Navigating with GPS: Calculating Azimuth, Distance, and Altitude
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Chapter 1: Introduction to GPS Calculations
Imagine you’re at a specific location on Earth and wish to point towards another place or aim a telescope. If you possess the GPS coordinates — latitude, longitude, and elevation — of both locations, this guide will illustrate an algorithm to compute two directional angles that align your position (Point A) with the target (Point B). Additionally, it will determine the distance between the two points. You can utilize an online calculator built with open-source JavaScript that implements this algorithm.
The Input Data
To execute the calculation, the algorithm requires three coordinates for each of the two points, summing up to six values. The coordinates are defined as follows:
- Latitude: Indicates how far north or south of the equator the point is. Positive values denote locations north, while negative values indicate south.
- Longitude: Denotes how far east (positive) or west (negative) the point is from the prime meridian in Greenwich, England.
- Elevation: Represents the height above (positive) or below (negative) mean sea level, measured in meters.
Geostationary Satellites
This algorithm is also applicable for targeting geostationary satellites. These satellites maintain an orbit directly above the equator at an altitude of 35,786 km. Since latitude and elevation are predetermined, you only need to provide the satellite's longitude in the online calculator, which includes a geostationary satellite option for convenience. This feature is frequently used for adjusting satellite dishes.
The Output Results
Upon entering the two sets of coordinates, the algorithm computes the following:
- Azimuth: The compass direction of Point B relative to Point A, measured in degrees from 0 (north) to 360 (complete circle).
- Altitude: The angle above (positive) or below (negative) the horizon of Point B from the perspective of Point A.
- Distance: The straight-line distance (not considering the Earth's curvature) between Point A and Point B, expressed in kilometers.
How the Calculation Works
Initially, the algorithm converts both points to Cartesian coordinates (x, y, z). This conversion is complicated by the Earth's oblate spheroid shape, where the equatorial diameter is larger than the polar diameter.
To visualize, consider a scenario where you are located at a point on the Earth's surface, with the center of the Earth as a reference point. The geodetic latitude (φ) differs from the geocentric latitude (θ), which is crucial for accurate calculations.
The conversion to Cartesian coordinates is carried out using the World Geodetic System (WGS 84) standard. For instance, the function GeocentricLatitude assists in this conversion, while EarthRadiusInMeters determines the distance from the Earth’s center to the surface based on geodetic latitude.
The algorithm then utilizes the function LocationToPoint to derive the Cartesian coordinates of a point based on its geodetic latitude, longitude, and elevation. It also computes the normal vector at that point, which indicates the upward direction perpendicular to the horizon, essential for adjusting elevation.
Finally, the coordinate system is rotated so that Point A is aligned with the equator and prime meridian, simplifying the calculations for azimuth and altitude. The x-axis points upward, the y-axis points east, and the z-axis points north.
Here is the rotation function:
def RotateGlobe(point_a):
# Function to rotate the coordinate system
pass
After adjusting the coordinate system, a line can be drawn from Point A to Point B in three-dimensional space. Through trigonometry and vector dot products, the calculator computes the azimuth and altitude angles as well as the straight-line distance between the points.
All the source code is accessible in the links provided below.
Resources
Online Calculator: This tool allows you to perform calculations directly in your browser.
GitHub Repository: Access the complete HTML and JavaScript code for the calculator.
Sample Applications
Here are three notable examples of how individuals have applied this JavaScript code:
- International Space Station Pointer: Jan Steinberg and his team at Bonn-Rhein-Sieg University created a device that consistently points towards the International Space Station by calculating its orbit and using the device's location on the ground.
- The Pterrascope: Rick Hemmings from Mendocino, California, designed a sculpture that accurately points to various global cities, termed the "pterrascope."
- Moonrise Photography: Ole Jørgen Nordhagen from Sola, Norway, used the calculator to determine the optimal times and locations for capturing photos of the Moon and Sun behind specific terrestrial landmarks.
Other Applications
If you have a drone transmitting its GPS data, you can automate the calculations for determining its direction and distance. One user has shared that they use this method to locate their drone when it lands out of sight. Additionally, sailboat racers utilize this tool for route planning, and technicians employ it for aligning microwave dishes.
If you've used this code in your projects, feel free to share your experiences! I'm always interested in learning about the innovative applications of this technology.
The first video demonstrates how to measure distance and altitude using GPS technology, providing insights into practical applications.
The second video tackles a LeetCode challenge focused on finding the highest altitude using JavaScript, showcasing problem-solving techniques in programming.