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Fast Vector Rotation Using Quaternions

10 Jan · Robert Eisele

Fast Vector Rotation Using Quaternions

A complete derivation of the efficient formula for rotating a 3D vector by a unit quaternion. Starting from the quaternion sandwich product,...

Diophantine Equations

Extending the Verified Range for \(A!B!=C!\) to \(B \leq 10^{1,000,000,000}\)

Polynomials

Polynomial Equations

Time Series Forecasting

Intraday Traffic Forecasting

Counting Principles

Logic Minimization

Introduction to the Quine-McCluskey Algorithm

Rotations

Quaternions

Complex Numbers

Introduction to Complex Numbers

Rational Numbers

Introduction to Rational Numbers

Real Numbers

PID Controllers

Introduction to PID Controllers

Angles and Arcs

Is an Angle Between Two Other Angles?

Intervals and Inequalities

Simplifying and Optimizing Interval Conditions

Bit Manipulation

Ratios and Proportions

Calculating Exact Age Ratios Between Two People

Similarity

The Fascinating Geometry Behind A-Series Paper

Coordinate Transformations

Mapping a Square to a Circle

Computational Geometry

Polygon Geometry

Drawing an Upright Star Polygon

Circle Geometry

Analytic Geometry

Intersection Point of Two Lines

Arithmetic Functions

Sum of Divisors

Regression

Introduction to Regression

Repunits

Why Perfect b-ary Trees Are Repunits

Surface Geometry

Introduction to the First Fundamental Form

Curves and Splines

Linear Algebra over GF(2)

Solving Lights Out Using Linear Algebra

Instance-Based Learning

Introduction to k-Nearest Neighbors

Vehicle Kinematics

Vector Paths

Converting SVG Circular Arcs to G-Code G2 and G3

Special Functions

Introduction to the Gamma Function

Differential Calculus

Vector Operations

Linear Programming

Introduction to Linear Programming

Randomness

Make a Fair Coin from a Biased Coin

Parsing

Introduction to the CYK Algorithm

Inverse Kinematics

Affine Transformations

Extracting Translation, Rotation, and Scaling from Homogeneous Transformations

Path Interpolation

Partial polyline drawing

Order Statistics

Stirling's Approximation

Introduction to Stirling's Approximation

Set-Theoretic Foundations

Continuity

Introduction to Continuity

Limits

Introduction to Limits

Function Classes

Introduction to Monotone, Bounded Variation, and Convex Functions

Monte Carlo Integration

Introduction to Monte Carlo integration

Integral Calculus

Algebraic Structures

Introduction to Algebraic Structures

Number Systems

Introduction to Number Systems

Integer Arithmetic

Directional Statistics

Rasterization

Introduction to Rasterization Algorithms

Vector Basics

Introduction to Vectors

Fast Exponentiation

Introduction to Exponentiation by Squaring

Prime Numbers

Natural Numbers and Integers

Introduction to Natural Numbers and Integers

Divisibility and GCD

Introduction to Greatest Common Divisor

Modular Arithmetic

Introduction to Modulo and Divisibility

Set Theory

Introduction to Sets

Collision Detection

Introduction to 2D Hit-Testing

Matrix Operations

Introduction to Matrix Multiplication

Shader Functions

GLSL Functions

Graph Algorithms

Introduction to Minimum Spanning Trees

Sequence Alignment

Introduction to Needleman-Wunsch Algorithm

Spectral Graph Theory

Spectral Theory

Perfect Squares

Determine if a number is a perfect square

Numerical Methods

Randomized Algorithms

Introduction to the Fisher-Yates Shuffle

Classification

Introduction to Linear Discriminant Analysis

Statistics

How to plot a covariance error ellipse

Sequences

Introduction to Sequences

Local Regression

Introduction to LOESS and LOWESS

Fixed-Point Iteration

The square root of two is a loop

Root Finding

Introduction to Newton's Method

Newton-Type Methods

Introduction to the Gauss-Newton Method

Digital Filters

Series

Introduction to Series

Continued Fractions

Introduction to Continued Fractions

Probabilistic Data Structures

Introduction to Bloomfilters

Taylor Series

Introduction to Taylor Series

Power Series

Introduction to Power Series

Metaheuristics

Introduction to Simulated Annealing

Gradient Methods

Introduction to Gradient Descent

Linear Systems

Introduction to Gauss-Jordan Elimination

Matrix Inversion

Introduction to Matrix Inverse

Matrices

Introduction to Matrices

Rolling Variance

Efficient and Accurate Rolling Variance

Cryptography

Introduction to Diffie-Hellman Key Exchange

Searching

Introduction to Binary Search

Lambert W Function

Introduction to Lambert W function

Numerical Stability

Introduction to the Log-Sum-Exp Trick in Machine Learning

Solutions to Coding Challenges