Zhuming Bi - Computer Aided Design and Manufacturing

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 2.7The 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 PLin an arbitrary direction can be formed by three known points P 1, P 2, and P 3. More generically, it can also be described as

(2.3)

where a 1, b 1, c 1 , and d 1are 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 2.8The 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.

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 2.1shows 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 2.1Coordinate 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( d x, d y, d z) to a new point P ′( x′ , y′ , z′ ).
Scale	In case of scaling, every coordinate value of P( x , y , z ) is multiplied by a constant. If the constants are the same along three axes, this corresponds to a uniform scaling (i.e. C x= C y= C z). Otherwise, it is a non‐uniform scaling .
Rotation	A rotation refers to the rotation around a specified axis with an angle (i.e. θ x, θ y, or θ zalong the x , y , and z axes, respectively). A generic rotation along a specific axis can be decomposed as a series of aforementioned rotations.
Mirror	The mirror of an object is defined with respect to a reference plane, i.e. O‐YZ, O‐XZ, and O‐XY planes, respectively.
Projection	The transaction for projection computes the coordinates P′( x ′, y ′, z ′) of a point P( x , y , z ) projected on a plane with a distance d to the observer.

A point only includes positional data while an object includes both positional and orientational data in space. In addition, it may be convenient to represent the position and orientation by a local coordinate system (LCS) attached to an object. Figure 2.9shows such a CS, which is called an object coordinate system ( O b− X b Y b Z b).