Читать книгу Computer Aided Design and Manufacturing - Zhuming Bi - Страница 39

Any geometric element in a three‐dimensional space has six degrees of freedom (DOF). To describe the spatial relation between two geometric elements in solids, a reference coordinate system must be established. Figure 2.3 shows three most commonly used coordinate systems (CSs), i.e. a Cartesian coordinate system, cylindrical coordinate system, and spherical coordinate system.

Figure 2.3 Three commonly used coordinate systems. (a) Cartesian coordinate system. (b) Cylindrical coordinate system. (c) Spherical coordinate system.

A Cartesian CS consists of three axes that are mutually perpendicular to each other. A position in the Cartesian CS is defined by its distances to the origin (x, y, z) projected on three axes (X, Y, Z). A cylindrical CS consists of two linear axes (X and Z) and one rotational axis. Correspondingly, a position in the cylindrical CS is defined by two scalar variables and one angular variable, i.e. (r, θ, z). A spherical CS consists of two rotational axes and one translational axis. The position of a point in the spherical CS is specified by three variables: the radial distance (R) of that point from a fixed origin, its polar angle (α) measured from a fixed zenith direction, and the azimuth angle (β) of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane, (R, α, β).

A user of a CAD system should always be aware that a reference CS is essential for geometric modelling of objects in computers. As shown in Figure 2.4, any commercial CAD software comes with a default coordinate system in modelling templates. The default coordinate system is usually a Cartesian CS. In Figure 2.4, the default Cartesian CS is O – XYZ and the planes of O – XY, O – XZ, O – YZ are defined as a default front plane, top plane, and right plane, respectively.

Figure 2.4 Default Cartesian CS in a CAD system.