- Coordinates for 3D CAD Modeling
- Geometric Entities
- 4.1 Manually Bisecting a Line or Circular Arc
- 4.2 Drawing Tangents to Two Circles
- 4.3 Drawing an Arc Tangent to a Line or Arc and Through a Point
- 4.4 Bisecting an Angle
- 4.5 Drawing a Line through a Point and Parallel to a Line
- 4.6 Drawing a Triangle with Sides Given
- 4.7 Drawing a Right Triangle with Hypotenuse and One Side Given
- 4.8 Laying Out an Angle
- 4.9 Drawing an Equilateral Triangle
- 4.10 Polygons
- 4.11 Drawing a Regular Pentagon
- 4.12 Drawing a Hexagon
- 4.13 Ellipses
- 4.14 Spline Curves
- 4.15 Geometric Relationships
- 4.16 Solid Primitives
- 4.17 Recognizing Symmetry
- 4.18 Extruded Forms
- 4.19 Revolved Forms
- 4.20 Irregular Surfaces
- 4.21 User Coordinate Systems
- 4.22 Transformations
- Key Words
- Chapter Summary
- Skills Summary
- Review Questions
- Chapter Exercises

## 4.22 Transformations

A 3D CAD package uses the default Cartesian coordinate system to store information about the model. One way it may be stored is as a matrix (rows and columns of numbers) representing the vertices of the object. Once the object is defined, the software uses mathematical methods to transform the matrix (and the object) in various ways. There are two basic kinds of transformations: those that transform the model itself (called geometric transformations) and those that merely change the view of the model (called viewing transformations).

### Geometric Transformations

The model stored in the computer is changed using three basic transformations (or changes): moving (sometimes called translation), rotating, and scaling. When you select a CAD command that uses one of these transformations, the CAD data stored in your model are converted mathematically to produce the result. Commands such as Move (or Translate), Rotate, and Scale transform the object on the coordinate system and change the coordinates stored in the 3D model database.

Figure 4.78 shows a part after translation. The model was moved over 2 units in the X-direction and 3 units in the Y-direction. The corner of the object is no longer located at the origin of the coordinate system.

**4.78** Translation. *This model has been moved 2 units in the X-direction and 3 units in the Y-direction.*

Figure 4.79 illustrates the effect of rotation. The rotated object is situated at a different location in the coordinate system. Figure 4.80 shows the effect of scaling. The scaled object is larger dimensionally than the previous object.

**4.79** Rotation. *This model has been rotated in the X-Y plane.*

**4.80** Scaling. *This model has been scaled to 1.5 times its previous size.*

### Viewing Transformations

A viewing transformation does not change the coordinate system or the location of the model on the coordinate system; it simply changes your view of the model. The model’s vertices are stored in the computer at the same coordinate locations no matter the direction from which the model is viewed on the monitor (Figure 4.81).

**4.81** Changing the View. *Note that the location of the model relative to the coordinate axes does not change in any of the different views. Changing the view does not transform the model itself.*

Although the model’s coordinates do not change when the view does, the software does mathematically transform the model database to produce the new appearance of the model on the screen. This viewing transformation is stored as a separate part of the model file (or a separate file) and does not affect the coordinates of the stored model. Viewing transformations change the view on the screen but do not change the model relative to the coordinate system.

Common viewing transformations are illustrated in Figure 4.82. Panning moves the location of the view on the screen. If the monitor were a hole through which you were viewing a piece of paper, panning would be analogous to sliding the piece of paper to expose a different portion of it through the hole. Zooming enlarges or reduces the view of the objects and operates similar to a telephoto lens on a camera. A view rotation is actually a change of viewpoint; the object appears to be rotated, but it is your point of view that is changing. The object itself remains in the same location on the coordinate system.

**4.82** Common View Transformations. *Panning moved the view of the objects in (a) to expose a different portion of the part in (b). In (c), the view is enlarged to show more detail. In (d), the view is rotated to a different line of sight. In each case, the viewing transformation applies to all the objects in the view and does not affect the location of the objects on the coordinate system. (Notice that the position relative to the coordinate system icon does not change.)*

Viewing controls transform only the viewing transformation file, changing just your view. Commands to scale the object on the coordinate system transform the object’s coordinates in the database.

Examine the six models and their coordinates in Figure 4.83. Which are views that look different because of changes in viewing controls? Which look different because the objects were rotated, moved, or scaled on the coordinate system?

**4.83** Geometric or Viewing Transformation? *Three of these models are the same, but the viewing location, zoom, or rotation has changed. Three have been transformed to different locations on the coordinate system.*

You will use the basic geometric shapes and concepts outlined in this chapter to build CAD models and create accurate freehand sketches. The ability to visualize geometric entities on the Cartesian coordinate system will help you manipulate the coordinate system when modeling in CAD.