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Spatial Coordinates Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Cartesian Three-space

An extension of rectangular coordinates into three dimensions is Cartesian three-space (Fig. 6-2), also called xyz-space . The independent variables are usually plotted along the x and y axes; the dependent variable is plotted along the z axis. Each axis is perpendicular to the other two. They all intersect at the origin , which is usually the point where x = 0, y = 0, and z = 0.

Three-Space and Vectors Spatial Coordinates Cartesian Three-space

Fig. 6-2. Cartesian three-space, also called xyz -space.

The scales in Cartesian three-space are all linear. This means that, along any given individual axis, equal distances represent equal changes in value. But the divisions (that is, the spaces between hash marks) on different axes do not necessarily have to represent the same increments. For example, the x axis might be designated as having 1 unit per division, the y axis 10 units per division, and the z axis 5 units per division.

Points in Cartesian three-space are represented by ordered triples ( x , y , z ). As with ordered pairs, there are no spaces between the variables and the commas when denoting an ordered triple; they’re all scrunched up together.

Cylindrical Coordinates

Figure 6-3 shows two systems of cylindrical coordinates for specifying the positions of points in three-space.

In the system shown in Fig. 6-3A, we start with Cartesian xyz -space. Then an angle θ is defined in the xy -plane, measured in degrees or radians (usually radians) counterclockwise from the positive x axis or reference axis. Given a point P in space, consider its projection P′ onto the xy -plane, such that line segment PP′ is parallel to the z axis. The position of P is defined by the ordered triple ( θ,r,h). In this ordered triple, θ represents the angle measured counterclockwise between P′ and the positive x axis in the xy -plane, r represents the distance or radius from P′ to the origin, and h represents the distance (altitude or height) of P above the xy -plane. This scheme for cylindrical coordinates is preferred by mathematicians, and also by some engineers and scientists.

Three-Space and Vectors Spatial Coordinates Cylindrical Coordinates

Fig. 6-3. (A) Mathematician’s form of cylindrical coordinates for defining points in three-space. (B) Astronomer’s and navigator’s form of cylindrical coordinates for defining points in three-space.

In the system shown in Fig. 6-3B, we again start with Cartesian xyz -space. The xy -plane corresponds to the surface of the earth in the vicinity of the origin, and the z axis runs straight up (positive z values) and down (negative z values). The angle θ is defined in the xy -plane in degrees (but never radians) clockwise from the positive y axis, which corresponds to geographic north. Given a point P in space, consider its projection P′ onto the xy -plane, such that line segment PP′ is parallel to the z axis. The position of P is defined by the ordered triple ( θ,r,h), where θ represents the angle measured clockwise between P′ and geographic north, r represents the distance or radius rom P′ to the origin, and h represents the distance (altitude or height) of P above the xy -plane. This scheme is preferred by navigators and astronomers.

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