Coordinate Geometry

Creating a three-dimensional (3D) model of an object in a computer involves a coordinate system and algebraic calculations necessary to manipulate (change shape, scale, orientation, and location) the modeled object. Every spatial object, such as a 3D model of a house, is based on coordinates referenced to a specific coordinate system. The most popular system of coordinates used in almost every 3D modeling software is the Cartesian coordinate system, described by three coordinates, x, y, and z, corresponding to the three dimensions of space. In the Cartesian coordinate system, all three axes are perpendicular to each other, and the system is defined by its origin (0, 0, 0) and the orientation of the x, y, and z axes. The simplest way of referencing one object to another object is when the axes of the first object are parallel to the axes of the second object. Two other conditions still must be met. First, one needs to know the exact position between the origins of two objects. Second, one also needs to know the scale or rather the distance unit that was used in both the coordinate systems. If, for example, one house is modeled using feet and the other house is modeled using meters, then the houses will be positioned wrongly in relation to each other, and one house will be approximately three times smaller than the other house since 1 meter equals approximately 3.28 feet (Figure 1).

Projecting geographic objects such as land masses or water bodies from a 3D model of the [Page 578]Earth to a 2D map is more complex than translating a 3D object from one coordinate system to another because the shape of the Earth is rather irregular compared with well-defined shapes such as a sphere or a cube. But as in the process of referencing two 3D objects, projecting geographic objects from a 3D model to a 2D map involves a coordinate system that acts as a referencing system, allowing one to translate the shape, size, and location of objects, albeit with some compromises, called distortions. Maps describe the surface of the Earth on a flat, 2D plane, which cannot be done without distortions. The distortions in shape, size, location, and, in consequence, in the angle and direction of lines, increase dramatically when the size of the mapped area exceeds a certain limit. As soon as the model grows and covers entire countries or a continent, the distortions rise to a significant problem, since the referencing between the real world and the modeled world gets rather imprecise.

Figure 1 Scale and distance unit implications of a coordinate system for 3D models

Source: Author.

Figure 2 Geocentric coordinate system

Source: Figure created by Peter Skotte, Centre for 3D GeoInformation, Aalborg University.

There is a fairly simple and obvious solution for this problem as long as one wants to reference the real world to a virtual model, also called a model map. In a computer, there is a possibility of describing the world in all three geometric dimensions without distortions caused by moving objects from 3D to 2D representation. Global positioning systems are doing it already, and virtual globe applications are getting there. The coordinate systems that enable modeling geographic entities in 3D are called geocentric coordinate systems, because they have their origin in the center of the globe (Figure 2). Contemporary modeling software does not use [Page 579]a geocentric coordinate system, but there are algorithms that can convert from one coordinate system to another.

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