Chapter Four Planes

ObjectivesUpon completion of this chapter the student will be able to:

• Describe the difference between a surface and a plane.• List the four ways that the boundaries of a plane may be defined.• List the three types of planes encountered in descriptive geometry.• Use AutoCAD to determine if two objects reside on the same plane.• Determine the location of a point on a plane.• Determine the location of a line on a plane.• Locate the piercing point of a line and a plane using the auxiliary view and

cutting plane methods.

• Use the cutting plane method in AutoCAD to determine the intersection of two planes.

• Locate the piercing point using the point command and app osnap option.• Locate the intersection of two planes.• Determine the angle between two planes.

Defining a PlaneThere are four ways in which a plane’s boundaries

can be secured.

• a line and a point• three points• two intersecting lines• two parallel lines.

Line and a Point

• The line and the point must both reside on the plane that they are defining. • The point cannot reside on the line. If the point is on the line then only one

possible boundary can be established.

Line and a Point Example

Three Points

• All three points must reside on the plane that they are defining.

• It must not be possible to connect all three points with a single, straight, line segment.

Three Points Example

Two Intersecting LinesTwo intersecting lines or line segments drawn at any length and intersecting at any angle may be used to define the boundaries of a plane if both lines reside on the plane that they are defining.

Two Intersecting Lines Example

Two Parallel Lines

To define a plane using two parallel lines, both line segments must reside on the plane that they are defining.

Two Parallel Line Example

Normal Planes

• If a plane is parallel to one of the three principal planes of projection (frontal, horizontal, profile) then the plane is referred to as a normal plane.

• There are three subcategories of normal planes: horizontal, frontal, and profile.

Normal Plane Example

Inclined Plane• When a plane is not classified as a normal

plane but appears as a line (edge view) in one of the principal views and distorted in the remaining two, then this plane is known as an inclined plane.

• The true shape and size of an inclined plane can only be found by constructing one or more auxiliary views

Inclined Plane Example

Oblique Planes

• When a plane does not appear in edge view in any of the three principal views then it is known as an oblique plane.

• The true shape and size of an oblique plane can only be determined in one or more auxiliary views

Oblique Planes Example

XY Plane• When an object is created in AutoCAD, it is

created on the XY-plane. • Theoretically, the XY-plane extends

indefinitely in both directions along the X- and Y-axes, but does not contain a thickness. Without a thickness an infinite number of XY-planes could be stacked upon one another.

Sketch Plane

• A sketch plane, in Mechanical Desktop, is one that extends infinitely in two directions and is used to construct the two-dimensional outline of a part’s features. The orientation of the X- and Y-axis on the sketch plane

Locating Point on a Plane

• A point will either coexist with a plane or outside the boundaries of the plane (above or below an established plane).

• When a plane containing a point is projected into a view where the plane appears as a line, then the point contained on that plane will appear as a point on the line.

Locating Lines on a Plane• The principles, techniques, and theories

discussed in the previous section regarding the location of points on a plane can also be applied when transferring the location of a line from one view to another.

Locating the Piercing Point of a Line and a Plane

• This point can either fall within or somewhere outside the established boundaries of the plane.

• If a line does not coexist on the plane or is parallel to that plane, then a piercing point will exist; because in theory a line extends indefinitely in both directions unless a boundary has been established.

Locating the Intersection of Two Planes

• When two or more planes intersect, they will share more than one common point.

• In fact, they will share a series of points that may be connected by a straight-line (line of intersection), and every point along that line will be common to both planes.

Determining the Angle between Two Planes

• When two planes intersect, they form an angle known as the dihedral angle (see Figure 4–57).

• This angle appears in true size in any view where the two planes appear in edge view, or any view in which the line of intersection appears as a point

Determining the Angle between Two Planes Example