Types of Coordinate Systems for 2D and 3D CAD Modeling/Drawings
Computer-aided design (CAD) drawings are produced and stored in relationship to a coordinate system, especially Cartesian coordinate system, polar coordinate system, cylindrical coordinate system, and spherical coordinate system.
Regardless of the CAD software system that will be used in 2D and 3D CAD modeling or to produce CAD drawings, it is important to understand the basics of the most widely used coordinate systems in CAD softwares.
Majority of CAD systems use the right hand rule, as applied to coordinate systems. Although it is quite rare, some CAD systems use the left-hand rule.
To get a clear illustration of the right-hand rule, do this: as shown in Figure 1 below, stretch the thumb of your right hand—to represent the direction of the positive x-axis; stretch the index finger of your right hand—to represent the direction of the positive y-axis; and stretch the other remaining fingers of your right hand to represent the direction of the z-axis.
Figure 1: The right hand rule, as applied in coordinate systems (Source: Technical Drawing with Engineering Graphics, 15th edition.)
The right hand rule is also related to the popular Cartesian coordinate system which can be used to express drawings in 2D (x, y) and 3D (x, y, z). When using a computer for 2D (two-dimensional) or 3D (three-dimensional) modeling, the face of your computer screen represents the 2D or x–y plane, and the z-axis represents the axis pointing directly towards you, as indicated in Figure 2 below.
Figure 2: Z-axis pointing towards your direction, as computer screen face represents the 2D or x–y plane. (Source: Technical Drawing with Engineering Graphics, 15th edition.)
Two-dimensional (2D) CAD systems use only the x– and y– coordinates of the Cartesian coordinate system, while 3D CAD systems use the x-, y-, and z– coordinates. When representing a 2D system in a 3D CAD system, the line of view is along the z-axis. Figure 3 below shows an orthographic view or 2D drawing produced with only x and y values, with the z-coordinate set at 0.
Figure 3: Computer screen face showing a CAD drawing in 2D or x–y plane. (Source: Technical Drawing with Engineering Graphics, 15th edition.)
Orthographic views show only two of three coordinate directions, with the line of view generally considered to be along one axis—usually the z-axis. Two-dimensional CAD drawings are the same: they are produced with and represented by x and y coordinates, while the z-axis is the line of view.
As stated, in a CAD system, the 2D (x–y) plane is aligned with the computer screen, while the z-axis is pointing horizontally and directly towards the person using the computer. However, in machining and many other applications, the z-axis is regarded as the vertical axis. Regardless of the name given to an axis, the coordinate axes (x, y, and z) must be perpendicular to each other (mutually perpendicular).
It is more important to understand how to use axes or coordinates in a model/drawing than to name the direction of default axes and planes. As shown in Figure 4 below, the structure of a 3D object is identified by its x, y, and z coordinates, with the location (0, 0, 0) taken as the starting point from which other points are plotted.
Figure 4: The coordinates for a 3D drawing (Source: Technical Drawing with Engineering Graphics, 15th edition.)
Coordinate systems/formats used to specify locations or points
Although 2D and 3D models/drawings are stored in a single Cartesian coordinate system, a CAD user may be drawn into a situation whereby they would be required to specify locations of some features using other coordinate systems.
The most distinctive of these CAD geometry coordinate systems or location methods are absolute coordinates, relative coordinates, polar coordinates, cylindrical coordinates, and spherical coordinates:
- Absolute coordinates are locations or points that are at a distance from a common point of origin in a Cartesian System. Locations are established using values on the x-, y-, and even z– axes. For example, in Figure 4 above, the absolute coordinate (3.5, 8, 4) represents a location that is 3.5 units away from the x-axis origin (0), 8 units away from the y-axis origin (0), and 4 units away from the z-axis origin (0). In other words, we can say that the location is (3.5, 8, 4) away from the origin (0, 0, 0). In Figure 5 below, the point of origin (point B) is (0, 0) and there are five absolute coordinates located away from the point of origin (0, 0): A (0, ─4), C (─6, 3), D (6, 4), E (2, ─2), and F (─3, ─5). Generally, absolute coordinates express the position/location of the points of an object with respect to an origin of a given coordinate system.
Figure 5: Absolute coordinates in 2D (Source: Siyavula)
- Relative coordinates are locations that are expressed in terms of their relative distances away from a reference point that is not the point of origin. Instead of specifying a location from the actual point origin, a relative coordinate can be used to specify a location in terms of the location’s distance away from a previous location.
- Polar coordinates are 2D coordinate systems in which individual locations or points are defined in terms of an angle (in degrees) and distance away from an axis—the axis could be any of x, y, or z Polar coordinates are absolute if they express a location or point in terms of its angle away from an axis and distance from the origin; on the other hand, polar coordinates are relative if they express a location in terms of its angle and distance away from another location that is not the origin.
Figure 6: Polar coordinate (Source: Geogebra)
- Cylindrical coordinates are locations that are specified in terms of a radius (r), an angle (θ), and distance a (z) which is usually in the z-axis direction. Any values attached to these terms are relevant for conveying information about locations or points that are on the edge of a cylinder. The radius tells the distance of the point from the center (or origin); the angle expresses the angular inclination of the point away from an axis (for instance, the x-axis shown in Figure 7 below) along which the point is located; and the distance expresses the height of the point on a cylinder. The difference between cylindrical coordinates and polar coordinates is that cylindrical coordinates include a height distance in the z-direction.
Figure 7: Cylindrical coordinate (Source: Mathinsight.org)
- Spherical coordinates are locations that are expressed by a radius (ρ), an angle (θ) from the x-axis, and another angle () from a 2D (example: either y–z or x–y) plane. Spherical coordinates express the position of a point on a sphere, with the origin of the coordinate system at the center of the sphere, and the radius indicating the size of the sphere.
Figure 8: Spherical coordinate (Source: Millersville.edu)