Reasoning
Read the definitions, follow the arguments and examine the conditions of each result.
Background: Euclidean triangles, Vectors and dot products, Radians. Start with the model for a brief introduction, follow the guided exploration for a closer look, or work through the arguments and experiment for a detailed study.
What you can learn
- Identify geodesics in three constant-curvature models.
- Measure spherical angles using tangent directions.
- Distinguish conformal pictures from distance-preserving maps.
Reading step 1 of 5: Choose a metric before measuring
Step 1 · Derivation · Selected
Choose a metric before measuring
A geodesic is locally straight with respect to a surface’s metric. In the plane, the relevant segments are ordinary straight lines. On a sphere they are arcs of great circles. In the Poincaré disk they are diameters or circular arcs meeting the unit circle orthogonally.
The same screen coordinates can therefore represent different distances. The model name, curvature and coordinate convention are part of every measurement. This exhibit uses a plane of curvature zero, a sphere of curvature 1/R² and a hyperbolic disk of curvature −1; it does not infer the curvature of physical space.
Sources: [M1] Discrete Differential Geometry: An Applied Introduction; [M4] Geometry with an Introduction to Cosmic Topology — Section 5.4
Step 2 · Derivation
Measure a spherical triangle
For unit directions a and b, the central angle is arccos(a·b), and the spherical distance is R times that angle. The dot product is clamped to its mathematical range before evaluation to protect against roundoff. Coincident and nearly antipodal endpoints require separate handling.
An interior angle is measured between the two geodesic tangent directions at a vertex. Three-dimensional chord directions give a different angle and cannot replace them. The positive coordinate-axis directions form a unit-sphere triangle with three angles of π/2, three sides of π/2 and area π/2. A common-hemisphere restriction selects short arcs and a consistent enclosed region.
Conditions and assumptions
- Unit direction vectors; nondegenerate short-arc triangle in the permitted hemisphere.
Sources: [M1] Discrete Differential Geometry: An Applied Introduction; [M4] Geometry with an Introduction to Cosmic Topology — Section 5.4
Step 3 · Theorem
Relate angle excess to area
For the permitted constant-curvature geodesic triangles, Gauss–Bonnet gives α + β + γ − π = KA. Positive spherical curvature produces an angle excess; negative hyperbolic curvature produces a defect. Flat triangles have angle sum π.
Choose the three hemispheres whose intersection is the spherical triangle. Their pairwise intersections are lunes with areas 2R²α, 2R²β and 2R²γ: each covers its angle divided by 2π of the full sphere. Across an antipodal pair, the lune-membership counts add to three in the triangle and its antipodal copy, and to one elsewhere. Integrating these counts gives 2R²(α + β + γ) = 2πR² + 2A, hence the angle-excess formula. Scaling R preserves the angles and multiplies the area by R².
Theorem. A valid constant-curvature geodesic triangle has angle excess equal to curvature times area.
Conditions and assumptions
- A simple, nondegenerate geodesic triangle bounding the stated region.
- The sphere uses short arcs within a common hemisphere.
- Angles are in radians.
Sources: [M1] Discrete Differential Geometry: An Applied Introduction; [M4] Geometry with an Introduction to Cosmic Topology — Section 5.4
Step 4 · Derivation
Read the disk and hyperboloid
The disk preserves angles between tangent directions, but it does not preserve lengths. In particular, a radial distance from the center to Euclidean radius r is 2 atanh(r), which increases without bound as r approaches one. The interaction limit r ≤ 0.94 is a numerical and viewing restriction, not the hyperbolic boundary itself.
The paired hyperboloid satisfies X₀² − X₁² − X₂² = 1 with X₀ positive. Distances there use the Lorentz bilinear form. Measuring ordinary Euclidean lengths on the pictured three-dimensional surface would describe a different geometry. Disk and hyperboloid distances provide two coordinate calculations for the same pair of points.
Theorem. The Poincaré disk metric assigns a specific hyperbolic distance to each pair of interior points.
Conditions and assumptions
- |u| < 1 and |v| < 1.
- Curvature is −1.
Sources: [M4] Geometry with an Introduction to Cosmic Topology — Section 5.4; [M10] Hyperbolic Geometry — MA448 lecture notes; [M11] Geometry III/IV — Lecture notes
Step 5 · Numerical experiment
Compare area calculations honestly
The disk area is calculated from angle defect and is labeled with that method. Substituting it back into the same formula cannot independently test Gauss–Bonnet. For a spherical triangle, the direction-vector determinant expression supplies a separate area calculation to compare with the measured angle excess.
The experiment saves valid triangles and radial disk samples, including coordinates, metric distances and display distances. Degenerate configurations have no reported triangle area. The sphere comparison tests two computational paths for a known theorem; it does not turn a numerical agreement into a new proof.
Numerical experiment. Spherical excess can be compared with an area calculation from the three unit direction vectors.
Conditions and assumptions
- The valid short-arc spherical triangle selected by this exhibit.
- The determinant calculation is independent of the angle-sum calculation.
Sources: [M4] Geometry with an Introduction to Cosmic Topology — Section 5.4; [M9] CSE 168 — Materials lecture
References and review status
Project attribution: BIPH Math Society. The exposition was prepared with tool assistance.
Review status: Not reviewed. Independent review by a teacher or society member has not yet been recorded. Numerical checks do not replace mathematical review.
[M1] Discrete Differential Geometry: An Applied Introduction
Keenan Crane
Version dated 29 January 2025. Chapter 2 and Sections 5.5–5.6 provide the background for triangular surfaces, angle defect and discrete Gauss–Bonnet. The exhibit’s explanations and implementation are original project work. Source checked 16 September 2026.
[M4] Geometry with an Introduction to Cosmic Topology — Section 5.4
Michael P. Hitchman
Author’s textbook treatment of hyperbolic geometry and triangle angle defect. The parent textbook supplies the surrounding Poincaré disk model and geometric context. Distances and angles in the exhibit use the metric of the named model. Source checked 16 September 2026.
[M9] CSE 168 — Materials lecture
UC San Diego CSE 168 course, Spring 2017
PDF page 65 gives the solid-angle tangent ratio, credited to Van Oosterom and Strackee. The project evaluates this ratio with atan2 and retains the denominator sign to select the correct quadrant, including permitted triangles with area above πR². Unit-vector solid angle is multiplied by R² and compared separately with angle excess. Source checked 16 September 2026.
[M10] Hyperbolic Geometry — MA448 lecture notes
Caroline Series
Introduction p. xi (PDF page 12) describes the hyperboloid and Lorentz metric; p. xii relates it to the Klein model. The paired three-dimensional view uses that metric rather than Euclidean surface lengths. Source and locator checked 16 September 2026.
[M11] Geometry III/IV — Lecture notes
Anna Felikson
Durham University, 2020–2021. Section 7.2, Theorem 7.3, p. 113 derives distance from the pseudo-scalar product. On the unit upper hyperboloid this gives cosh d = X₀Y₀ − X₁Y₁ − X₂Y₂. Source checked by direct PDF retrieval on 16 September 2026.