Analysis and calculus · Maths EE idea · Solid
Curves and solids of constant width
A research question to start from
Which curves other than the circle have constant width, and why do all curves of constant width w have perimeter πw?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Geometry with a surprising theorem (Barbier's) that you can prove and check on examples.
Mathematics you would need
- Support functions and widths
- Parametric curves
- Arc length
- Reuleaux polygons
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Construct Reuleaux triangles and polygons and compute their perimeters and areas.
- Prove Barbier's theorem for Reuleaux polygons, then discuss the general case.
- Compare areas for a given width.
Scope and difficulty
Solid. Solid.
Pitfalls
- Only the Reuleaux triangle.
- General theorem quoted without explanation.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Barbier's theorem constant width perimeter; Reuleaux polygon area. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).