Geometry and topology · Maths EE idea · Ambitious
Two models of hyperbolic geometry
A research question to start from
How do the Poincaré disc and upper half-plane models represent the same hyperbolic geometry, and how does the angle sum of a triangle behave in them?
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
A deep idea with concrete calculations; comparing models is a natural analysis.
Mathematics you would need
- Euclid's parallel postulate
- Circles orthogonal to a boundary
- Möbius transformations (basic)
- Hyperbolic distance
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
- Describe lines and distance in both models.
- Show a map between the models preserves lines.
- Compute angle sums of triangles and relate to area.
Scope and difficulty
Ambitious. Ambitious; restrict to what you can derive.
Pitfalls
- Escher pictures instead of mathematics.
- Formulas quoted without understanding.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Poincare disc model upper half-plane; hyperbolic triangle angle sum area. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).