Constant width for MBorcherds
- Ryan Hirst
I am a student, too. Let's go to recess, ok? These questions and tasks are for the Math People. I say, Mr Borcherds' worksheet http://www.geogebratube.org/material/show/id/3597, alas, is not a shape of constant width. Proof: drag the given points. The only purpose I have for finding errors is to fix them. So let me try my hand at a solution:
I say, the black curve through P and Q is a curve of constant width (diameter). I included a Hint above for how, following Euler, we might define curves of constant width in general. For this worksheet, I followed Mr. Borcherds construction, applying constraints and corrections as they arose. Let me step through his worksheet http://www.geogebratube.org/material/show/id/3597, and my thought process. I divide "not a curve of constant width" into two CASES:
- Manipulation of the given points changes the diameter: Let one point pass another. Then a) the figure changes size. The question remains, how to construct a family of figures of constant diameter? Q: Does such a family exist? If so, what are the constraints? b) the new figure no longer has a distinct diameter in each direction. By definition, this is not a curve of constant width. Assertion: There exists a curve of constant width from the same generating points, with the new (maximum) diameter. (Task: Construct such a figure.)
- The points can be manipulated so that the figure has a variable (maximum) diameter. (Proof: Drag the given points so that the dashed figure is convex.) Assertion: I say, there exists such a figure of constant diameter, from the same given points and construction method. Task: (Construct such a figure.)
- The figure is handed: Given vector AB, point C gives the direction of construction. Suppose I begin with C on the other side of AB. What is the maximum magnitude of arc AC, on the circle with center B?
- Click the black curve, then drag (for example) C until it coincides with E. What is the magnitude of ∡CDE? ... Arc CE? ..and ∡EDC? ...and arcEC? Demonstrate your answer, using the given figure (above). Now demonstrate the answer in your own construction. Does GGB agree?
- What happens to point C when you try to pull it past A?
- Let a circle pass through three given points K, L, M. What is the length of the arc in the limiting figure as K, L, M approach one another, and coincide? Demonstrate your answer with GGB.
- In the figure above, consider the arc from C to A, on the circle with center B and radius r. When is | arcCA| > π r? Demonstrate your answer with your own construction in GGB.