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2.2.3 Coordinate Transformation of Points
ОглавлениеWhen 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′). | ||
| 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. Cx = Cy = Cz). Otherwise, it is a non‐uniform scaling. | ||
| Rotation | A rotation refers to the rotation around a specified axis with an angle (i.e. θx, θy, or θz along 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. |
