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By David W. Henderson and Daina Taimina

Before we discuss the mathematics of the pump we must describe the geometry of the spiral arms. Focus on one of the four arms:
Imagine that the red circle in the picture is a spool that has a red thread wrapped around it in such a way that when fully wound the end of the thread is at the point A. Now imagine unwinding the thread, keeping the spool fixed, and keeping the thread pulled taut. The end of the thread traces the outer edge of the spiral arm. This curve is called the involute of a circle (in this case, the red circle).
Instead of keeping the spool fixed and unwinding the thread, we could rotate the spool and pull the thread taut in the same direction. It is this later view that we will use in analyzing the spiral pump.
Examine the picture of the spiral pump below. Now imagine that there is red thread rolled around the left spool and then pulled taut and wrapped around the right spool as indicated in the picture. Place an orange dot on the thread at the place that two arms touch each other. Now, instead of unwrapping the thread we will turn the two spools at the same rate always keeping the thread taut. Since the spiral arms are in the shape of an involute, the orange dot will follow the outer edge of both spiral arms. Thus, as the spools rotate, the spiral arms will stay in contact.
Algebraic description of the involute of a circle:
Consider the following drawing:
Let t be the amount of thread unwrapped. Since this amount of thread was wrapped around the spool to the point A, the length of red arc is thus t. The angle that the red arc subtends is (in radian measure) t/a, where a is the radius of the axle. We now describe the location of the point P in terms of its distance r from the origin (the center of the spool) and the angle θ (subtended by the green arc).






The first equation follows from the Pythagorean theorem and the second equation follows from the observation that the difference between the angle θ that subtends the green arc and the angle t/a that subtends the orange arc is the angle in the red right triangle whose tangent is

. 

When t is much larger than a, then these equations become very closely:
Or, , which is the equation of an Archimedes Spiral. Because of this, the Archimedes Spiral is often confused with the involute of a circle.
Links to websites with dynamic descriptions of involutes.
http://mathworld.wolfram.com/Involute.html
http://wwwgap.dcs.stand.ac.uk/~history/Curves/Involute.html
http://www.2dcurves.com/spiral/spirali.html