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 [url]http://www.geogebratube.org/material/show/id/3597[/url], alas, is not a shape of constant width. [math]\;\;[/math][i]Proof:[/i] 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 [b]Hint[/b] 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 [url]http://www.geogebratube.org/material/show/id/3597[/url], and my thought process. I divide "not a curve of constant width" into two CASES: [list=1] [*] Manipulation of the given points changes the diameter: Let one point pass another. Then [math]\;\;[/math] [b]a)[/b] the figure changes size. The question remains, [i]how to construct a family of figures of constant diameter?[/i] [math]\;\;[/math][b]Q:[/b] Does such a family exist? If so, what are the constraints? [math]\;\;[/math][b] b) [/b]the new figure no longer has a distinct diameter in each direction. By definition, this is not a curve of constant width. [math]\;\;[/math][b]Assertion:[/b] There exists a curve of constant width from the same generating points, with the new (maximum) diameter. [math]\;\;[/math]([b]Task:[/b] Construct such a figure.) [*] The points can be manipulated so that the figure has a variable (maximum) diameter. [math]\;\;[/math] ([i]Proof: [/i]Drag the given points so that the dashed figure is convex.) [math]\;\;[/math][b]Assertion:[/b] I say, there exists such a figure of constant diameter, from the same given points and construction method. [math]\;\;[/math][b]Task:[/b] (Construct such a figure.) [/list] I found the following questions and puzzles pertinent: What are the constraints on the arcs/moving points? In GGB, I have given every arc by one of two custom methods. Why? Suppose I wish to rotate the figure in place, in a fixed box about a fixed pivot. I will cut a groove into the figure, and let it slide along the pivot. What is the shape (locus) of the groove? I have passed a diameter through the figure, rather than using a bounding box. Why? Hints and Challenges: [list] [*]The figure is [i]handed:[/i] 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. [/list] [b]Extra Credit:[/b] [i]In angular measure, what is the mathematical relationship between orientation (heading) and arc length? How can this relationship be made perfectly definite?[/i] Though proofs were omitted, I have provided working demonstrations of correct answers to all the questions and tasks. _________ I disapprove of trick questions. Let me be plain: the angle system in GGB is pretend. I know how to measure anyway. Lucky for me, I guess. Consider always the student.