# Descriptive Geometry/Distances

## Distance from a Point to a Line[edit]

The true distance between a line and a point is the shortest distance between them. First, project an auxiliary view across a folding line parallel to the line in order to find its true length. Make sure to project the point over as well. Second, create another auxiliary view across a fold perpendicular to the line to find it in point view. The distance between the two points is the true distance.

## Distance from a Line to a Line[edit]

### Parallel Lines[edit]

* Parallel Lines:* Find the true length of the parallel lines (finding the true length of one of the lines will find the true length for the other) by projecting them across a parallel folding line. Next, find the point view of the two lines by making a perpendicular folding line and projecting the lines again. The distance between the two points is the true distance between the two parallel lines.

### Skew Lines[edit]

* Skew Lines:* Find the true length of either one of the two lines but still project both lines over the folding line. Then find the point view of the line that is in true length. Draw a line perpendicular to the line that passes through the point, this is the distance between the skewed lines.

## Distance from a Line to a Plane[edit]

In order to find the distance between a line and a plane, the first step is to get the line in point view. This can be done by creating a folding line perpendicular to the line in true length and using transfer distances to construct the view.

If the line and plane are parallel, this construction will also put the plane in edge view. the distance between the line and the plane would be the length of the line drawn from the line in point view to the plane in edge view. The plane in edge view will appear as a line, but it can be extended infinitely in any direction.

If the line and plane are not parallel (i.e. the plane is not in edge view when the line is in point view) then the line will intersect the plane at some point and the distance will be zero. You can verify this by finding the plane in edge view and seeing that the line will eventually pierce the plane.

## Distances at an Angle from the Horizontal, as Applied to Line Segments[edit]

Finding a distance at an angle from the horizontal involves finding the distance between two points that do not share a plane perpendicular to the direction of view. To find this distance, one begins by connecting the two given points with a line segment. Then, one constructs the line first in point view so that one can then construct in in true length, which gives the true distance between the two points.

**Example Problem**

*Meteorological Station A stands on a mountain 3250m above sea level. Meteorological Station B sits on a 4500m peak 10km away from Station A. The weather service is interested in using the two stations in conjunction with a weather balloon at 7000m as a triangulation device. The weather balloon must be placed such that it completes an equilateral triangle. Where could the weather service place the balloon, and what would the distance be from the balloon and each of the weather stations?*

## Distance from a Point to a Solid[edit]

Finding the shortest distance from a point to a solid in this case means finding the shortest distance between a given point and the closest point on the nearest face to the solid, assuming the solid has planar faces. It is therefore useful when doing this problem to isolate the elements that one is interacting with, in this case, the nearest face and the point in question. This simplifies the problem to finding the shortest distance between a point and a plane.

To find the shortest distance between a point and a plane, one must first find a view of the plane in edge view. By showing the plane in edge view, one is further simplifying the problem to that of the shortest distance between a point and a line. Once you have the plane in edge view, you must then create a line that intersects the plane perpendicularly. After transferring this line back to the previous views one can then use the piercing point construction to find the point on the plane closest to the given point, and hence the closest point on the solid to the given point.

**Example Problem**

*Superllama is on the run from a cubic kilometer of Dr. von Shortcake's evil flying demon babies. Each flying demon baby occupies one cubic meter of air. The cube's base is 2km off of the ground, and is angled such that four of the edges points a cardinal direction. If Dr. von Shortcake freezes Superllama in place at an altitude of 10km and 8km away in top view from the center of the cube which is located N60degE relative to Superllama, where is the flying demon baby located that gets to Superllama first?*

## Distance from a Line to a Solid[edit]

To solve for the shortest distance between a line and a solid, the problem must be treated as one of solving for the shortest distance between a line and a plane. The face of the solid nearest to the line in question should be treated as this separate plane. The process then becomes fairly straightforward: all you have to do is perform the construction for finding the shortest distance from a line to a plane (which, as is explained above, is itself just simplified to the construction for finding the shortest distance between a point and a line).

To reiterate, the construction for finding the shortest distance from a line to a plane is as follows: First you must construct a view in which the line is shown in point view. If the line and the plane are parallel, the plane should be shown in edge view in the same view in which the line is shown in point view (otherwise, the line and plane will eventually intersect and so the shortest distance between them is zero, as is also mentioned above). All that must be done then is to connect the line in point view to the edge view of the plane with a perpendicular line. This perpendicular line is in true length and is the shortest distance between the line and the plane, which in this case means it is also the shortest distance between the line and the solid. You may then project the shortest distance line to the other views if desired by using transfer distances.

In a case where the solid is a solid with circular faces, such as a cylinder, to find the shortest distance between the line and the solid you may abstract the solid as a line - its center line - and use the construction for finding the shortest distance between a line and a point. You simply have to add the thickness, the diameter of the solid back in to the final view to get the actual scaled distance between the solid and the line.

### Example Problem[edit]

A famous Italian bell tower is in danger of collapsing. It is approximated in the drawing as an oblique cylinder. In an attempt to shore up the tower, engineers have placed an angled support into the ground next to the tower and now want to connect the tower to the support using guy wires. To start out the engineers want to know what the shortest possible length of a guy wire connecting the tower to the support can be; in other words, what the shortest distance between the tower and the support is. Find this distance. Use a scale of 1 mm equals 5 feet; the radius of the bell tower is 10 feet (2 mm).

*Solution:* In order to solve this problem, all you have to do is treat the center line of the cylinder as a line, and then use the construction for finding the shortest distance between a point and a line, as is described above.

## Distance from a Plane to a Solid[edit]

The shortest distance between a plane and a solid can be found by treating the nearest face of the solid to the plane as a separate plane. This way the problem is simplified to finding the shortest distance between two planes, and that is the construction you have to do to find the shortest distance between the original plane and the solid. Once again, the planes must be parallel to do this construction, as otherwise they will eventually intersect, making the shortest distance between them zero.

The construction for finding the shortest distance between two planes in the end simplifies into finding the shortest distance between a line and a plane. You must first create a view in which both planes are shown in edge view, which as the planes are parallel means no more than creating an edge view for just one of the planes. Then you pick an arbitrary point on one of the edge view lines from which you can draw a perpendicular line to the other edge view line. This line is in true length and is the shortest distance between the two planes. To project the shortest distance line to the other views, you must perform piercing point constructions to find where the points at which the shortest distance line meets the planes are.

### Example Problem[edit]

An architect has a sun-shading device placed above one side of the shed roof of his building. The shading device is parallel to the surface of the roof. What is the shortest possible length of a support wire connecting the shading device to the roof (shortest distance from the plane of the shading device to the face of the solid roof)?

*Solution:* This problem is somewhat of a trick question. It tests how well you understand the concepts that drive the problem. In this case the plane is uniformly parallel to the closest face of the solid. The front face of the solid is already shown in true shape in the front view, as it is parallel to the folding line in both front and top views. This means the plane's edge view in the front view is shown in true length, as it is parallel to one of the sides of the front face of the solid. Since all of this is true, this means that the shortest distance from the plane to the closest face of the solid is the same at all perpendiculars from the plane to the face. So, the shortest distance from the plane to the face is already shown inherently in that front view; all you have to do is measure that distance if you so please.

## Distance from a Solid to a Solid[edit]

When solving for the shortest distance between two solids, treat both as lines, using the line to line method. Take two cylinders for example, or a cone(s), and solve for the shortest distance between their axes. Then take into account, the depth of the solid; the distance from the center to the surface of the solid.