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# Geometry for Modeling and Design

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## Geometric Entities

### Points

Points are geometric constructs. Points are considered to have no width, height, or depth. They are used to indicate locations in space. In CAD drawings, a point is located by its coordinates and usually shown with some sort of marker like a cross, circle, or other representation. Many CAD systems allow you to choose the style and size of the mark that is used to represent points.

Most CAD systems offer three ways to specify a point:

• Type in the coordinates (of any kind) for the point (see Figure 4.13). 4.13 Specifying Points. Point 1 was added to the drawing by typing the absolute coordinates 3,4,7. Point 2 was added relative to Point 1 with the relative coordinates @2,2,2.

• Pick a point from the screen with a pointing device (mouse or tablet).

• Specify the location of a point by its relationship to existing geometry (e.g., an endpoint of a line, an intersection of two lines, or a center point).

Picking a point from the screen is a quick way to enter points when the exact location is not important, but the accuracy of the CAD database makes it impossible to enter a location accurately in this way.

### Lines

A straight line is defined as the shortest distance between two points. Geometrically, a line has length but no other dimension such as width or thickness. Lines are used in drawings to represent the edge view of a surface, the limiting element of a contoured surface, or the edge formed where two surfaces on an object join. In a CAD database, lines are typically stored by the coordinates of their endpoints.

For the lines shown in Figure 4.14, the table below shows how you can specify the second endpoint for a particular type of coordinate entry. (For either or both endpoints, you can also snap to existing geometry without entering any coordinates.) 4.14 Specifying Lines. (a) This 2D line was drawn from endpoint (3,2) to (6,6). (b) This 3D line was drawn from endpoint (3,2,0) to (5,4,6).

### Planes

Planes are defined by any of the following (see Figure 4.15):

• Three points not lying in a straight line

• Two parallel lines

• Two intersecting lines

• A point and a line 4.15 Defining a Plane. The highlighted entities in each image define a plane.

The last three ways to define a plane are all special cases of the more general case—three points not in a straight line. Knowing what can determine a plane can help you understand the geometry of solid objects and use the geometry as you model in CAD.

For example, a face on an object is a plane that extends between the vertices and edges of the surface. Most CAD programs allow you to align new entities with an existing plane. You can use any face on the object—whether it is normal, inclined, or oblique—to define a plane for aligning a new entity.

Defining planes on the object or in 3D space is an important skill for working in 3D CAD. The software provides tools for defining new planes (see Figure 4.16). The options for these tools are based on the geometry of planes, as defined in the preceding list. Typical choices allow the use of any three points not in a line, two parallel lines, two intersecting lines, a point and a line, or being parallel to, perpendicular to, or at an angle from an existing plane. 4.16 Defining a Plane in CAD. A point and a line (the edge between two surfaces in this case) were used to define a plane in this Pro/ENGINEER model.

A plane may serve as a coordinate-system orientation that shows a surface true shape. You will learn more about orienting work planes to take advantage of the object’s geometry later in this chapter.

### Circles

A circle is a set of points that are equidistant from a center point. The distance from the center to one of the points is the radius (see Figure 4.17). The distance across the center to any two points on opposite sides is the diameter. The circumference of a circle contains 360° of arc. In a CAD file, a circle is often stored as a center point and a radius. 4.18 AutoCAD Circle Construction Options (Autodesk screen shots reprinted courtesy of Autodesk, Inc.)

Most CAD systems allow you to define a circle by specifying any one of the following:

• the center and a diameter

• the center and a radius

• two points on the diameter

• three points on the circle

• a radius and two entities to which the circle is tangent

• three entities to which the circle is tangent.

These methods are illustrated in Figure 4.19.

### Arcs

An arc is a portion of a circle. An arc can be defined by specifying any one of the following (see Figure 4.20):

• a center, radius, and angle measure (sometimes called the included angle or delta angle)

• a center, radius, and chord length

• a center, radius, and arc length

• the endpoints and a radius

• the endpoints and a chord length

• the endpoints and arc length

• the endpoints and one other point on the arc (3 points) 4.20 Defining Arcs. Arcs can be defined many different ways. Like circles, arcs may be located from a center point or an endpoint, making it easy to locate them relative to other entities in the model. 4.21 AutoCAD Arc Construction Options (Autodesk screen shots reprinted courtesy of Autodesk, Inc.)