Geometry: metric distances and triangle area

Question

How do screen distances differ from metric distances, and how do triangle angles relate to area?

Mathematical background

The plane, sphere and Poincaré disk use different metrics. For the valid constant-curvature geodesic triangles in this exhibit, angle excess equals curvature times area.

The disk’s radial metric distance grows as ln((1+r)/(1−r)), while the displayed radial coordinate is r.

\alpha+\beta+\gamma-\pi=KA

Sources: [M1], [M4]

d(u,v)=\operatorname{arcosh}\left(1+\frac{2|u-v|^2}{(1-|u|^2)(1-|v|^2)}\right)

Sources: [M4]

Read the mathematical derivations.

Hypothesis or expected result

Theoretical expectations

The unit coordinate-axis spherical triangle has three right angles and area π/2. Scaling its radius to 2 preserves those angles and multiplies its area by four.

Disk distances are expected to depart increasingly from screen distances toward the unit-circle boundary. The unit boundary remains outside the model domain.

Parameters and assumptions

Use one planar right triangle, the coordinate-axis spherical triangle at radii 1 and 2, and disk triangles with vertices 0, r and ir.
Disk radii are 0.1,0.3,0.5,0.7,0.9,0.94. All input coordinates are saved.
Sphere arcs are short great-circle arcs in the permitted common hemisphere.

Each case’s complete parameter object appears below and in its JSON file. Display rounding is used only in this article; the saved data retain full numeric precision.

Method and numerical settings

Run the shared geometry kernel for each case and retain coordinates, geodesic distances, angles and area-method labels.

Compare spherical angle excess with the independent unit-vector solid-angle formula. Compare disk and hyperboloid distances using their own metrics.

Methods reported by the executed kernel

Euclidean coordinate distance; Euclidean coordinate determinant / 2. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.
R acos(clamped dot); equivalent atan2(cross norm, dot) near coincident or antipodal directions; Independent unit-vector solid angle × R². Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.
Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.

Software identifier: 0.1.0+0e6017e6a747.dirty.02665ba9098c
Git revision: 0e6017e6a747d3c610b956612b4125e232283001
Uncommitted worktree state: Present when the batch was generated
Source SHA-256: 02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b
Runtime: v26.5.1

Results from an actual run

This saved batch contains 9 completed calculations. The table and downloads below refer to those actual runs.

The largest recorded absolute diagnostic residual was 4.44089e-16. 9 of 9 residual-bearing cases were within their own reported numerical tolerance.

Across 9 recorded readings, Angle sum ranged from 1.694232 to 4.712389.

Across 9 recorded readings, Triangle area ranged from 0.01999933 to 6.283185.

Across 9 recorded readings, Distance AB ranged from 0.2006707 to 3.476099.

Across 9 recorded readings, Distance BC ranged from 0.2847369 to 6.260959.

Spherical radius 1: angle sum 4.712389 radians; area 1.570796 model units²; independent area residual 0. The area uses the vector expression, while the comparison uses the tangent angles.

Spherical radius 2: angle sum 4.712389 radians; area 6.283185 model units²; independent area residual 0. The area uses the vector expression, while the comparison uses the tangent angles.

Disk radius r = 0.1: radial metric distance 0.2006707; triangle area 0.01999933; disk/Lorentz distance residual 5.55112e-17. Area here is derived from angle defect.

Disk radius r = 0.3: radial metric distance 0.6190392; triangle area 0.1795163; disk/Lorentz distance residual 0. Area here is derived from angle defect.

Disk radius r = 0.5: radial metric distance 1.098612; triangle area 0.4899573; disk/Lorentz distance residual 2.22045e-16. Area here is derived from angle defect.

Disk radius r = 0.7: radial metric distance 1.734601; triangle area 0.9112313; disk/Lorentz distance residual 4.44089e-16. Area here is derived from angle defect.

Disk radius r = 0.9: radial metric distance 2.944439; triangle area 1.361618; disk/Lorentz distance residual 4.44089e-16. Area here is derived from angle defect.

Disk radius r = 0.94: radial metric distance 3.476099; triangle area 1.44736; disk/Lorentz distance residual 4.44089e-16. Area here is derived from angle defect.

Download summary CSV Download complete summary JSON

Recorded radial disk distances

Vertical: Distance from the origin

Hyperbolic metric distanceEuclidean screen distance

Lines connect the recorded cases to aid comparison. Intermediate settings are not additional experiments.

Actual case outputs. JSON retains every observable and the complete computed data.
Case	Selected measurements	Residual and tolerance	Data
1. Planar right triangle	Angle sum: 3.141593 rad Triangle area: 0.5 model units² Distance AB: 1 model units Area method: Euclidean coordinate determinant / 2	0 Tolerance: 2.41421e-10	JSON · CSV
2. Spherical coordinate-axis triangle · R=1	Angle sum: 4.712389 rad Triangle area: 1.570796 model units² Distance AB: 1.570796 model units Area method: Independent unit-vector solid angle × R² Independent area residual: 0	0 Tolerance: 2.57080e-10	JSON · CSV
3. Spherical coordinate-axis triangle · R=2	Angle sum: 4.712389 rad Triangle area: 6.283185 model units² Distance AB: 3.141593 model units Area method: Independent unit-vector solid angle × R² Independent area residual: 0	0 Tolerance: 7.28319e-10	JSON · CSV
4. Disk radial triangle · r=0.1	Angle sum: 3.121593 rad Triangle area: 0.01999933 model units² Distance AB: 0.2006707 model units Area method: Area derived from angle defect Disk / Lorentz distance residual: 5.55112e-17	5.55112e-17 Tolerance: 1.28474e-10	JSON · CSV
5. Disk radial triangle · r=0.3	Angle sum: 2.962076 rad Triangle area: 0.1795163 model units² Distance AB: 0.6190392 model units Area method: Area derived from angle defect Disk / Lorentz distance residual: 0	0 Tolerance: 1.90160e-10	JSON · CSV
6. Disk radial triangle · r=0.5	Angle sum: 2.651635 rad Triangle area: 0.4899573 model units² Distance AB: 1.098612 model units Area method: Area derived from angle defect Disk / Lorentz distance residual: 2.22045e-16	2.22045e-16 Tolerance: 2.68070e-10	JSON · CSV
7. Disk radial triangle · r=0.7	Angle sum: 2.230361 rad Triangle area: 0.9112313 model units² Distance AB: 1.734601 model units Area method: Area derived from angle defect Disk / Lorentz distance residual: 4.44089e-16	4.44089e-16 Tolerance: 3.83394e-10	JSON · CSV
8. Disk radial triangle · r=0.9	Angle sum: 1.779975 rad Triangle area: 1.361618 model units² Distance AB: 2.944439 model units Area method: Area derived from angle defect Disk / Lorentz distance residual: 4.44089e-16	4.44089e-16 Tolerance: 6.20123e-10	JSON · CSV
9. Disk radial triangle · r=0.94	Angle sum: 1.694232 rad Triangle area: 1.44736 model units² Distance AB: 3.476099 model units Area method: Area derived from angle defect Disk / Lorentz distance residual: 4.44089e-16	4.44089e-16 Tolerance: 7.26096e-10	JSON · CSV

Interpretation

The sphere cases keep the same three unit directions as radius changes. The disk cases keep the form (0, r, ir) while changing r. Compare values within those controlled families; the three model defaults do not represent an isometric triangle shared by all geometries.

An area calculated directly from angle defect cannot independently validate the defect formula. The spherical cross-check uses a separate calculation; the disk area remains labeled as derived.

The radial disk cases compare multiple coordinate representations of the same model, not measurements of physical space.

Limitations and possible errors

The radial interaction cutoff is a numerical and viewing restriction.
The study excludes degenerate, antipodal and invalid triangles. Screen distances on the pictured hyperboloid are not Lorentz distances.

Warnings recorded during these runs

A vertex is near the interaction cutoff. Disk screen lengths strongly understate metric lengths there.

Reproduce this study

Open Geometry and restore the case’s model, sphere radius and vertex coordinates.
Inspect the saved JSON and CSV for the actual metric and area-method metadata before comparing cases.

Open geometry

1. Planar right triangle

Recorded 17 September 2026 at 00:39 UTC. 37 exported data rows.

{
  "model": "plane",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      1,
      0,
      0
    ],
    [
      0,
      1,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Euclidean coordinate distance; Euclidean coordinate determinant / 2. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.

Angle sum: 3.141592654 rad
Triangle area: 0.5 model units²Euclidean coordinate determinant / 2
Curvature K: 0 model units⁻²
Distance AB: 1 model units
Distance BC: 1.41421356 model units
Distance CA: 1 model units
Angle A: 1.57079633 rad
Angle B: 0.785398163 rad
Angle C: 0.785398163 rad
Area method: Euclidean coordinate determinant / 2

Download complete case JSON · Download case CSV

2. Spherical coordinate-axis triangle · R=1

Recorded 17 September 2026 at 00:39 UTC. 322 exported data rows.

{
  "model": "sphere",
  "radius": 1,
  "points": [
    [
      1,
      0,
      0
    ],
    [
      0,
      1,
      0
    ],
    [
      0,
      0,
      1
    ]
  ]
}

Model version: geometry-1.0.0

R acos(clamped dot); equivalent atan2(cross norm, dot) near coincident or antipodal directions; Independent unit-vector solid angle × R². Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.

Angle sum: 4.71238898 rad
Triangle area: 1.570796327 model units²Independent unit-vector solid angle × R²
Curvature K: 1 model units⁻²
Distance AB: 1.57079633 model units
Distance BC: 1.57079633 model units
Distance CA: 1.57079633 model units
Angle A: 1.57079633 rad
Angle B: 1.57079633 rad
Angle C: 1.57079633 rad
Area method: Independent unit-vector solid angle × R²
Area derived from angle excess: 1.570796327 model units²
Independent area residual: 0

Download complete case JSON · Download case CSV

3. Spherical coordinate-axis triangle · R=2

Recorded 17 September 2026 at 00:39 UTC. 322 exported data rows.

{
  "model": "sphere",
  "radius": 2,
  "points": [
    [
      1,
      0,
      0
    ],
    [
      0,
      1,
      0
    ],
    [
      0,
      0,
      1
    ]
  ]
}

Model version: geometry-1.0.0

Angle sum: 4.71238898 rad
Triangle area: 6.283185307 model units²Independent unit-vector solid angle × R²
Curvature K: 0.25 model units⁻²
Distance AB: 3.14159265 model units
Distance BC: 3.14159265 model units
Distance CA: 3.14159265 model units
Angle A: 1.57079633 rad
Angle B: 1.57079633 rad
Angle C: 1.57079633 rad
Area method: Independent unit-vector solid angle × R²
Area derived from angle excess: 6.283185307 model units²
Independent area residual: 0

Download complete case JSON · Download case CSV

4. Disk radial triangle · r=0.1

Recorded 17 September 2026 at 00:39 UTC. 333 exported data rows.

{
  "model": "hyperbolic",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      0.1,
      0,
      0
    ],
    [
      0,
      0.1,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Disk: 2 asinh(chord / geometric mean of boundary factors); hyperboloid: Lorentz displacement norm; Area derived from angle defect. Angles use outgoing unit tangents. Rendering samples do not approximate the reported lengths or areas.

Angle sum: 3.12159332 rad
Triangle area: 0.01999933337 model units²Area derived from angle defect
Curvature K: -1 model units⁻²
Distance AB: 0.200670695 model units
Distance BC: 0.284736857 model units
Distance CA: 0.200670695 model units
Angle A: 1.57079633 rad
Angle B: 0.775398497 rad
Angle C: 0.775398497 rad
Area method: Area derived from angle defect
Disk / Lorentz distance residual: 5.551e-17
Radial distance B: 0.200670695 model units
Radial distance C: 0.200670695 model units

Download complete case JSON · Download case CSV

5. Disk radial triangle · r=0.3

Recorded 17 September 2026 at 00:39 UTC. 333 exported data rows.

{
  "model": "hyperbolic",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      0.3,
      0,
      0
    ],
    [
      0,
      0.3,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Angle sum: 2.962076305 rad
Triangle area: 0.1795163484 model units²Area derived from angle defect
Curvature K: -1 model units⁻²
Distance AB: 0.619039208 model units
Distance BC: 0.901599473 model units
Distance CA: 0.619039208 model units
Angle A: 1.57079633 rad
Angle B: 0.695639989 rad
Angle C: 0.695639989 rad
Area method: Area derived from angle defect
Disk / Lorentz distance residual: 0
Radial distance B: 0.619039208 model units
Radial distance C: 0.619039208 model units

Download complete case JSON · Download case CSV

6. Disk radial triangle · r=0.5

Recorded 17 September 2026 at 00:39 UTC. 333 exported data rows.

{
  "model": "hyperbolic",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      0.5,
      0,
      0
    ],
    [
      0,
      0.5,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Angle sum: 2.651635327 rad
Triangle area: 0.4899573263 model units²Area derived from angle defect
Curvature K: -1 model units⁻²
Distance AB: 1.09861229 model units
Distance BC: 1.68069977 model units
Distance CA: 1.09861229 model units
Angle A: 1.57079633 rad
Angle B: 0.5404195 rad
Angle C: 0.5404195 rad
Area method: Area derived from angle defect
Disk / Lorentz distance residual: 2.220e-16
Radial distance B: 1.09861229 model units
Radial distance C: 1.09861229 model units

Download complete case JSON · Download case CSV

7. Disk radial triangle · r=0.7

Recorded 17 September 2026 at 00:39 UTC. 333 exported data rows.

{
  "model": "hyperbolic",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      0.7,
      0,
      0
    ],
    [
      0,
      0.7,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Angle sum: 2.230361347 rad
Triangle area: 0.9112313064 model units²Area derived from angle defect
Curvature K: -1 model units⁻²
Distance AB: 1.73460106 model units
Distance BC: 2.8339393 model units
Distance CA: 1.73460106 model units
Angle A: 1.57079633 rad
Angle B: 0.32978251 rad
Angle C: 0.32978251 rad
Area method: Area derived from angle defect
Disk / Lorentz distance residual: 4.441e-16
Radial distance B: 1.73460106 model units
Radial distance C: 1.73460106 model units

Download complete case JSON · Download case CSV

8. Disk radial triangle · r=0.9

Recorded 17 September 2026 at 00:39 UTC. 333 exported data rows.

{
  "model": "hyperbolic",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      0.9,
      0,
      0
    ],
    [
      0,
      0.9,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Angle sum: 1.779974996 rad
Triangle area: 1.361617658 model units²Area derived from angle defect
Curvature K: -1 model units⁻²
Distance AB: 2.94443898 model units
Distance BC: 5.20123293 model units
Distance CA: 2.94443898 model units
Angle A: 1.57079633 rad
Angle B: 0.104589334 rad
Angle C: 0.104589334 rad
Area method: Area derived from angle defect
Disk / Lorentz distance residual: 4.441e-16
Radial distance B: 2.94443898 model units
Radial distance C: 2.94443898 model units

Download complete case JSON · Download case CSV

9. Disk radial triangle · r=0.94

Recorded 17 September 2026 at 00:39 UTC. 333 exported data rows.

{
  "model": "hyperbolic",
  "radius": 1,
  "points": [
    [
      0,
      0,
      0
    ],
    [
      0.94,
      0,
      0
    ],
    [
      0,
      0.94,
      0
    ]
  ]
}

Model version: geometry-1.0.0

Angle sum: 1.694232479 rad
Triangle area: 1.447360174 model units²Area derived from angle defect
Curvature K: -1 model units⁻²
Distance AB: 3.47609869 model units
Distance BC: 6.2609587 model units
Distance CA: 3.47609869 model units
Angle A: 1.57079633 rad
Angle B: 0.0617180763 rad
Angle C: 0.0617180763 rad
Area method: Area derived from angle defect
Disk / Lorentz distance residual: 4.441e-16
Radial distance B: 3.47609869 model units
Radial distance C: 3.47609869 model units

Download complete case JSON · Download case CSV

Each run’s download address is available in the summary JSON. The printed article preserves the methods, measurements and limitations.

References and review status

Project attribution: BIPH Math Society. Exposition and implementation were prepared with tool assistance.

Review status: Not reviewed. Independent teacher or member review has not yet been recorded.

[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.