{"id":444,"date":"2025-04-07T11:49:05","date_gmt":"2025-04-07T11:49:05","guid":{"rendered":"https:\/\/uat.esri.com\/en-us\/software-engineering\/blog\/?post_type=blog&p=444"},"modified":"2025-04-16T17:20:25","modified_gmt":"2025-04-16T17:20:25","slug":"positions-of-gps-satellites-in-3d","status":"publish","type":"blog","link":"https:\/\/www.esri.com\/en-us\/software-engineering\/blog\/articles\/positions-of-gps-satellites-in-3d","title":{"rendered":"Positions of GPS Satellites in 3D"},"content":{"rendered":"\n

Calculating the 3D positions of GPS satellites can be challenging due to the complex trigonometry involved in translating 2D coordinates plus angular directions to a 3D space. In this article, let’s explore how to calculate the positions and display satellite locations in 3D, real-time, and even in Augmented Reality!<\/p>\n\n\n\n

This is a revision of a previous post<\/a> I wrote in the Esri Community – Native SDKs Blog in 2021. The main concepts remain the same, while some new features are added to the project, and the codebase is migrated to adopt the new 200.x ArcGIS Maps SDK for Swift.<\/em><\/p>\n\n\n\n

Concepts<\/strong><\/h2>\n\n\n\n

Handheld GPS\/GNSS receivers are widely used devices that leverage signals from satellites to determine precise geographic locations. The receivers are used in various occasions such as navigation, outdoor recreation, surveying, and scientific research. And they can connect to computers and phones easily. They often use the NMEA protocol to communicate location data and other satellite-derived information in a standardized format. This protocol allows GPS receivers to be integrated with mapping software, expanding their usability beyond standalone navigation to include real-time data analysis and visualization.<\/p>\n\n\n

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A typical representation of a satellite tracking view on a mobile device.<\/figcaption><\/figure><\/div>\n\n\n

Many handheld GNSS\/GPS receivers support tracking satellite positions in 2D (schematically demonstrated in the diagram above). They typically use $GPGSV<\/code> sentences from the NMEA (National Marine Electronics Association) specifications, which describe satellite positions through elevation and azimuth angles. These angles enable the calculation of approximate satellite positions in the sky by factoring in the receiver’s location coordinates, Earth\u2019s radius, GPS satellite orbit altitude, elevation angle, and azimuth angle. On certain devices, each satellite shows up as a signal bar; other devices use 2 concentric circles to depict the earth and the satellites’ orbit, and the satellites are plotted on the graph. These graphics are OK in terms of giving people a general sense on where the satellites are currently located, but are neither intuitive, nor informative in 3D space. <\/p>\n\n\n\n

To gain that 3D perspective, let’s look more at the elevation and azimuth angles provided in the $GPGSV<\/code> sentences from the NMEA specs.<\/p>\n\n\n\n

What are these angles? Imagine yourself standing in an open field (depicted by the gray plane in the image below), holding both a GPS receiver and a compass. Azimuth<\/strong> (the blue vector) tells us what direction to face at, and Elevation<\/strong> (the red vector) tells us how high up in the sky to look at. Turning around from the true north and raising the head, you can gaze at the approximate direction of the satellite in the sky with these 2 angles. These 2 values are simple to interpret for someone standing on the surface of the earth. However, things get a bit trickier when it comes to mapping software.<\/p>\n\n\n

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Concepts of Azimuth and Elevation angles in 3D space<\/figcaption><\/figure><\/div>\n\n\n

How to represent azimuth and elevation on a map, or a 3D globe? There are a few known variables. The device location is represented as a pair of latitude-longitude (lat-lon<\/em>) coordinates, which is a point on the earth’s surface. We also know where we should look at with these 2 angles. In a simplified way, the earth can be treated as a sphere (rather than an ellipsoid, which would complicate the math by quite a bit). Thus, we don’t need to take the variation of the radius of the earth into account, as it is always 6371 km.<\/p>\n\n\n\n

The real problem is – how to convert our lat-lon<\/em> coordinates on the surface of the globe into the 3D coordinate system of a sphere. Some trigonometry has to be done.<\/p>\n\n\n\n

Methods<\/strong><\/h2>\n\n\n\n

For the technical discussion below, I’ll use Swift and related Apple APIs to demonstrate the idea.<\/p>\n\n\n\n

Geometry, Matrix and Rotations<\/h3>\n\n\n\n

The first thought was to convert all the lat-lon<\/em> coordinates into 3D Cartesian coordinates of the sphere. However, it is quite complicated and error-prone to transform all the angles into xyz<\/em>‘s. The lat-lon<\/em> coordinates aren’t too bad – just a few rectangular to polar conversions. The real pain is the azimuth and elevation angles – to represent them in Cartesian, much vector and polar math need to be done. In contrast, the problem can seemingly be expressed succinctly in a spherical coordinate system. Which led me towards thinking of vector operations and matrices. The resulting coordinates of the satellite are the summation of 2 vectors: the one from the center of the earth to the device location, and the other one from the device location to the satellite in the orbit, represented by azimuth and elevation angles.<\/p>\n\n\n\n

Finding it hard to visualize the spatial relationships of these vectors in the spherical coordinates, I decided to sketch the whole system in 3D to make it clearer. Using the online geometry sketching tool GeoGebra<\/a>, I was able to create a spatial representation of the satellite and the earth.<\/p>\n\n\n

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Diagram to show what is an Elevation Angle<\/figcaption><\/figure><\/div>\n\n\n

In order to calculate the coordinates, vanilla Cartesian transformations aren’t enough. Quaternions are introduced to express a position and orientation in space. SIMD library is used so that small vectors and matrices are easily expressed and quickly computed.<\/p>\n\n\n\n

Quartenions<\/h4>\n\n\n\n

Quaternions are defined by a scalar (real) part, and three imaginary parts collectively called the vector part. They are often used in graphics programming as a compact representation of the spatial rotation of an object in three dimensions. One of the advantage is that they don’t suffer from singularity points (or gimbal locks), which are hard to avoid in Cartesian space when compounding rotations. Furthermore, they are quicker to compute than the representations by matrices. For instance, the lat-lon<\/em> coordinates can be effectively represented by quaternions.<\/p>\n\n\n\n

Consider a unit vector of the earth radius, with starting point at the center of the earth and terminal point at 0\u00b0 East on the equatorial plane. By rotating this vector along the y-axis by latitude<\/em> degrees, and z-axis by longitude<\/em> degrees, we will have the vector pointing at the lat-lon<\/em> position on the surface of the globe.<\/p>\n\n\n\n

SIMD<\/h4>\n\n\n\n

Swift has a handy simd<\/code><\/a> library. simd<\/code> provides integer and floating-point types and functions for small vector and matrix computations. The functions provide basic arithmetic operations, element-wise mathematical operations, and geometric and linear algebra operations. It supports up to 4\u00d74 elements in a matrix, which is a perfect fit for quaternions.<\/p>\n\n\n\n

The Quest to Find \u201cThe 3rd Axis\u201d<\/h3>\n\n\n
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Tap the image to see the animation<\/a><\/figcaption><\/figure><\/div>\n\n\n

It is hard to describe it in words, so please use the visualization on GeoGebra<\/a> to see it in action. Toggle the graphics on the left pane to see their spatial relationships.<\/p>\n\n\n\n

With 3 axes, 4 quaternions and all the mysterious steps, how to ensure the result is correct? Luckily, with the visuals created on GeoGebra, I can get the azimuth and elevation values reversely from the result by specifying the other parameters, namely latitude, longitude, earth radius and orbit radius. So, if my calculation is correct, the resulting satellite coordinates should be the same as on GeoGebra.<\/p>\n\n\n\n

Algorithm<\/h3>\n\n\n\n
\/\/\/ Calculates a satellite's position from its elevation and azimuth angles.\n\/\/\/ - Parameters:\n\/\/\/ - deviceLocation: The location of the device that receives satellite info.\n\/\/\/ - elevation: The elevation angle.\n\/\/\/ - azimuth: The azimuth angle.\n\/\/\/ - satelliteAltitude: The satellite's altitude above ground level.\n\/\/\/ - earthRadius: The earth's radius.\n\/\/\/ - Returns: A point representing the position of the satellite.\nfunc satellitePosition(\n    deviceLocation: Point,\n    elevation: Measurement<UnitAngle>,\n    azimuth: Measurement<UnitAngle>,\n    satelliteAltitude: Measurement<UnitLength> = .init(value: 20200, unit: .kilometers),\n    earthRadius: Measurement<UnitLength> = .init(value: 6371, unit: .kilometers)\n) -> Point {\n \n    \/\/ Degree to radian conversions.\n    let longitude = Measurement<UnitAngle>(value: deviceLocation.x, unit: .degrees).converted(to: .radians).value\n    let latitude = Measurement<UnitAngle>(value: deviceLocation.y, unit: .degrees).converted(to: .radians).value\n    let elevation = elevation.converted(to: .radians).value\n    let azimuth = azimuth.converted(to: .radians).value\n\n    \/\/ 1. Basic trigonometry\n    let r1 = earthRadius.value\n    let r2 = r1 + satelliteAltitude.value\n    let elevationAngle = elevation + .pi \/ 2\n    let inscribedAngle = asin(r1 * sin(elevationAngle) \/ r2) \/\/ Law of Sines\n    let centralAngle = .pi - elevationAngle - inscribedAngle \/\/ Internal angles sum = pi\n\n    \/\/ 2. Magical quaternions\n    let latLonAxis = simd_normalize(\n        sphericalToCartesian(r: r1, theta: .pi \/ 2 - latitude, phi: longitude)\n    )\n \n    \/\/ Find the vector between device location and North point (z axis)\n    var thirdAxis = simd_normalize(\n        simd_double3(0, 0, 1 \/ latLonAxis.z) - latLonAxis\n    )\n \n    \/\/ Rotate to the azimuth angle\n    let quaternionAzimuth = simd_quatd(\n    angle: 1.5 * .pi - azimuth,\n        axis: latLonAxis\n    )\n    thirdAxis = quaternionAzimuth.act(thirdAxis)\n \n    \/\/ Rotate the lat-lon vector to elevation angle, with the rotated third axis\n    let quaternionElevation = simd_quatd(\n        angle: -centralAngle,\n        axis: thirdAxis\n    )\n    let resultVector = quaternionElevation.act(latLonAxis)\n\n    \/\/ 3. Make result\n    let result = cartesianToSpherical(vector: resultVector)\n    let resultLatitude = Measurement(\n        value: .pi \/ 2 - result.theta,\n        unit: UnitAngle.radians\n    ).converted(to: .degrees).value\n\n    let resultLongitude = Measurement(\n        value: result.phi,\n        unit: UnitAngle.radians\n    ).converted(to: .degrees).value\n\n    return Point(\n        x: resultLongitude,\n        y: resultLatitude,\n        z: satelliteAltitude.converted(to: .meters).value,\n        spatialReference: .wgs84\n    )\n}\n<\/code><\/pre><\/div>\n\n\n\n
The method to get the satellite’s position can be split into 3 parts – basic trigonometry, quaternion rotation, and coordinate system conversion.<\/p>\n\n\n
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\"Annotated
Various angles for calculation, visualized<\/figcaption><\/figure><\/div>\n\n\n
First, we need some basic trigonometry on the elevation plane<\/em>, i.e. the plane where the elevation angle exists. Using Law of Sines, we can get the value of the central angle<\/em>, assuming both the earth radius and the satellite orbital distance are constants.<\/p>\n\n\n\n
Second, the rotations.<\/p>\n\n\n\n