# Planes, lines and distances: looking for engineering problems

In a week from now, I am going to teach a one-time class to first-year engineering students about 3D geometry, more specifically: distances between points and lines/planes, and the common perpendicular between two skew lines.

Are there any cool engineering use cases (construction, chemistry, physics, computer science...) that I can use to illustrate some of these mathematical problems? I have looked a while on the internet, but cannot seem to find any good ones. And all textbooks I have consulted stay on the mathematical side of the spectrum, without applications.

Thanks!

Doing piping layout for a new chemical processes utilizes 3D modeling software a lot of the time now. I would think trying to show how pipes can and can't physically be laid out could use those concepts.

A typical situation is that one large pipe starts from a source somewhere, and a number of smaller pipes branch from the large pipe and can run in any direction, having to cross over/under other pipes. Trying to run a new piping network in the midst of an existing piping network can get pretty challenging.

The situation I see that might be plausible to set up is to make a fictional pipe network and then analyze how (or if it's even possible) to run a new header and branch piping for an expansion project.

Figure 1. Deceptively complex? Image source: The Daily Mail.

Analysis of the classic cable pylon alone would probably take a full class.

For any of your students that fall in love with these structures then I recommend The Pylon Appreciation Society.

• Thanks for your suggestion! This got me thinking: since I only have limited time to spend on the topic of distances between lines, I might go for a physics application in which I calculate the magnitude of the magnetic field that is caused by a high-voltage line. The magnitude is dependent on the inverse of the distance. – CedricDeBoom Feb 16 '20 at 9:20

Just of list of things that come to mind:

• Piping for oil refineries & water distribution systems for domestic water supply & hydro-electric power stations.
• Multi storey car parks - above or below ground.
• Underground mines using sub horizontal tunnels, vertical to sub vertical shafts & spiral ramps.
• The design of roads in hilly or mountainous terrain. If the terrain is not too steep roads can just go over the terrain. In steep terrain switch back roads are needed.
• Piping & ventilation systems in high rise builds
• Intercontinental ballistic missile interception, as was proposed in the 1980s - Strategic Defense Initiative.

There are lots of uses. Distance between two points comes up so often that i couldnt even count the uses. One particularily cool way to demonstrate it is raytracing, or sphere tracing. Raytracing is all about finding line plane intersections. And sphere tracing about closest point on a primitive. While raytracing seems like a gimic for graphics it is a valid simulation for many optics problems, as well as any number of other stochastic physics simulations.

Many engineering problems today use CAD. But the CAD application couldnt work without intersection finding. You can find lots of interesting examples in analytical geometry books for engineers. These are nolonger commonly thaught since our CAD allready handle those. But the examples in books are as valid and interesting today as they were yesterday. Look one up.

Distance between 2 points can be a demonstrated well by iteratively finding a solution to a problem that you dont know how to solve analytically. But can use gradient descent optimisation to solve by minimizing the distance.

But there are more good examples.

• I really like the raytracing example! Since I have a computer science background, I immediately relate to that topic. I have no experience with CAD design, but I guess I'll have a look at some of the algorithms that are used there. Thanks again! – CedricDeBoom Feb 16 '20 at 9:22