2025-11-29 01:53:05
In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius 1, the area equals the triangle excess
Area = E = interior angle sum − π.
Small triangles have interior angle sum near π. But you could, for example, have a triangle with three right angles: put a vertex on the north pole and two vertices on the equator 90° longitude apart.
In hyperbolic geometry, the sum of the interior angles of a triangle is always less than π. In a space with curvature −1, the area equals the triangle defect, the difference between π and the angle sum.
Area = D = π − interior angle sum.
Again small triangles have an interior angle sum near π. Both spherical and hyperbolic geometry are locally Euclidean.
The interior angle sum can be any value less than π, and so as the angle sum goes to 0, the triangle defect, and hence the area, goes to π.
The figure below has interior angle sum 0 and area π in hyperbolic geometry.
Strictly speaking this is an improper triangle because the three hyperbolic lines (i.e. half circles) don’t intersect within the hyperbolic plane per se but at ideal points on the real axis. But you could come as close to this triangle as you like, staying within the hyperbolic plane.
Note that the radii of the (Euclidean) half circles doesn’t change the area. Any three semicricles that intersect on the real line as above make a triangle with the same area.
The post A triangle whose interior angles sum to zero first appeared on John D. Cook.2025-11-28 22:54:46
Let ℍ be the upper half plane, the set of complex real numbers with positive imaginary part. When we measure distances the way we’ve discussed in the last couple posts, the geometry of ℍ is hyperbolic.
What is a circle of radius r in ℍ? The same as a circle in any geometry: it’s the set of points a fixed distance r from a center. But when you draw a circle using one metric, it may look very different when viewed from the perspective of another metric.
Suppose we put on glasses that gave us a hyperbolic perspective on ℍ, draw a circle of radius r centered at i, then take off the hyperbolic glasses and put on Euclidean glasses. What would our drawing look like?
In the previous post we gave several equivalent expressions for the hyperbolic metric. We’ll use the first one here.
Here the Fraktur letter ℑ stands for imaginary part. So the set of points in a circle of radius r centered at i is
Divide the expression for d(x + iy, i) by 2, apply sinh, and square. This gives us
which is an equation for a Euclidean circle. If we multiply both sides by 4y and complete the square, we find that the center of the circle is (0, cosh(r)) and the radius is sinh(r).
So to recap, if we put on our hyperbolic glasses and draw a circle, then switch out these glasses for Euclidean glasses, the figure we drew again looks like a circle.
To put it another way, a hyperbolic viewer and a Euclidean viewer would agree that a circle has been draw. However, the two viewers would disagree where the center of the circle is located, and they would disagree on the radius.
Both would agree that the center is on the imaginary axis, but the hyperbolic viewer would say the imaginary part of the center is 1 and the Euclidean viewer would say it’s cosh(r). The hyperbolic observer would say the circle has radius r, but the Euclidean observer would say it has radius sinh(r).
For small r, the hyperbolic and Euclidean viewpoints nearly agree because
cosh(r) = 1 + O(r²)
and
sinh(r) = r + O(r³)
Note that if you asked a Euclidean observer to draw a circle of radius 100, centered at (0, 1), he would say that the circle will extend outside of the half plane. A hyperbolic observer would disagree. From his perspective, the real axis is infinitely far away and so he can draw a circle of any radius centered at any point and stay within the half plane.
Now what if we looked at circles centered somewhere else? The hyperbolic metric is invariant under Möbius transformations, and so in particular it is invariant under
z ↦ x0 + y0z.
This takes a circle with hyperbolic center i to a circle centered at x0 + i y0 without changing the hyperbolic radius. The Euclidean center moves from cosh(r) to y0 cosh(r) and the radius changes from sinh(r) to y0 sinh(r).
The post A circle in the hyperbolic plane first appeared on John D. Cook.2025-11-27 21:14:40
The previous post described a metric for the Poincaré upper half plane. The development is geometrical rather than analytical. There are also analytical formulas for the metric, at least four that I’ve seen.
It’s not at all obvious that the four equations are equivalent, or that any of them matches the expression in the previous post.
There are equations for expressing arcsinh, arccosh, and arctanh in terms of logarithms and square roots. See the bottom of this post. You could use these identities to show that the metric expressions are equal, but I don’t know of a cleaner way to do this than lots of tedious algebra.
Before diving into the calculations, you might want some assurance that you’re trying to prove the right thing. Here’s some Python code that generates random pairs of complex numbers and shows that the four expressions give he same distance.
import numpy as np
def d1(z1, z2):
return 2*np.arcsinh( abs(z1 - z2) / (2*(z1.imag * z2.imag)**0.5) )
def d2(z1, z2):
return np.arccosh(1 + abs(z1 - z2)**2 / (2*z1.imag * z2.imag) )
def d3(z1, z2):
return 2*np.arctanh( abs( (z1 - z2)/(z1 - np.conjugate(z2)) ) )
def d4(z1, z2):
return 2*np.log( (abs(z2 - z1) + abs(z2 - np.conjugate(z1)))/(2*np.sqrt(z1.imag * z2.imag)) )
np.random.seed(20251127)
for n in range(100):
z1 = np.random.random() + 1j*np.random.random()
z2 = np.random.random() + 1j*np.random.random()
assert( abs(d1(z1, z2) - d2(z1, z2)) < 1e-13 )
assert( abs(d2(z1, z2) - d3(z1, z2)) < 1e-13 )
assert( abs(d3(z1, z2) - d4(z1, z2)) < 1e-13 )
Perhaps you’re convinced that the four expressions are equal, but why should any of them be equivalent to the definition in the previous post?
The previous post pointed out that the metric is invariant under Möbius transformations. We can apply such a transformation to move any pair of complex numbers to the imaginary axis. There you can see that the cross ratio reduces to the ratio of the two numbers.
More generally, if two complex numbers have the same real part, the distance between them is the log of the ratio of their imaginary parts. That is, if
then
if x, y1, and y2 are real and y2 > y1 > 0.
Here’s a little Python code that empirically shows that this gives the same distance as one of the expressions above.
def d5(z1, z2):
assert(z1.real == z1.real)
return abs( np.log( z1.imag / z2.imag ) )
for n in range(100):
x = np.random.random()
z1 = x + 1j*np.random.random()
z2 = x + 1j*np.random.random()
assert( abs(d1(z1, z2) - d5(z1, z2)) < 1e-13 )
So now we have five expressions for the metric, all of which look different. You could slug out a proof that they’re equivalent, or get a CAS like Mathematica to show they’re equivalent, but it would be more interesting to find an elegant equivalence proof.
Update: Although the four expressions at the top of the post are analytically equal, they are not all equally accurate for numerical evaluation. I did a little testing and found the arctahn method to be the least accurate and the rest roughly equally accurate.
The post Equal things that don’t look equal first appeared on John D. Cook.2025-11-27 02:28:14
One common model of the hyperbolic plane is the Poincaré upper half plane ℍ. This is the set of points in the complex plane with positive imaginary part. Straight lines are either vertical, a set of points with constant imaginary part, or arcs of circles centered on the real axis. The real axis is not part of ℍ. From the perspective of hyperbolic geometry these are ideal parts, infinitely far away, and not part of the plane itself.
We can define a metric on ℍ as follows. To find the distance between two points u and v, draw a line between the two points, and let a and b be the ideal points at the end of the line. By a line we mean a line as defined in the geometry of ℍ, what we would see from our Euclidean perspective as a half circle or a vertical line. Then the distance between u and v is defined as the absolute value of the log of the cross ratio (u, v; a, b).
Cross ratios are unchanged by Möbius transformations, and so Möbius transformations are isometries.
Another common model of hyperbolic geometry is the Poincaré disk. We can use the same metric on the Poincaré disk because the Möbius transformation
maps the upper half plane to the unit disk. This is very similar to how the Smith chart is created by mapping a grid in the right half plane to the unit disk.
Update: See the next post for four analytic expressions for the metric, direct formulas involving u and v but not the ideal points a and b.
The post Hyperbolic metric first appeared on John D. Cook.2025-11-25 09:25:29
The opening line of William Gibson’s novel Neuromancer is famous:
The sky above the port was the color of a television, tuned to a dead channel.
When I read this line, I knew immediately what he meant, and thought it was a brilliant line. Later I learned that younger readers didn’t know what he was saying.
My mind went to an old black-and-white television, one that received analog broadcasts, and that displayed “snow” when tuned to a channel that had no broadcast signal. Someone whose earliest memories of television are based on digital color broadcast might imagine the sky above the port was solid blue rather than crackly gray.
Gibson discusses how his book has aged in a preface to a recent edition. He says that science fiction that is too prescient would be received poorly.
Imagine a novel from the sixties whose author had somehow fully envisioned cellular telephony circa 2004, and had worked it, exactly as we know it today, into the fabric of her imaginary future. Such a book would have seemed highly peculiar in the sixties … in ways that would quickly overwhelm the narrative.
He then goes on to say
I suspect that Neuromancer owes much of its shelf life to my almost perfect ignorance of the technology I was extrapolating from. … Where I made things up from whole cloth, the colors remain bright.
I find it odd that many judge a work of science fiction by what it “got right.” I don’t read science fiction as a forecast; read it to enjoy a story. I don’t need a book to be prescient, but until reading Gibson’s remarks it hadn’t occurred to me that fiction that is too prescient might not be enjoyable fiction, at least for its first readers.
The post TV tuned to a dead channel first appeared on John D. Cook.2025-11-25 01:06:51
Suppose Alice runs a confidential restaurant. Alice doesn’t want there to be any record of who visited her restaurant but does want to get paid for her food. She accepts Monero, and instead of a cash register there are two QR codes on display, one corresponding to her public view key A and the other corresponding to her public spend key S.
A customer Bob walks into the restaurant and orders a burger and fries. When Bob pays Alice, here’s what’s going on under the hood.
Bob is using software that generates a random integer r and multiplies it by a point G on an elliptic curve, specifically ed25519, obtaining the point
R = rG
on the curve. The software also multiplies Alice’s view key A, a point on the same elliptic curve, by r, then runs a hash function H on the produce rA that returns an integer k.
k = H(rA).
Finally, Bob’s software computes the point
P = kG + S
and sends Alice’s cash register, i.e. her crypto wallet, the pair of points (P, R). The point P is a stealth address, an address that will only be used this one time and cannot be linked to Alice or Bob [1]. The point R is additional information that helps Alice receive her money.
Alice and Bob share a secret: both know k. How’s that?
Alice’s public view key A is the product of her private view key a and the group generator G [2]. So when Bob computes rA, he’s computing r(aG). Alice’s software can multiply the point R by a to obtain a(rG).
rA = r(aG) = a(rG) = aR.
Both Alice and Bob can hash this point—which Alice thinks of as aR and Bob thinks of as rA—to obtain k. This is ECDH: elliptic curve Diffie-Hellman key exchange.
Next, Alice’s software scans the blockchain for payments to
P = kG + S.
Note that P is on the blockchain, but only Alice and Bob know how to factor P into kG + S because only Alice and Bob know k. And only Alice can spend the money because only she knows the private key s corresponding to the public spend key S where
S = sG.
She knows
P = kG + sG = (k + s)G
and so she has the private key (k + s) corresponding to P.
[1] Bob sends money to the address P, so there could be some connection between Bob and P on the Monero blockchain. However, due to another feature of Monero, namely ring signatures, someone analyzing the blockchain could only determine that Bob is one of 16 people who may have sent money to the address P, and there’s no way to know who received the money. That is, there is no way, using only information on the blockchain, who received the money. A private investigator who saw Bob walk into Alice’s restaurant would have additional information outside the blockchain.
[2] The key assumption of elliptic curve cryptography is that it’s computationally infeasible to “divide” on an elliptic curve, i.e. to recover a from knowledge of G and aG. You could recover a by brute force if the group were small, but the elliptic curve ed25519 has on the order of 2255 points, and a is some integer chosen randomly between 1 and the size of the curve.
The post How stealth addresses work in Monero first appeared on John D. Cook.
