Now that you can use auxiliary views to create desired views of geometric entities, you can solve many descriptive geometry problems. There are basically only four separate manipulations you can generate with auxilliary views:
When presented with a problem, these are the only tools you have to solve it graphically. Before beginning any of them to solve the problem, you must determine first what view is required. Creating that view is a matter of using one or more of the four previouly listed steps.
We will solve the first set of problems graphically, and deal with intersections through the use of auxiliary views. We have already discussed intersections of two lines. Remember that if two lines intersect, they have a single point in common, and that point must project from view to view. Now let us consider the intersection of a line and a plane in space.
As entities intersect in 3D space, it is often desirable to show the intersections with correct visibility.
Realize that the intersection between a line and a plane in space is still a single point. This single point is defined as the piercing point, the point at which the line pierces the plane. To solve the problem graphically, first determine what view must be drawn to find the solution.
Using aids to visualize the problem is helpful. A pencil can represent the line and a clear plastic drafting triangle can represent the plane. Hold the pencil so it passes through or intersects the opening in the triangle. To find the single point on the line which lies on the plane, how must these objects be viewed?
It should be clear that in a view in which the plane appears as a line, the single point the line has in common with the plane can be indentified easily. The construction of views to obtain that view has been previously reviewed. Once the piercing point is located, it can be projected back into the given views.
A "two-view" method can also be used to find the piercing point of a line and a given plane. The construction to solve the problem is quite simple, but, it is more difficult to grasp conceptually than the auxilliary view method.
Two planes in space intersect to form a single line. If the planes are assumed to extend indefinitely in all directions, the line of intersection will have infinite length. If the planes are defined as having a certain size and shape, the line of intersection will be finite, and will appear as a visible line where it lies within the boundaries of both planes.
To find the line of intersection using auxilliary views, you must construct a view where either one of the given planes appears in edge view. In that view, the line defined by the two points that lie on the boundary of the plane that appears forshortened and the plane that appears as a line, define the line of intersection. This is basically a piercing point problem you do twice. Find where any line on one plane pierces the other plane. Now repeat that step and find where a different line on either plane pierces the other plane. The two points you have will define the line of intersection between the two planes.
The true angle between any two lines can be measured in a view where both lines are true length. To construct that particular view, choose one of the two lines and construct a view where it appears as a point. If neither of the given lines is true length, this will take two views. Remember to project both lines into each view. Once the point view of one of the lines is constructed, the other line will appear foreshortened (unless they are parallel). An additional auxiliary view parallel to the foreshortened line in that view will show both lines TL. Measure the angle between the lines in that view.
If the two lines lie on the same plane (i.e. if they intersect), an alternate approach usually can reduce the number of views required to solve to problem. The intersecting lines define a plane. Draw that plane true size, and every line on the plane will be TL, and the angle can be measured in that view.
The true angle between a line and a plane can be measured in a view where the line is TL and the plane appears as a line (or edge). There are two different approaches to construct this desired view.
To solve using the plane method, first construct the TS view of the plane. If the plane does not appear as a line in the given problem, the edge view must be constructed before you can obtain the TS view. Project the line into each view as well. Once the TS view of the plane is constructed, any adjacent view will show that plane as a line. Construct an auxiliary view parallel to the foreshortened line in the view where the plane appears TS. Projection onto this view will show the TL line, and the plane will appear as a line. Measure the angle in that view.
To solve using the line method, first construct the point view of the line. The plane will appear foreshortened in this view. Once the point view of the line is constructed, any adjacent view of the line will show it TL. Determine the auxiliary view direction, which will also show the plane as a line by constructing a TL line on the plane in the view where the given line appears TL. An auxiliary view perpendicular to this line will show the plane as a line, and the line will be TL. Measure the angle in that view.
The true angle between two planes can be measured in a view where both planes appear as lines. How does the line of intersection between two planes appear when both planes appear as lines?
To measure the true angle between two planes, a view must be constructed where both planes appear as lines. In that view, the line of intersection will appear as a point. Once the line of intersection is determined, it is simply a matter of constructing the required views to show that line as a point.
The shortest distance from a point to a line is always a line through the given point that is perpendicular to the line. This is very similar to a true angle problem, where the true angle must be 90 degrees.
To find the shortest distance from a point to a line, construct a view where the line is TL. A line drawn from the given point perpendicular to the line will be a view of the shortest distance from the point to the line. If the shortest distance must be measured, that line must be projected into a view where it appears TL.
An alternative method also can be used. A line and a point define a plane. Construct a view where that plane appears TS. Every line on that plane is TL, and angles between lines are true. A line can be drawn from the given point perpendicular to the given line and measured in that view.
The shortest distance from a point to a plane is always a line through the given point that is perpendicular to the plane.
To find the shortest distance from a point to a plane, construct the edge view of the plane. A line from the point drawn perpendicular to the plane will be the shortest line from the point to the plane. The line will always be TL and can be measured in that same view.
The shortest distance between two lines is always a line which is perpendicular to both lines.
To find the shortest distance between two lines, construct a view where either of the given lines appears as a point. Construct a line, from that point view, perpendicular to the other given line shown in that view. In order for the line you construct to be perpendicular to both given lines, it must be TL in this view.
Remember that to determine whether or not two lines are perpendicular, only one of them has to be TL. If the angle between them is 90 degrees in a view where either of the lines is TL, they are perpendicular.