Engineering Graphics, Class 13Descriptive Geometry

Mohammad I. KilaniMechanical Engineering Department

University of Jordan

Projecting a line into other views

Given the front and right side projections for a line a-b, the projection of line a-bin the top view is found by using the alignment rule and the skip-a-view rule on its endpoints.

Extend projection lines into the top view from the end points in the front view.

Find the distances x and y from the line end points in the right side view to the fold line F/R, and transfer them into the top view.

Finding the true length of a line

The true length of a line can be found using the true length rule. A line will appear in its true length in an auxiliary view taken such that that the fold line is parallel to the current projection.

Note that the parallel fold line generates the maximum possible length of the projected line, which is the true length of the line.

Constructing a point view of a line

A point view of a line is useful in finding the distances between lines and points and distances between lines and lines.

Using the point view rule, a point view of a line is found in a view generated by a fold line drawn perpendicular to the true length line.

Finding the distance between a line and a point in space

The true distance between a line and a point in space will be evident in a view that shows a point view of the line.

Finding the true distance between two parallel lines

If two lines are parallel, their projections will be parallel in all views.

To find the true distance between two parallel lines, a view is needed to show the end view (point view) of both lines simultaneously. In this view the real distance between the lines will be apparent

Finding the true distance between two nonparallel (skewed) lines

If two lines are not parallel, they will appear nonparallel in at least one view.

To determine the true distance between two nonparallel lines, find a point view of one of the lines. In this view the real distance between the lines will be apparent.

Note that if the apparent points of intersections align in two views, then the lines actually intersect.

Projecting a plane into another view

A plane may be located by three non-collinear points on it.

To transfer a plane into another view, find three points on the plane, and project them into the desired view using the skip-a-view rule.

Constructing an edge view of a plane surface

If the end view (point view) of a line in the plane is shown in a view, then this view shows the edge view of the plane.

To secure a true length line on the plane, a line may be added on the plane in any view, arranged to be parallel to an adjoining folding line, and having its termination at the plane boundaries.

Projecting the added line terminations to the corresponding boundaries in the adjoining view relocated the added line in that view. Moreover, it will be a true length line, as it was made to be parallel to the fold line in the proceeding view.

Finding the true distance between a plane surface and a point

Finding the true distance between a given point and a plane in space requires a view that shows the edge view of the plane, and the point in the same view.

The distance from the point to the plane is the path perpendicular to the plane, and passing through the point.

Finding the true angle between two planes (dihedral angle)

The edge of intersection between two planes is a line that is common to both planes. The end view of that line will provide the edge view of both planes simultaneously, and the angle between them will be evident.

Determining the visibility of lines

To determine which line of an apparent intersection of two lines is closer to the viewer, project the exact crossover point of the lines to the adjoining view.

In the adjoining view, determine which of the lines is closer to the folding line. This line is in front of the other line in the first view.

Determining the visibility of lines

To determine which line of an apparent intersection of two lines is closer to the viewer, project the exact crossover point of the lines to the adjoining view.

In the adjoining view, determine which of the lines is closer to the folding line. This line is in front of the other line in the first view.

Determining the piercing point by construction

The piercing point is the exact location of the intersection of a surface and a line. It may be determined by constructing an edge view of the plane surface. The edge view lies in the path of the line, indicating the piercing point location. This point may be transferred into the other views, and the visibility determined to find which part of the line is visible from the viewing direction.

Step 1 Step 2

Determining the piercing point by construction

Step 3 Step 4

Step 5

Determining the piercing point by line projection

An alternate method of determining the piercing point simply assumes a cutting plane that passes through the line and that appears in edge view in one of the given projection aligned with the projection of the line. This new plane cuts the original plane, leaving a “scar” on it. The intersection of the line and scar locates the piercing point.

Step 1. Assume the cutting plane to appear in edge view in the top view

Step 2. Determine the points X and Ywhere the cutting plane cuts the original plane. The piercing point is where the “scar” X -Y crosses the given line.

Step 3. Project the piercing point to the top view, and determine the visibility.

Finding the intersection of two planes by line projection

The intersection between the plane ABC and the plane abc below is the line connecting the piercing points of the lines AB and the line AC into the plane abc. These points may be found by line projection as illustrated in the previous slide. The visibility rule may be used to determine the hidden plane in each view.

Piercing points of a line into a cylinder

The cylindrical surface appears as a circular edge in the top view, and its intersection with the line is apparent in the top view as point X. This point may be located in the front view using the alignment rule.

The inclined surface appears as a straight edge in the front view, and its intersection with the line is apparent in this view as point Y. This point may be located in the top view by the alignment rule.

Step 2Given Step 1

Piercing points of a line into a sphere

Find a view where the line appears in its true length (front view in the example shown). The line will appear parallel to the folding line in the adjacent view (top view in the example).

Assume a cutting plane that passes through the line and that appears in edge view in the top view. Locate the points X and Y where the plane intersects the sphere in the top view. This also establishes the diameter of the circle of intersection between the sphere and the cutting plane.

Transfer the intersection circle into the front view and find the piercing points where the line crosses the circle of intersection. Locate the piercing points to the top view and determine the visibility

The revolution Method

The revolution method was used in descriptive geometry to solve spatial true length problems before the introduction of the auxiliary view method.

In the auxiliary view method, the observer changes his point of view to look perpendicular at the object’s inclined surface. In the revolution method the object is revolved about an axis until the inclined surface becomes aligned with one of the principal views.

Auxiliary View Method Revolution Method

Finding the true-length of a line by revolution (top view)

To find the true-length of line AB in the front view, revolve the line in the top view to align itwith frontal plane. The top view represents the circular base of a right cone, Line AB' is the outside element of the cone’s frontal plane, and appears in true-length in the front view.

1) Rotate the line in the top view to align with the front plane

2) Project the revolved line into the frontal plane to find True length

Finding the true-length of a line by revolution (front view)

To find the true-length of line CD in the front view, revolve the line in the front view to align it with top plane. The front view represents the circular base of a right cone, Line CD' is the outside element of the cone’s horizontal plane, and appears in true-length in the top view.

1) Rotate the line in the front view to align with the top plane

2) Project the revolved line into the top plane to find True length

Finding the true-length of a line by revolution (profile view)

To find the true-length of line EF in the profiole view, revolve the line in the front view to align it with profile plane. The front view represents the circular base of a right cone, Line EF' is the outside element of the cone’s profile plane, and appears in true-length in the profile view.

1) Rotate the line in the front view to align with the profile plane

2) Project the revolved line into the profile plane to find true length

Alternative points of revolution

In the preceding examples, the lines were revolved about one of their ends. Alternately, a line may be revolved about any point on its length as illustrated below to find its true length in the front view.

1) Rotate the line in the front view to align with the profile plane

2) Project the revolved line into the profile plane to find true length

Finding the true-size of a plane by revolution

A plane surface that appears as an edge in the front view may be found true size by a primary auxiliary view normal to the front view. Alternately, it may be revolved in the front view to show its true size in the top view.

The axis of revolution appears as a point in the front view and in its true length in the top view. Revolving the edge of the plane into the horizontal in the front view and projecting it to the top view yields the true size of the surface. The depth dimensions do not change.

Auxiliary View Method Revolution Method

Finding the true-size of a plane by revolution

When a plane appears as an edge in one of the principal views (top view), it can be revolved to be parallel to the adjacent view (front view).

The new front view is true-size when projected horizontally across from its original front view.

Finding the edge view of a plane by revolution

When a plane does not appear as an edge in one of the principal views, one can find an edge view of the plane by revolution.

draw a frontal line on the top view of the plane and project it to the front view to appear in true length. Revolve the plane until the true length line is vertical in the front view. The true length line projects as a point in the in the top view, and the plane appears as an edge.

Finding the true-size of a plane by double revolution

The revolution in the preceding slide may be used as a first step to find the true size of a plane when it does not appear as an edge in one of the principal views.

A second revolution positions the resulting edge view of the plane parallel to the frontal plane. Projecting the top views of the resulting plane to the front plane results in the true size of the plane.

Finding the true-size of a plane by auxiliary-view and revolution combination

When a plane does not appear as an edge in one of the principal views, a combination of an auxiliary view and a single revolution finds the plane in true size.

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