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Integration Hub

Universal pipeline engine — dynamically chain any functions across all sources

—

Total Functions

—

Pipeline Builder

Auto-Compose

Catalog — All Functions by Category

Compatibility Explorer — What can connect to what?

👥 Collaborate

Real-time sessions with screen-sharing, chat & feature sync

Direct Connection (WiFi)

Peer-to-peer, no server needed. Same network or copy-paste codes.

Status: Disconnected

Create Room

Room Name

Password (optional)

Join Room

Room ID or Name

Password (if required)

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Shared Calculation Feed

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📝 Scratchpad ▾

Functions

⊞ Matrix Lab

Interactive matrix computation & visualisation

Matrix Builder

Rows

Cols

Operations on Matrix A

Matrix B (same size)

Result

— run an operation —

Linear System Ax=b

Vector b (comma-separated)

Expression Evaluator

Enter a matrix expression

Heatmap Visualisation

◉ Statistics Studio

Descriptive stats, distributions, hypothesis tests, regression

Dataset Input

Data (comma-separated numbers)

Descriptive Statistics

N
—

Mean
—

Median
—

Std Dev
—

Min
—

Max
—

Skewness
—

Kurtosis
—

IQR
—

Mode
—

Histogram

Bins

Two-Sample Tests

Sample A

Sample B

Linear Regression

X values

Y values

Distribution Calculator

Distribution

Param 1

Param 2

x

◈ Finance Calculator

Loans, investments, options, DCF, bonds, portfolio analysis

Loan / Mortgage

Principal ($)

Annual rate (%)

Term (years)

Payments/year

Compound Interest / Investment

Initial ($)

Annual rate (%)

Years

Monthly contrib ($)

Bond Pricing

Face value ($)

Coupon rate (%)

Years to maturity

YTM / mkt rate (%)

ROI / CAGR

Initial value

Final value

Years

Black-Scholes Options

Spot price S

Strike K

Time (years)

Risk-free rate r

Volatility σ

NPV / IRR (DCF)

Discount rate (%)

Cash flows (year 0,1,2… comma-separated)

Retirement Planner

Current savings ($)

Monthly contrib ($)

Annual return (%)

Years to retire

Retire annual need ($)

Inflation (%)

Currency Rates (live)

Amount

From

To

⇌ Unit Converter

Convert between hundreds of units — or write unit-aware expressions directly

⚡ Unit-Aware Expression Evaluator

Type any expression with mixed units — e.g. 5 km + 3 mi, 60 kg * 9.81 m/s^2, 100 degC to degF

Quick Convert

Value

From unit

To unit

Length

Mass

Temperature

°C

°F

K

°R (Rankine)

°De (Delisle)

Area

Volume

Speed

Energy / Power

Joules

kWh

Cal (kcal)

BTU

eV

ft·lbf

Digital Storage

📏 Unit Conversions

Comprehensive unit conversion across 20+ categories

Category

Value

From

→

To

All Conversions

🪐 Solar System Simulator

N-body gravitational simulation — place bodies, set velocities, watch orbits evolve

Playback

Speed: 1.0x

Trails Labels Vectors

G constant: 1.0

Time: 0.00 | Bodies: 0 | Energy: —

Add Body

Name

Mass

X

Y

Vx

Vy

Color

Bodies

✏️ Edit:

Name

Mass

X

Y

Vx

Vy

Color

Radius

Fixed (immovable)

Interaction

Click to select bodies, drag canvas to pan

Collision Detection

Enable collision detection Merge on collision (combine mass)

Simulation auto-pauses on collision

Save / Load

Save/load custom arrangements to browser storage

Scroll to zoom · Drag to pan · Click to select

⚛ Physics Toolkit

Constants, mechanics, electromagnetism, thermodynamics, relativity, quantum

Physical Constants (click to copy)

Kinematics

Choose equation

Electromagnetism

Thermodynamics

Special Relativity

Velocity v (m/s)

Rest mass m₀ (kg)

Proper time τ (s)

Rest length L₀ (m)

Quantum Mechanics

Wave Physics

Frequency f (Hz)

Wave speed v (m/s)

∞ Number Theory

Primes, factorisation, sequences, modular arithmetic, cryptography

Primality & Factorisation

Number n

Sieve of Eratosthenes

Find primes up to

GCD / LCM / Bezout

a

b

Modular Arithmetic

base

exponent

modulus

Integer Sequences

N terms

Sequence

Continued Fractions

Number (decimal or expression)

Chinese Remainder Theorem

Remainders (comma-sep)

Moduli (pairwise coprime)

RSA Key Generation (educational)

Prime p

Prime q

⊻ Logic & Bit Operations

Boolean algebra, truth tables, bitwise ops, base conversion, gates

Interactive Bit Manipulator (32-bit)

Decimal

Hex

Binary (space-grouped OK)

Octal

Bitwise Operations

A (decimal)

B (decimal)

Truth Table Generator

Boolean expression (use A,B,C and &&, ||, !, ^)

Variables (comma-sep)

Base Converter

Value

Input base

Logic Gate Simulator

Input A

Input B

Checksum / Hash (simple)

⚄ Probability & Simulation

Combinatorics, random sampling, dice, Monte Carlo, Markov chains

Combinatorics Calculator

n

k / r

Dice Roller

Number of dice

Sides

Rolls

Birthday Problem

People n

Days/year

Monte Carlo π Estimator

Points to sample

Random Sampling

Population (comma-separated or range a–b)

Sample size

Conditional Probability / Bayes

P(A)

P(B|A)

P(B|¬A)

Markov Chain Simulator

Transition matrix (rows comma-sep, rows semicolons)

Initial state

Steps

Expected Value & Variance

Values (comma-sep)

Probabilities (comma-sep, must sum to 1)

Waves & Optics

Quantum Mechanics

Classical Mechanics

Fluid Mechanics

Chemistry

Periodic table, stoichiometry, thermochemistry, equilibrium, pH

Periodic Table

Click an element for details.

Molar Mass Calculator

Chemical formula

pH / pOH Calculator

Type

Value

Param 2 (Ka/Kb, conc.)

Stoichiometry / Dilution

Thermochemistry

Gas Laws

Solution Chemistry

Electrochemistry

Equilibrium

Atomic Structure

△ Geometry Studio

2D & 3D shapes, coordinate geometry, transformations, curves

Shape Calculator

Shape

Coordinate Geometry

x₁

y₁

x₂

y₂

Polygon (arbitrary)

Vertices as x1,y1;x2,y2;… (closed)

Geometry Canvas

Transformations

Points (x,y;…)

Param 1

Param 2

Conic Sections

Conic type

h

k

a/r

b/p

Function Plotter

Multi-function, implicit curves, asymptotes, tangent lines, derivative overlay, parametric, polar

x: to y: to

∫ from to

ℂ Complex Plane Explorer

Argand diagram, transformations, Mandelbrot/Julia sets

Complex Number z

Re(z)

Im(z)

Operations

Re(w)

Im(w)

Fractal Mode

Max iterations

∂ ODE Solver + Phase Portrait

Vector fields, solution curves, multiple numerical methods

Equation dy/dx = f(x,y)

f(x,y) — use x, y, math functions

x₀

y₀

x end

Step h

Method

2nd Order: y'' = f(x,y,y')

f(x,y,v) where v=y'

y₀

y'₀

Phase Portrait (dy/dx = f)

x range

y range

Click canvas to add solution curves

∿ Fourier Series Builder

Live harmonic synthesis, DFT/FFT, waveform reconstruction

Harmonics (drag sliders)

FFT Analysis

Signal data (comma-separated samples)

Waveform Presets

Terms N

Period T

↗ Linear Algebra Visualiser

Drag vectors, animate transforms, eigenvectors, SVD

2×2 Transformation Matrix

a

b

c

d

Eigen Analysis

SVD Decomposition

3D Vector Operations

u

v

Drag vectors on canvas · matrix shown in red/blue

⌖ Numerical Methods

Root finding, integration, interpolation — animated step-by-step

Root Finding

f(x)

a (left bracket)

b / x₀ (initial)

Tolerance

Max iterations

Numerical Integration

f(x)

a

b

n (intervals)

Interpolation

Known points (x,y;x,y;…)

Evaluate at x

Method

Differentiation (numerical)

f(x)

at x =

h (step)

Cryptography

Classical ciphers, frequency analysis, Diffie-Hellman, modern hashes

Text Input / Output

Plaintext / Ciphertext

Key / Parameter

Classical Ciphers

Frequency Analysis

Hash Functions

Diffie-Hellman Key Exchange

Prime p

Generator g

Alice's secret a

Bob's secret b

Base Encoding

One-Time Pad

Signal Processing

Convolution, filters, z-transform, windowing, spectral analysis

Signal Generator

Waveform

Frequency (Hz)

Amplitude

Sample rate

Duration (s)

Phase (rad)

Convolution & Correlation

Signal h(t) — kernel (comma-sep)

FIR Filter Design

Filter type

Cutoff f₁ (normalised)

Cutoff f₂ (band)

Taps N

Window

Windowing Functions

z-Transform (common pairs)

◎ Graph Theory

Build graphs, BFS/DFS, shortest path, MST, Euler/Hamilton

Graph Input

Edges (u-v or u-v:weight, one per line)

Directed?

Layout

Algorithms

Source node

Target node

Graph Properties

Click nodes to select · drag to reposition

Adjacency Matrix

📄 Documents

Full-featured rich text editor with equations, sharing & forum publishing

📄

Sign in to access your documents

Rich text editor with LaTeX equations, sharing & forum publishing.

Saved

0 words 0 chars rev 0

⌨ Code IDE

Python 3.11 (Pyodide) — full stdlib + packages

Not loaded

Programs

📦 Packages

🔗 Export as Custom Function

Export the current code as a custom function in the Computum library. The function appears under the Custom category in the function menu.

Function name

Category

Description

Auto-save

0 lines | 0 chars

Console Output

Python 3.11+ runtime (Pyodide v0.27.3). Press Run to start. Install packages from the package manager.

Fractal Explorer

Interactive fractal rendering & exploration

Fractal Type

Type

Max Iterations

Color Scheme

Resolution

View Bounds

Center Re

Center Im

Zoom

Julia C (Re)

Julia C (Im)

Presets

Fractal Dimensions Reference

Rendered Fractal (click to zoom in)

Hover over the fractal to see coordinates

🌀 Chaos & Attractor Lab

Explore dynamical systems, bifurcations & strange attractors

Attractor Type

System

Steps

dt

Downsample

Lorenz Parameters

σ (sigma)

ρ (rho)

β (beta)

x₀

y₀

z₀

Attractor Visualization

Time Series

◯ Topology Explorer

Euler characteristic, Betti numbers, knot invariants & surfaces

Euler Characteristic Calculator

Vertices (V)

Edges (E)

Faces (F)

Polyhedra Reference

Knot Invariants

Knot

Knot Parametric Curve

Knot Type

Simplicial Complex Analyzer

Simplices (JSON)

3D Visualization

Fundamental Groups Reference

S¹ → Z (infinite cyclic) | T² → Z×Z | S² → {e} (trivial) | RP² → Z/2Z | Klein bottle → ⟨a,b | aba⁻¹b⟩ | Möbius strip → Z

Data Science

Pandas-like DataFrame with statistics & visualization

Import Data

Paste CSV

Paste JSON

Manual Entry

Columns (comma-sep)

Rows

Data Table

Import data to begin

Chart

Output

Ready

Neural Network

Design, train, and visualize neural networks from scratch

Network Architecture

Layer Type

Units / Filters

Activation

L1 Regularization

L2 Regularization

Dropout Rate

Training Configuration

Optimizer

Loss Function

Learning Rate

Epochs

Batch Size

Momentum (SGD)

Beta1/Beta2 (Adam)

Ready

Preset Architectures

Training Data

Input (JSON array of arrays)

Target (JSON array of arrays)

Network Architecture Visualization

Training Loss

Prediction

Input (JSON array)

No prediction yet

Weight Visualization

Model Import / Export

CAS+ Step-by-Step

Computer Algebra System with detailed solution steps

Equation Solving (Step-by-Step)

Expression / Equation

Variable

Differentiation (Step-by-Step)

Expression

Variable

Integration (Step-by-Step)

Expression

Variable

Definite?

Algebra (Step-by-Step)

Expression

Linear System Solver

Matrix A (rows as arrays)

Vector b

Apply Transformation Rule

Expression

Rule

Solution Hints

Expression

Solution Steps

Enter an expression and choose an operation

Formula Database

Search

Quick Reference

📓 Notebook

Jupyter-style cell-based computation

Spreadsheet

A1-notation formula engine

A1 fx

🔬 Simulation & Modeling

Physics, population, epidemiology, circuits, economics

Projectile

v0 (m/s)

Angle

Pendulum

L (m)

Angle

Spring-Mass

Mass

k

x0

Circuit RLC

V

R

C

L

Population (Logistic)

P0

r

K

tMax

SIR Model

N

beta

gamma

I0

Economics

GDP

Growth

Inflation

Unemploy

△ Interactive Geometry Lab

Euclidean constructions · measurements · transformations · coordinate geometry

(—, —)

BASICS CONSTRUCT MEASURE TRANSFORM GRID

Click to place a point

Properties

Select an object to inspect it.

Objects (0)

Advanced Statistics

ANOVA, Bayesian, Time Series, PCA

One-Way ANOVA

Bayesian Inference

Prior

Sensitivity

FPR

Time Series AR(1)

phi

PCA

CAD Pro

Professional 2D technical drawing — layers, snap, dimensioning, DXF/SVG export

Draw

Edit

Measure

Symbols

History

X: 0.00 Y: 0.00 (mm) Tool: Line — click first point 0 objects · 0 selected · zoom 100%

Layers

Properties

No selection. Use Select tool to pick an object.

Quick Help

Coordinate input
100,50 — absolute
@50,0 — relative
@100<45 — polar (dist<angle)
Mouse
Left: draw/select · Right: cancel tool
Wheel: zoom · Middle-drag: pan
Keys
Esc: cancel · Del: delete · Ctrl+Z: undo
Ctrl+A: select all · Ctrl+S: save project

LaTeX Editor

Write LaTeX with live KaTeX preview

Documents

Function Docs

Browse all functions

Select a function to view docs.

Tutorials

Interactive step-by-step lessons

Themes

Choose from 10+ themes

Light

Dark

Solarized

Dracula

Nord

Monokai

Ocean

Rose

Forest

Sunset

Plugins

Extend Computum

Install Plugin

Installed

Calculators

Mortgage, BMI, retirement, tax, resistor, carbon

Mortgage

Principal $

Rate

Years

BMI

Weight kg

Height m

Retirement

Age

Target

$/mo

Return

Saved $

Tax

Income $

Resistor

Band1

Band2

Mult

Tol

Carbon

Driving mi/yr

Flights

kWh

Gas

Math Art

Spirographs, L-systems, tessellations, mandalas, and more

Algorithm Visualizer

Watch sorting algorithms

Competitive Math

Practice problems

Score: 0/0

Select a problem

Version History

Snapshots and rollback

Files

0 files

No file selected

Word wrap Line #

No file

🌐 3D Plotter

Interactive 3D surface, parametric, vector field & contour plots

Plot Type

Surface z = f(x,y)

Expression

x min

x max

y min

y max

Resolution

Colorscale

Parametric Surface

x(u,v)

y(u,v)

z(u,v)

u: [min,max]

v: [min,max]

Vector Field F(x,y,z)

Fx(x,y,z)

Fy(x,y,z)

Fz(x,y,z)

Grid points per axis

3D Parametric Curve

x(t)

y(t)

z(t)

t min

t max

3D Scatter Data

Points [[x,y,z], ...]

Quick Presets

Analysis

Plot a surface to see analysis…

⚡ Optimizer

Gradient descent, Nelder-Mead, L-BFGS, linear programming, constrained & global optimization

Unconstrained Minimization

Objective f(x) — single variable or f([x0,x1,…])

Starting x (or x0,x1,…)

Learning rate (GD)

Max iterations

Tolerance

Multivariable — f(x,y)

Objective f(x,y)

x₀

y₀

Linear Programming (Simplex)

Objective coefficients (maximize cᵀx)

Constraint rows Ax ≤ b (one per line: a1,a2 ≤ b)

Constrained Optimization

Minimize f(x,y)

Subject to g(x,y) = 0 (equality constraint)

And h(x,y) ≤ 0 (inequality, optional)

x₀

y₀

Global Optimization

Function f(x) (may have multiple local minima)

Search min

Search max

Convergence History

∇ PDE Solver

Finite-difference solvers for heat, wave, Laplace, Burgers, and Schrödinger equations

Equation

Parameters

Diffusivity α

Grid points Nx

Time steps Nt

Domain x: [0, L]

Time end T

Initial Condition u(x,0)

Boundary conditions

Numerical Scheme

Solution at t

t = T

Energy & Conservation

Run solver to see…

∬ Interval Arithmetic

Rigorous computation with guaranteed error bounds — never lose track of precision

Interval Calculator

Expression (use interval notation: [a,b])

Interval Functions

Significance Arithmetic

Value with uncertainty

Significant digits

Continued Fractions

Number to expand

Max terms

Interval Visualization

Precision Tracking Chain

Enter a computation chain to track error propagation

∑ Unified CAS Engine

Symbolic algebra, calculus, ODE/PDE solving, Risch integration, pattern matching, assumption-based simplification — all CAS engines in one

SymPy (Pyodide): initializing...

Symbolic Input

Variable

Assumptions

Assume about variable (e.g. "x > 0", "n is integer")

Pattern Matching

Pattern (use _ for wildcard)

Replacement

ODE Solver

ODE (e.g. y'' + y = 0)

Initial y(0)

Initial y'(0)

Range [0, T]

Steps

Symbolic Result (LaTeX)

ODE Solution Plot

Step-by-Step Solution

📊 Data & Entities

Live data APIs, curated datasets, entity framework, and knowledge lookup

Entity Lookup

Entity type

Name or query

Quick Data

Live Data API

Data source

Query (optional — e.g. country name)

Data Explorer

Data Visualization

📖 Documentation

Auto-generated API reference, tutorials, and examples

🧮 Miscellaneous Calculators

50 everyday calculators for finance, health, conversion, and more

Category

🧮

Select a calculator from the list

⏱ Session Manager

Create, switch, and inspect your calculation sessions

🔤 Advanced Algebra

Polynomial operations, systems of equations, inequalities, and more

Polynomial Roots

Enter polynomial in x (e.g. x^3 - 6x + 4)

Polynomial Factor

Enter polynomial to factor (e.g. x^2 - 4)

Expand Expression

Expand (e.g. (x+1)^3)

Simplify Expression

Simplify (e.g. 2x + 3x - x)

Symbolic Derivative

Expression, variable (e.g. x^3 + 2x, x)

Symbolic Integral

Expression, variable (e.g. x^2, x)

Solve Equation

Enter equation (e.g. x^2 - 4 = 0)

Quadratic Formula

Enter a, b, c for ax² + bx + c = 0

a

b

c

System of Linear Equations (2x2)

a₁x + b₁y = c₁ a₂x + b₂y = c₂

a1,b1,c1

a2,b2,c2

📐 Complex Residue Calculator

Residue theorem, contour integrals, pole analysis

Residue at a Pole

f(z) expression, pole Re, pole Im, order

Pole Re

Pole Im

Order

Residue Theorem: Sum of Residues

Sum residues to get contour integral: ∮ f(z) dz = 2πi × Σ Res(f, z_k)

Enter residues (comma-separated):

Laurent Series Coefficients

f(z) centered at z₀ (e.g. 1/(z*(z-1)))

Center z₀ (real)

Terms

Quick Reference

Residue Theorem: If f is analytic inside and on a simple closed contour C except for isolated singularities z₁,...,z_n inside C, then:
∮_C f(z) dz = 2πi × Σ_k=1ⁿ Res(f, z_k)

Simple pole: Res(f, z₀) = lim_z→z₀ (z - z₀) f(z)
Double pole: Res(f, z₀) = lim_z→z₀ d/dz [(z - z₀)² f(z)]

Common residues:
• Res(1/(z-a), a) = 1
• Res(1/(z-a)^n, a) = 0 for n > 1
• Res(e^z/z, 0) = 1
• Res(sin(z)/z, 0) = 0

📐 Vector Operations

Vector arithmetic, dot/cross products, projections, angles

Vector Arithmetic

Vector A (comma-separated)

Vector B

Dot & Cross Product

Uses vectors A and B from above

Magnitude & Angle

Uses vectors A and B from above

Projection

Project A onto B

Unit Vector & Normalization

Normalize vector A

2D Vector Visualization

📊 Series Explorer

Taylor series, power series, convergence, and visualization

Taylor / Maclaurin Series

f(x) expression (use x as variable)

Center a

Number of terms

View range [-R, R]

Quick Series

Convergence Test

Enter terms a_n as expression in n (e.g. 1/n^2)

Series Visualization

Series Terms

🔀 Function Composer

Compose, transform, and visualize function combinations

Function Composition f ∘ g

f(x) - outer function

g(x) - inner function

Evaluate Composition at x

x value

Function Transformations

f(x) =

a (vertical)

b (horizontal)

h (shift x)

k (shift y)

Transform: a · f(b(x - h)) + k

Quick Compositions

Graph

🎯 Eigenvalue Visualizer

Eigenvalues, eigenvectors, diagonalization, and spectral decomposition

2x2 Matrix Eigenanalysis

Enter matrix [[a,b],[c,d]]

3x3 Matrix Eigenanalysis

Enter 3x3 matrix (row by row)

Quick Matrices

Eigenspace Visualization

Transformation Visualization

♪ Music Math Tools

Frequencies, scales, chords, tuning systems, and acoustics

Note Frequency Calculator

MIDI note number (A4 = 69, C4 = 60)

Scale Generator

Root note (0=C, 1=C#, ..., 11=B) and scale type

Root (0-11)

Scale

Chord Builder

Root MIDI note + chord type

Root MIDI

Chord

Interval Calculator

Two frequencies (Hz)

f1 (Hz)

f2 (Hz)

Equal Temperament Reference

Harmonic Series

Fundamental frequency (Hz)

Number of harmonics

⚡ Circuit Calculator

Ohm's law, series/parallel, voltage dividers, RC/RL/RLC analysis, op-amps

Ohm's Law Triangle

V (Volts)

I (Amps)

R (Ohms)

Power Calculator

Voltage (V)

Current (A)

Series Resistors

Resistances (comma-separated, Ohms)

Parallel Resistors

Resistances (comma-separated, Ohms)

Voltage Divider

V_in (V)

R1 (Ω)

R2 (Ω)

RC Time Constant

R (Ω)

C (F)

V_in (V)

LED Resistor Calculator

V_source (V)

V_LED (V)

I_LED (mA)

Capacitor Energy & Charge

Capacitance (F)

Voltage (V)

⭐ Astronomy Tools

Celestial mechanics, orbital mechanics, stellar physics, cosmology

Kepler's Third Law

For orbits around a central body

Semi-major axis a (AU)

Central mass (M☉)

Orbital Velocity

Altitude (km)

Body radius (km)

GM (km³/s²)

Escape Velocity

Mass (kg)

Radius (m)

Stellar Luminosity

L = 4πσR²T&sup4; (Stefan-Boltzmann)

Radius (R☉)

Temperature (K)

Distance Modulus

m - M = 5 log₁₀(d/10pc)

Apparent mag m

Absolute mag M

Hubble's Law

v = H₀ × d

Distance (Mpc)

H₀ (km/s/Mpc)

Solar System Quick Reference

♥ Medical Calculators

Clinical calculations, dosing, unit conversions, vital statistics

BMI Calculator

Weight (kg)

Height (cm)

Body Surface Area (BSA)

Weight (kg)

Height (cm)

Drug Dosage Calculator

Dose (mg/kg)

Weight (kg)

Concentration (mg/mL)

Frequency (per day)

Ideal Body Weight (Devine)

Height (cm)

Sex

Mean Arterial Pressure (MAP)

Systolic BP (mmHg)

Diastolic BP (mmHg)

Unit Conversions (Medical)

Value

⬭ Non-Euclidean Geometry

Hyperbolic · Spherical · Taxicab · Curvature comparison

Poincaré Disk (Hyperbolic)

Click inside the disk to place points. Hyperbolic geodesics (arcs ⊥ boundary) connect consecutive points. Angle sum < π.

Click inside the disk to add points.

Hyperbolic Triangle Solver

Enter side lengths a, b, c (in hyperbolic units). Computes angles via hyperbolic law of cosines. Angle sum < π, area = π − (α+β+γ).

a

b

c

Taxicab (Manhattan) Geometry

d = |Δx| + |Δy|. Circles are diamonds. Compare with Euclidean distance.

Click "Show taxicab circle" to begin.

Spherical Triangle Solver

Enter side lengths a, b, c (angular, in degrees, 0–180). Computes angles via spherical law of cosines. Angle sum > π, area = R²·E where E = α+β+γ−π.

a (°)

b (°)

c (°)

Curvature Comparison

Visualize the same triangle (α=β=γ=60°) under three geometries. Adjust the triangle scale to see how angle sum deviates.

Triangle scale (side length)

Euclidean (K=0)

Spherical (K>0)

Hyperbolic (K<0)

Reference: Key Formulas

Hyperbolic distance (Poincaré disk):
d(z₁,z₂) = arcosh(1 + 2|z₁−z₂|² / ((1−|z₁|²)(1−|z₂|²)))

Hyperbolic law of cosines:
cosh(c) = cosh(a)·cosh(b) − sinh(a)·sinh(b)·cos(γ)

Hyperbolic area = π − (α+β+γ) (angle defect)

Spherical law of cosines:
cos(c) = cos(a)·cos(b) + sin(a)·sin(b)·cos(γ)

Spherical area = R²·(α+β+γ−π) (spherical excess)

Taxicab distance: d = |Δx| + |Δy|

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