15.2 Basic concepts of hyperbolic geometry

Hyperbolic geometry challenges our Euclidean intuition. It's a world where parallel lines aren't what we expect and triangles break the 180-degree rule. This mind-bending geometry opens doors to new mathematical insights and real-world applications.

In hyperbolic space, circles grow exponentially, and triangles have a built-in "angle deficit." These quirks lead to fascinating properties and problem-solving techniques unique to this geometry. It's like discovering a new language for describing space and shapes.

Fundamental Principles and Comparisons to Euclidean Geometry

Principles of hyperbolic geometry

• Hyperbolic geometry is a non-Euclidean geometry operates on a hyperbolic plane
  • Hyperbolic plane has a constant negative Gaussian curvature creates a saddle-shaped surface
• In hyperbolic geometry, the parallel postulate of Euclidean geometry is replaced by the hyperbolic parallel postulate
  • Given a line $l$ and a point $P$ not on $l$, there are infinitely many lines through $P$ do not intersect $l$ (ultraparallel lines)
• The sum of the angles in a hyperbolic triangle is always less than 180°
  • As the area of a hyperbolic triangle increases, the sum of its angles approaches zero
  • Example: A hyperbolic triangle with angles 30°, 45°, and 60° has an angle sum of 135°, less than 180°

Hyperbolic vs Euclidean geometry

• Euclidean geometry operates on a flat plane, while hyperbolic geometry operates on a hyperbolic plane with negative curvature
  • Euclidean plane has zero Gaussian curvature (flat surface)
  • Hyperbolic plane has negative Gaussian curvature (saddle-shaped surface)
• In Euclidean geometry, the parallel postulate states given a line and a point not on the line, there is exactly one line through the point parallel to the given line
  • In hyperbolic geometry, there are infinitely many lines through the point do not intersect the given line (ultraparallel lines)
• The sum of the angles in a Euclidean triangle is always 180°, while in a hyperbolic triangle, the sum is always less than 180°
  • Example: Euclidean triangle with angles 60°, 60°, and 60° has an angle sum of 180°
  • Hyperbolic triangle with the same angles has an angle sum less than 180°
• In Euclidean geometry, the circumference and area of a circle grow linearly with the radius
  • Circumference: $C = 2\pi r$, Area: $A = \pi r^2$
• In hyperbolic geometry, the circumference and area of a circle grow exponentially with the radius
  • Circumference: $C = 2\pi \sinh(r)$, Area: $A = 4\pi \sinh^2(r/2)$, where $\sinh$ is the hyperbolic sine function

Properties and Problem Solving in Hyperbolic Geometry

Properties of hyperbolic shapes

• Hyperbolic lines are geodesics on the hyperbolic plane, the shortest paths between two points
  • Hyperbolic lines can be modeled using the Poincaré disk model or the upper half-plane model
    • Poincaré disk model: Lines are represented by circular arcs perpendicular to the disk's boundary
    • Upper half-plane model: Lines are represented by vertical lines or semicircles with centers on the x-axis
• Hyperbolic triangles have the property the sum of their angles is always less than 180°
  • The area of a hyperbolic triangle is proportional to its angle defect, $\pi$ minus the sum of its angles
    • Example: A hyperbolic triangle with angles 30°, 45°, and 60° has an angle defect of $\pi - (30° + 45° + 60°) = 45°$
• Hyperbolic triangles with the same angle sum are congruent, regardless of their side lengths
  • Contrast to Euclidean geometry, where triangles with the same angle sum can have different side lengths
    • Example: Two Euclidean triangles with angles 30°, 60°, and 90° can have different side lengths
    • In hyperbolic geometry, all triangles with these angles are congruent

Applications in hyperbolic geometry

• Use the hyperbolic parallel postulate to determine the number of lines through a point do not intersect a given line
  • Example: In the Poincaré disk model, draw a line and a point not on the line, then construct infinitely many lines through the point do not intersect the given line
• Calculate the sum of the angles in a hyperbolic triangle and compare it to the Euclidean case
  • Example: Given a hyperbolic triangle with angles 45°, 60°, and 75°, the angle sum is 180°, less than the Euclidean sum of 180°
• Apply the relationship between the area and angle defect of a hyperbolic triangle to solve problems
  • Example: If a hyperbolic triangle has an angle defect of 30°, its area is proportional to $\frac{30°}{180°} \pi \approx 0.52$
• Utilize models such as the Poincaré disk or upper half-plane to visualize and solve problems in hyperbolic geometry
  • Example: In the upper half-plane model, construct a hyperbolic line between two points and measure its length using the hyperbolic metric $ds = \frac{\sqrt{dx^2 + dy^2}}{y}$
• Compare and contrast the properties of figures in hyperbolic geometry with their Euclidean counterparts to gain a deeper understanding of the differences between the two geometries
  • Example: Investigate the behavior of circles in hyperbolic geometry and compare their circumference and area growth to Euclidean circles