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# Geometry for Modeling and Design

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## 4.15 Geometric Relationships

When you are sketching, you often imply a relationship, such as being parallel or perpendicular, by the appearance of the lines or through notes or dimensions. When you are creating a CAD model you use drawing aids to specify these relationships between geometric entities.

Two lines or planes are parallel when they are an equal distance apart at every point. Parallel entities never intersect, even if extended to infinity. Figure 4.48 shows an example of parallel lines. 4.48 The highlighted lines are parallel.

Two lines or planes are perpendicular when they intersect at right angles (or when the intersection that would be formed if they were extended would be a right angle), as in Figure 4.49. 4.49 The highlighted lines are perpendicular.

Two entities intersect if they have at least one point in common. Two straight lines intersect at only a single point. A circle and a straight line intersect at two points, as shown in Figure 4.50. 4.50 The highlighted circle intersects the highlighted line at two different points.

When two lines intersect, they define an angle as shown in Figure 4.51. 4.51 An angle is defined by the space between two lines (such as those highlighted here) or planes that intersect.

The term apparent intersection refers to lines that appear to intersect in a 2D view or on a computer monitor but actually do not touch, as shown in Figure 4.52. When you look at a wireframe view of a model, the 2D view may show lines crossing each other when, in fact, the lines do not intersect in 3D space. Changing the view of the model can help you determine whether an intersection is actual or apparent. 4.52 Apparent Intersection. From the shaded view of this model in (a), it is clear that the back lines do not intersect the half-circular shape. In the wireframe front view in (b), the lines appear to intersect.

Two entities are tangent if they touch each other but do not intersect, even if extended to infinity, as shown in Figure 4.53. A line that is tangent to a circle will have only one point in common with the circle. 4.53 Tangency. Lines that are tangent to an entity have one point in common but never intersect. 3D objects may be tangent at a single point or along a line.

When a line is tangent to a circle, a radial line from the center of the circle is perpendicular at the point of tangency, as shown in Figure 4.54. Knowing this can be useful in creating sketches and models. 4.54 A radial line from the point where a line is tangent to a circle will always be perpendicular to that line.

The regular geometry of points, lines, circles, arcs, and ellipses is the foundation for many CAD drawings that are created from these types of entities alone. Figure 4.55 shows a 2D CAD drawing that uses only lines, circles, and arcs to create the shapes shown. Figure 4.56 shows a 3D wireframe model that is also made entirely of lines, circles, and arcs. Many complex-looking 2D and 3D images are made solely from combinations of these shapes. Recognizing these shapes and understanding the many ways you can specify them in the CAD environment are key modeling skills. 4.55 A 2D Drawing Made of Only Lines, Circles, and Arcs 4.56 A 3D Model Made of Only Lines, Circles, and Arcs