5.2 Area and defect in Hyperbolic Geometry

Hyperbolic geometry bends the rules of flat space. In this realm, triangles have angle sums less than 180°, and shapes have more area than you'd expect. It's a world where parallel lines can diverge and circles grow faster than their radii.

The Gauss-Bonnet formula is our guide here. It connects a shape's curvature to its angles and area, revealing the unique properties of hyperbolic space. Understanding this helps us navigate this curved universe and its surprising geometrical truths.

Area and Defect in Hyperbolic Geometry

Area of hyperbolic shapes

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• Gauss-Bonnet formula relates total curvature of a surface to its Euler characteristic and boundary
  • For a polygon $P$ on a surface with constant Gaussian curvature $K$: $\iint_P K dA + \sum_{i=1}^n \alpha_i = 2\pi \chi(P)$
    • $dA$ represents the area element
    • $\alpha_i$ represents the exterior angles at the vertices
    • $\chi(P)$ represents the Euler characteristic of the polygon
• In hyperbolic geometry, $K = -1$, simplifying the formula to: $\text{Area}(P) = \sum_{i=1}^n \alpha_i - 2\pi \chi(P)$
• For a hyperbolic triangle with angles $\alpha, \beta, \gamma$: $\text{Area}(\triangle) = \pi - (\alpha + \beta + \gamma)$
• Euler characteristic of a polygon with $n$ vertices, $e$ edges, and $f$ faces: $\chi(P) = v - e + f$
• Examples:
  • Hyperbolic triangle with angles $\frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{5}$: $\text{Area}(\triangle) = \pi - (\frac{\pi}{4} + \frac{\pi}{3} + \frac{\pi}{5}) \approx 0.467\pi$
  • Hyperbolic square with all right angles: $\text{Area}(\square) = 4(\frac{\pi}{2}) - 2\pi(1) = 0$

Angular defect and surface curvature

• Angular defect at a vertex of a polygon is the difference between the sum of the angles around that vertex and $2\pi$
  • For a vertex $v_i$ with incident angles $\theta_1, \theta_2, \ldots, \theta_k$: $\text{defect}(v_i) = 2\pi - \sum_{j=1}^k \theta_j$
• Total angular defect of a polygon is the sum of the defects at each vertex
• In hyperbolic geometry, the angular defect is always positive because the sum of the angles in a hyperbolic triangle is always less than $\pi$
• Angular defect relates to the Gaussian curvature of the surface
  • For a polygon $P$ on a surface with constant Gaussian curvature $K$: $\text{Area}(P) = \frac{1}{K} \sum_{i=1}^n \text{defect}(v_i)$
• Examples:
  • Hyperbolic triangle with angles $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$: $\text{defect}(v_1) = 2\pi - \frac{11\pi}{12} \approx 1.083$
  • Hyperbolic pentagon with all angles $\frac{2\pi}{3}$: $\text{Total defect} = 5(2\pi - \frac{2\pi}{3}) = \frac{10\pi}{3} \approx 10.472$

Area of ideal hyperbolic triangles

• Ideal hyperbolic triangle has all three vertices on the boundary of the hyperbolic plane (at infinity)
• Angles of an ideal hyperbolic triangle are all zero
• Using the Gauss-Bonnet formula for a hyperbolic triangle: $\text{Area}(\triangle) = \pi - (\alpha + \beta + \gamma)$
• Substituting $\alpha = \beta = \gamma = 0$: $\text{Area}(\triangle) = \pi - (0 + 0 + 0) = \pi$
• Therefore, the area of an ideal hyperbolic triangle is always equal to $\pi$

Hyperbolic polygons and angular defect

To find the area of a hyperbolic polygon:

1. Calculate the exterior angles at each vertex
2. Use the Gauss-Bonnet formula to compute the area

To find the angular defect at a vertex:

1. Sum the angles incident to the vertex
2. Subtract the sum from $2\pi$

Remember in hyperbolic geometry:

• Sum of the angles in a triangle is less than $\pi$
• Angular defect is always positive
• Area of an ideal triangle is always $\pi$ Examples:
• Hyperbolic hexagon with all angles $\frac{2\pi}{3}$: $\text{Area}(\hexagon) = 6(\frac{2\pi}{3}) - 2\pi(1) = 2\pi$
• Hyperbolic triangle with angles $\frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{5}$: $\text{defect}(v_1) = 2\pi - (\frac{\pi}{3} + \frac{\pi}{4} + \frac{\pi}{5}) \approx 1.033$