front plane, top plane, and right plane, respectively.
Figure 2.4 Default Cartesian CS in a CAD system.
2.2.2 Reference Points, Lines, and Planes
Other than the references of coordinate systems, other elements such as points, lines, and planes can also be used as the references in geometric modelling. Moreover, geometric elements at a high level can be represented by a number of points at the lowest level. Figure 2.5 shows a point in the reference CS that is completely defined by giving its three coordinate values (x, y, z) along x‐, y‐, and z‐axes. For a given geometric element, the same amount of information can be defined in many different ways. For example, a point P is relevant to the high‐level elements such as lines or surfaces in the sense that a point can be an end‐point of these elements. Therefore, a reference point can be defined directly based on its relations to existing geometric elements.
Figure 2.5 The definition of a point from existing reference geometries. (a) Arc centre. (b) Centre of face. (c) Intersection of two lines. (d) Projection of a vertical line on a plane. (e) Interaction of a line and plane. (f) Interaction of three planes.
Figure 2.5 shows that a point can be defined in many different ways: Figure 2.5a to f shows that the point is defined as the centre of an arc, the centre of a face, the interaction of two lines that interact, the projected points of a vertical line on a plane, the intersection of a line and a plane, and the intersection of three planes, respectively.
Mathematically, a point P in the Cartesian coordinate system can be represented as
(2.1)
where x, y, and z are the coordinates and i, j, and k are the unit vectors along the axes of X, Y, and Z, respectively.
As shown in Figure 2.6b, a line L in the Cartesian coordinate system can be represented as the line segment connected by two points P1 (x1, y1, z1) and P2 (x2, y2, z2) as
(2.2)
where v is the vector along the line determined by P1 and P2, t is an independent variable, and P(x, y, z) is an arbitrary point on the line.
Figure 2.6 Using points to represent lines and planes. (a) Point. (b) Line. (c). Plane.
Similar to the definitions of a reference point in Figure 2.5, a reference line can be defined directly based on its relationship to existing geometric elements. Figure 2.7a to d shows that the line is defined as the connection between two end‐points, the intersecting lines of two faces, the rotational axis of a cylindrical surface, and the line passing through a point and perpendicular to a plane, respectively.
Figure 2.7 The definition of a reference line from existing geometric elements. (a) Line by two points. (b) Line by two interacting planes. (c) Line by a cylindrical face. (d) Line by passing a given point perpendicular to a plane.
As shown in Figure 2.6c, a plane PL in an arbitrary direction can be formed by three known points P1, P2, and P3. More generically, it can also be described as
(2.3)
where a1, b1, c1, and d1 are constant coefficients and (x, y, z) is an arbitrary point on PL.
Similar to the definitions of a reference line in Figure 2.7, a reference plane be defined directly based on its relationship to existing geometric elements. Figure 2.8a to d shows that the reference plane can be defined by specifying three points on the plane, one point and a line, two lines, and a distance to an existing paralleled plane, respectively.
Figure 2.8 The definition of a reference plane from existing geometric elements. (a) Plane by three points. (b) Plane by a through point and line. (c) Plane by two parallel or interacting lines. (d) Plane by an offset of an existing plane.
2.2.3 Coordinate Transformation of Points
When an object consists of multiple geometric elements, the position and orientation of each element affect the geometry of the object. To place an element at the correct position and orientation, the coordinate transformation is often required. Table 2.1 shows the common types of coordinate transformation for a point. The coordinate transformation is performed point by point and the common coordinate transformations include translation, scaling, rotation, mirroring, and projection. In Table 2.1, the second column gives the explanations of these coordination transformations, and the third column gives the mathematical representation and graphic illustration of each type of transformation.
Table 2.1 Coordinate transformation of a point.
| Transformation | Features | Illustration | |
| Translation | A translation is the simplest transformation and is the translation when the point P (x, y, z) is moved by the vector d(dx, dy, dz) to a new point P′(x′, y′, z′). |
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