The Hyperbolic Plane

The plane of constant negative curvature, drawn in the Poincaré disc. Build a triangle and compare its angles and area with a flat one, set a point beside a line and see which lines through it never meet the line, or measure a circle against 2πr.

Worked example — The Hyperbolic Plane

Sides, as distances in the plane	1.0000, 1.0000, 1.0000
Angle at A	52.643°
Angle at B	52.643°
Angle at C	52.643°
The three angles together	157.93°
Area	0.385
As a share of π	12.26%
A flat triangle with the same sides (Heron)	0.43

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — The Hyperbolic Plane

The hyperbolic plane is the complete, simply connected surface whose Gaussian curvature is the same negative number everywhere; here K = −1, so lengths are in units of the curvature radius and areas in its square. A triangle's area is π less its angle sum, and through a point not on a line pass infinitely many lines that never meet it. The Poincaré disc draws the whole infinite plane inside a circle: straight lines become arcs that meet the rim at right angles, angles are drawn true, and lengths shrink towards the rim, which is infinitely far away.

What each symbol means

K: the Gaussian curvature, −1 at every point of this plane
Π(d): the angle of parallelism: lines through a point at distance d from a line meet it only within Π(d) of the perpendicular

No triangle here holds more than π, however long its sides 🖖

Press Equilateral, side 1. The sides are straight lines of this plane, drawn as arcs because the disc is a map, and the angles add to 157.93°. The shortfall from 180° is the area: 180° − 157.93° = 22.07°, which in radians is 0.385. Now press Equilateral, side 10. Each angle has closed to 0.772°, the sum is 2.32°, and the area is 3.101, 98.71% of π, where a flat triangle with sides of 10 holds 43.30. Since the area is π less the angle sum, and angles cannot fall below zero, no triangle in this plane holds more than π, whatever its sides. The table shows the approach: side 5 already reaches 84.41% of π, side 20 reaches 99.99%. Only a triangle whose corners have gone off to infinity, with three angles of 0°, holds π exactly.

Through a point off a line, infinitely many lines never meet it 🖖

Press Parallels at distance 1. The black line is a straight line of this plane, and the point is 1 from it. Watch the line turning through the point. It meets the black line only while it is within 40.395° of the perpendicular, on one side or the other. Lobachevsky called that angle the angle of parallelism, Π(d), and in 1840 gave it as tan(Π/2) = e^−d. The two lines at exactly Π(d), in pink, close in on the black line towards its ends and never reach it, and every line between them in the remaining 99.21° misses it too. Euclid's geometry has exactly one parallel through the point; this one has infinitely many. Press Parallels at distance 3: Π(d) is down to 5.700°, and 168.60° of directions miss. Bring the point close instead and the fan narrows towards nothing, 55.05° at d = 0.5: near the line, the plane behaves almost as paper does.

A circle's rim grows exponentially, and most of a disc lies near its edge 🖖

Press Circle of radius 10. The circumference is 2π sinh 10 = 69,198.18, against 62.83 on paper, 1,101 times as long. The area inside, 2π(cosh 10 − 1) = 69,191.90, is almost the same number, and 63.22% of it lies within 1 of the rim. Far out, every step of 1 multiplies the rim by nearly e. Henderson and Taimina crochet a model of this plane with that rule: in every new row they turn N stitches into N + 1, so each row is the same factor longer than the last, and the fabric, holding more edge than a flat sheet has room for, has to fold and wave. The saddle in Gauss's Theorema Egregium ran long in the same way. Type r = 0.5 here and the ring gives a curvature of −1.0126 (the estimate is exact only as the ring shrinks), where the saddle's ring gave −0.884 at the same radius: this plane is curved −1 everywhere, the saddle only at its centre.

Problem solved in full

A triangle known only by its angles 4 steps
A triangle in this plane has angles of 30°, 45° and 60°. What is its area, and how long are its sides? On paper the question has no answer: every enlargement of a triangle keeps its angles. Press Equilateral, side 1 first, where the page gives an angle sum and an area.

Open this problem in the tool
1. On the preset the area is π less the angle sum, in radians:
  
  $A = π - (A + B + C) = (18 0^{\circ} - 157.9 3^{\circ}) \times \frac{π}{18 0 ^{\circ}} = 0.385$
2. The same rule gives this triangle's area at once:
  
  $A = (18 0^{\circ} - 3 0^{\circ} - 4 5^{\circ} - 6 0^{\circ}) \times \frac{π}{18 0 ^{\circ}} = \frac{π}{4} = 0.7854$
3. For the sides, this plane has a second law of cosines, which takes angles in and gives a side out:
  
  $cosh a = \frac{cos 3 0 ^{\circ} + cos 4 5 ^{\circ} cos 6 0 ^{\circ}}{sin 4 5 ^{\circ} sin 6 0 ^{\circ}} = \frac{0 . 866025 + 0 . 353553}{0 . 612372} = 1.991564 \Rightarrow a = 1.3121$
4. and the other two the same way, starting from B and from C:
  
  $cosh b = \frac{cos B + cos C cos A}{sin C sin A} \Rightarrow b = 1.6231, cosh c = \frac{cos C + cos A cos B}{sin A sin B} \Rightarrow c = 1.8131$
Answer

Area π/4 = 0.7854, sides 1.3121, 1.6231 and 1.8131. No other triangle has these angles: enlarge it and its area grows, so its angle sum has to fall. Similar triangles of different sizes do not exist in this plane, and a triangle's angles carry a length scale with them, the curvature radius.

$A = \frac{π}{4} = 0.7854, a, b, c = 1.3121, 1.6231, 1.8131$

Learning path

Geometry on a curved surface

References (2)

The angle of parallelism: through a point at distance p from a line, the lines within Π(p) of the perpendicular cut it and the rest do not, with tan ½Π(x) = e⁻ˣ: N. Lobachevsky, Geometrical Researches on the Theory of Parallels (Berlin, 1840), translated by G. B. Halsted, Scientiae Baccalaureus 1(3), February 1891 (scan at the University of Missouri Digital Library).
A model of the hyperbolic plane crocheted by increasing the stitches from N to N + 1 in every row, so each row is the same factor longer than the last; and the area of an ideal triangle, with all three corners at infinity, equal to π times the square of the plane's radius: D. W. Henderson and D. Taimiņa, "Crocheting the Hyperbolic Plane", The Mathematical Intelligencer 23(2) (2001), 17–28.

By the 42Math team · Reviewed against published sources. Corrections welcome.