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Analytic Geometry is the study of geometry using a coordinate system. It is important for many scientific fields, such as physics and aviation; it is also the foundation of every video game.


Introduction


There is a lot more to analytic geometry than what I include in this text. As I am a programmer with an interest in video games, I have focused this text primaily on functions of analytic geometry that are important to computer graphics and video game development.


Coordinate Systems


In analytic geometry we generally use the Cartesian coordinate system to study geometric properties in either two dimensions (the Euclidean plane) or in three dimensions (Euclidean space). These are often refered to as simply 2-D and 3-D or 2-space and 3-space (the latter are more often used in science.) Analytic geometry, of course, doesn't limit itself to these two systems, as there are a variety of other coordinate systems, as well as more than just three dimensions. For the sake of the discussion here, we will limit ourselves to these two systems.


In two dimensions, the Cartesian coordinate system is composed of a single, flat plane with two perpendicular (orthogonal) axes: x and y. The x-axis generally represents the horizontal and the y-axis the vertical. The actual orientation of the axes depends on the situation in which it is used; for example, in screen space (on a computer screen, that is), the y-axis is inverted (i.e., it points down). It is not uncommon to be given a coordinate system with different labels for the axes; for instance, planar systems are often given representations such as (i, j) or (a, b). In any case, it is assumed that the axes are in the same order.

Euclidean Plane
2-D Plane

In three dimensions, the Cartesian coordinate system is composed of three perpendicular (orthogonal) axes: x, y, and z. The x-axis generally represents the horizontal, the y-axis the vertical, and the z-axis depth. Again, the actual orientation of the axes depends on the situation. In spacial systems, we will often encounter coordinates given representations such as (a, b, c) or (i, j, k).

Euclidean Space
3-D Space


  Analytic Geometry: Points