- Objectives
- Overview
- 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
- Worksheets
- Review Questions
- Chapter Exercises
This chapter is from the book
This chapter is from the book
This chapter is from the book
4.13 Ellipses
An ellipse can be defined by its major and minor axis distances. The major axis is the longer axis of the ellipse; the minor axis is the shorter axis. Some ellipses are shown and labeled in Figure 4.38.
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4.38 Major and Minor Axes of Some Ellipses
An ellipse is created by a point moving along a path where the sum of its distances from two points, each called a focus of an ellipse (foci is the plural form), is equal to the major diameter. As an aid in understanding the shape of an ellipse, imagine pinning the ends of a string in the locations of the foci, then sliding a pencil along inside the string, keeping it tightly stretched, as in Figure 4.39. You would not use this technique when sketching, but it serves as a good illustration of the definition of an ellipse.
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4.39 Pencil and String Method. When an ellipse is created with the pencil-and-string method, the length of the string between the foci is equal to the length of the major axis of the ellipse. Any point that can be reached by a pencil inside the string when it is pulled taut meets the condition that its distances from the two foci sum to the length of the major diameter.
Most CAD systems provide an Ellipse command that lets you enter the major and minor axis lengths, center, or the angle of rotation for a circle that is to appear elliptical.
SPOTLIGHT
Locating the Foci of an Ellipse
To locate the foci of an ellipse, draw arcs with their centers at the ends of the minor axis and their radii equal to half the major axis. The intersection of each pair of arcs is a focus of the ellipse.
SPOTLIGHT
The Perimeter of an Ellipse
The perimeter, P, of an ellipse is a set of points defined by their distance from the two foci. The sum of the distances from any point on the ellipse to the two foci must be equal to the length of the major diameter. The perimeter of an ellipse may be approximated in different ways. Many CAD packages use infinite series to most closely approximate the perimeter. The mathematical relationship of each point on the ellipse to the major and minor axes may be seen in the approximation of the perimeter at right:
