The conchoid of Nicomedes and other engineering geometries

Before I became a Math major, I was an Engineering major, and the things that interested me the most in engineering were ‘non standard’ ways of generating power. Solar power is a classical example of ‘non standard’. Another thing that is really interesting is that ancient geometric principles are being used to generate that power. Though not all of those ways of generating power are considered power generation to most people; some are like the use of the conchoid Nicomedes to develop a heliostat to help generate solar power, and others are like the downtown library in Salt Lake City which uses an understanding of geometry and physics to heat and cool a massive structure.

The conchoid of Nicomedes can be thought of as a curve r=b*secant θ in polar coordinates.  This family of curves was discovered by Nicomedes, who was an ancient Greek mathematician. Wolfram/Alpha describes the conchoid as “…the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family for 0<a/b<1, a/b=1, and a/b>1. (For a=0, it obviously degenerates to a circle.)

ConchoidofNicomedes

ConchoidofNicomedesCurves The Conchoid of Nicomedes ¹

The conchoid of Nicomedes was popular during the 17th century, at least with mathematicians. Currently it is going through a slight revival in popularity, at least with heliostat designers.  For those of you wondering what a heliostat is, heliostat means stationary sun. It is used to project the suns rays wherever you might want them to be pointing, such as a solar energy collector.

An example of the conchoid of Nicomedes being used as a solar concentrator.²

How this works is that there are a series of gimbaled mounts (6) for the mirrors (3) on the poles holding them up and the motion of a single bar (5) is enough to get the whole array to face a new direction. The mirrors will focus the rays of the sun onto the solar collector (14)  and off you go with all that nice fresh solar energy. The solar collector is at the apex of the focus point of the conchoid and all the mirrors are following the path of the line of the conchoid, or put another way each mirror is tangential to a corresponding point on the curve of the conchoid. It really is an interesting way to use this type of math to harness power.

File:Salt Lake City Public Library -IMG 1754.JPG The downtown Library in SLC ³

The Salt Lake City library (downtown) also harnesses the power of the sun but most people don’t seem to realize it. The entire southwest side of the library is shaped like an enormous lens and if you can go in and look at what it is, it really is amazing. That lens captures the warmth of the sun and is used for both heating the entire structure, but also is used for cooling in the summer time. During the winter its operation is pretty basic: capture was much heat as possible. This heating can be calculated by summing the total energy captured by the area of the glass and dividing that total by the total area covered in glass to determine the energy captured by the windows. This equation is true for any window but the shape of the lens really helps out, because of the magnification equation.5 This equation uses the ratio of image and object distance to determine magnification. In this case an image is not being attempted to be made, but the magnification is still able to be used to magnify the available heat.

During the summer the true genius of this piece of geometrical engineering happens.  There are vents that run parallel to the surface of the water (they are dark brown). Those vents open and a corresponding set of vents on the roof line also open. The heat generated by the “lens” then escapes through the roof vents and creates a lower pressure inside the lens that the air from the lower vents rushes to fill. The air from the lower vents has been cooled by passing over the reflecting pool, and more air is pulled from the library creating a natural gentle breeze. This whole set up uses the heat of the sun to cool a huge space, all using ancient geometric principles.

1 http://mathworld.wolfram.com/ConchoidofNicomedes.html

2 http://www.google.com/patents/US7677241

3 http://commons.wikimedia.org/wiki/File:Salt_Lake_City_Public_Library_-IMG_1754.JPG

4 http://www.slcpl.org/branches/view/Main+Library

5 http://www.physicsclassroom.com/class/refrn/Lesson-5/The-Mathematics-of-Lenses

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