and X as seen from C. The conversion is based on the World Geodetic System standard WGS 84. In the calculator this is implemented in the function
GeocentricLatitude, shown here:
Likewise, the distance from the center of the Earth to a given point on its surface is variable. The function
EarthRadiusInMeters calculates this distance as a function of geodetic latitude, also using WGS 84:
As an interesting aside, geocentric and geodetic latitudes are the same when you are on the equator: both are 0 degrees. Both kinds of latitude are also the same if you are on either of the Earth’s poles: +90 or -90 degrees. But they disagree by different amounts at other locations on the Earth.
The calculator uses the function
LocationToPoint to calculate the Cartesian coordinates of a point given its geodetic latitude, longitude, and elevation. This function also calculates the normal vector at the given point, meaning the upward direction perpendicular to the observer’s horizon, to correctly adjust for elevation above mean sea level. The resulting vector components are all expressed in meters.
To complete the calculations, the algorithm uses a trick. It rotates the coordinate system so that the observation point (Point A) is at a pretend equator and prime meridian. In other words, imagine rotating a globe you are holding so that your observation site is now on the equator and directly facing you. The coordinates now are ideally situated for calculating azimuth and altitude angles as seen by the observer at Point A:
- The x-axis points straight up (out from the center of the globe).
- The y-axis points due east.
- The z-axis points due north.
Here is the rotation function
RotateGlobe rotates the coordinate system, you can draw a line from Point A on the rotated globe toward Point B on that same rotated globe, through three-dimensional space. This line will usually pass through the Earth, assuming both A and B are on or near the Earth’s surface. Using a combination of trigonometry functions and vector dot products, the calculator determines the azimuth angle and altitude angles. It uses the Pythagorean formula to determine the straight-line distance between the points.
All of the code is available in the links below.
- Online Calculator: allows you to perform the calculations in your browser.
Example #1: International Space Station Pointer
Jan Steinberg and his colleagues from the Bonn-Rhein-Sieg University of Applied Sciences used the algorithm to build a device that always points to the International Space Station. By calculating the orbit of the ISS and using the known location of the device on the ground, the computer converts the two locations into an azimuth and altitude angle. It uses stepper motors to orient the pointer in real time.
Example #2: The Pterrascope
Rick Hemmings from Mendocino, California used the Altitude/Azimuth calculator page to build a sculpture that points to various cities around the globe. This sculpture points directly toward the other cities at the correct azimuth and altitude angles, downward through the Earth. He coined the word “pterrascope” for this project.
Example #3: Moonrise Photo
Ole Jørgen Nordhagen from Sola, Norway used the Azimuth/Distance calculator to figure out when and where to take photographs of the Moon and Sun behind selected terrestrial objects.
- If you have a drone that broadcasts its GPS coordinates, and you have a device that tells you your own GPS coordinates, you can automate calculation of the direction and range of your drone. One person emailed me that he uses this technique to locate his drone when it lands beyond visible range.
- A man who races sailboats uses my tool to help plan routes through the required waypoints of a race.
- A microwave dish technician used my tool to help aim microwave relays from one tower to another.
- A satellite dish installer said my tool helped aim dishes to specific satellites. I added the “geostationary satellite” checkbox based on his suggestion, so he didn’t have to enter the elevation number manually.
- I originally wrote this tool so I could experiment with magnetic loop antennas for AM receivers I built. These antennas are highly sensitive to the direction of the broadcast tower. By rotating the receiver, I can tell which direction the signal is coming from. I could correlate this with known GPS coordinates for the broadcast towers to verify I was really receiving from the tower I thought I was.
Feel free to comment here to share how you use this code! I’m always interested in the creative things people do.