Study metric tensors, arc elements, and local area scaling. Solve upper half-plane form problems using exports, plots, examples, and clear guidance.
This graph shows how the metric coefficient 1/y² changes with height y. The selected y value is marked for quick local interpretation.
| x | y | dx | dy | E | F | G | ds² | ds | √det(g) |
|---|---|---|---|---|---|---|---|---|---|
| 0.0000 | 1.0000 | 1.0000 | 0.0000 | 1.000000 | 0.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 |
| 0.5000 | 2.0000 | 1.0000 | 1.0000 | 0.250000 | 0.000000 | 0.250000 | 0.500000 | 0.707107 | 0.250000 |
| -1.0000 | 3.0000 | 2.0000 | 1.0000 | 0.111111 | 0.000000 | 0.111111 | 0.555556 | 0.745356 | 0.111111 |
| 1.2000 | 0.8000 | 0.4000 | 0.9000 | 1.562500 | 0.000000 | 1.562500 | 1.515625 | 1.231107 | 1.562500 |
For the hyperbolic upper half-plane, the metric is
ds² = (dx² + dy²) / y²
This means the first fundamental form has coefficients:
E = 1 / y², F = 0, G = 1 / y²
So for a differential vector (dx, dy), the evaluated squared length is:
ds² = E dx² + 2F dxdy + G dy²
Since F = 0 here, it becomes:
ds² = (dx² + dy²) / y²
The determinant of the metric tensor is:
det(g) = EG - F² = 1 / y⁴
The local area density is:
√det(g) = 1 / y²
The first fundamental form describes how lengths and angles are measured on a surface or geometric model. In the upper half-plane model of hyperbolic geometry, the metric depends on height. As y becomes smaller, the metric coefficient grows, so equal Euclidean displacements represent larger hyperbolic lengths.
This calculator is useful for differential geometry, Riemannian geometry, and hyperbolic plane studies. It computes the metric tensor entries, evaluated arc element, determinant, and local area density. The tool also gives a graph and export options so you can inspect results more carefully.
Because the metric is diagonal in this model, calculations are direct but still informative. You can test how the same vector changes its hyperbolic length at different heights. This makes the calculator helpful for building geometric intuition and checking manual derivations.
It computes the first fundamental form for the hyperbolic upper half-plane at a selected point. It returns E, F, G, the evaluated line element, metric determinant, and local area density for the differential vector you provide.
The upper half-plane model only includes points above the real axis. The metric contains 1/y², so y must stay positive. At y = 0, the expression is undefined and the model boundary is reached.
They are the coefficients of the first fundamental form. In this model, E = 1/y², F = 0, and G = 1/y². These define the metric tensor and control local measurement of lengths and areas.
ds² is the squared hyperbolic line element for the given differential vector. It measures infinitesimal distance according to the hyperbolic metric, not the ordinary Euclidean metric of the coordinate plane.
The hyperbolic metric scales by 1/y². Near the boundary where y is small, equal Euclidean displacements become much larger hyperbolic displacements. Higher points shrink that scaling and produce smaller hyperbolic lengths.
√det(g) is the local area density of the metric. It tells you how Euclidean area elements are rescaled into hyperbolic area elements. In the upper half-plane model, this value is exactly 1/y².
No. This version evaluates the first fundamental form and related local quantities only. It does not solve geodesic equations, although its outputs are the ingredients needed for deeper geometric analysis.
It is useful in differential geometry coursework, hyperbolic geometry examples, metric tensor verification, and quick checks of handwritten derivations. It also helps compare local behavior of vectors at different heights in the model.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.