Geometry and topology · Maths EE idea · Solid
Triangles on a sphere
A research question to start from
How does the angle sum of a spherical triangle relate to its area, and what does this mean for great-circle routes?
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
Non-Euclidean geometry made concrete: one elegant theorem (Girard's) and applications you can compute.
Mathematics you would need
- Great circles and spherical distance
- Spherical triangles
- Girard's theorem
- Vectors in 3D
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
- Prove Girard's theorem with lunes.
- Derive the spherical law of cosines with vectors.
- Compute great-circle distances and compare with flat-map routes.
Scope and difficulty
Solid. Solid; keep navigation as an application, not the focus.
Pitfalls
- Geography essay with little proof.
- Mixing degrees and radians.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Girard's theorem spherical excess proof lunes; spherical law of cosines. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).