R3 – Complex Rotations – Example 3
This post provides an example of a rotation chain involving three consecutive rotations. It uses the Wave Number Rotation formula adapted from Rodrigues’ formula and shows the Quaternion equivalent. Wave Number Rotation Formula Example 3 Rotation 1 In this example, 3 rotations take place one after the other: Calculate the chain of rotations using the […]
R3 – Reversal, Remain and Zero Rules
This post covers the Reversal, Remain and Zero rules of rotation in R3. The Reversal Rule of Rotation in R3 The third universal rule of rotation in R3 is the Reversal Rule. This rule states that a rotation can be reversed by multiplying the result of the initial rotation by the same operator, but with […]
R3 – Euler and Tait-Bryan Angles
Introduction This description of Euler and Tait-Bryan angles in R3 is based on the description of Euler Angles in Wikipedia. According to Euler, any rotation, R, is a combination of 3 Euler angles of rotation. There are two types of Euler Angles. The first are the classic Euler angles, also known as proper angles. The […]
R3 – Using the Rotation Ball
Standard Orientation To use the Wave Number Rotation Ball, hold it in a standard orientation before making any rotations. The standard orientation has X^ facing, Z^ to the north and Y^ to the east. The fixed x, y and z-axes The x, y and z-axes themselves do not move when rotating around them. It is […]
R3 – Logs
Definition This post covers the definition of R3 logs, logarithm formulae and provides some examples. Wikipedia defines logs as follows: ‘In mathematics, logarithm (log) is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.’ Logs are useful in expressing […]
R3 – Exponentiation – Fractions, Flip Sign and Rules
This post covers the rules of R3 exponentiation. It also describes the use of fractions and flip sign with exponentiation along with some examples. Fractions Exponents in fractions represent multiplication roots. The post on R3 roots outlined how the Counter and Full roots of Opposite Values are always equal and how there is only 1 […]
R3 – Exponentiation – Definition
Introduction This post covers the definition of R3 Exponentiation in Wave Numbers. In Wikipedia, exponentiation is defined as follows: ‘Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as “b raised to the power of n”. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: bn = […]
R3 – Cube Roots
Similar to R3 Division, cube roots in R3 are solved using simultaneous equations. Simultaneous Equations Needed The following is a method for finding R3 cube roots using simultaneous equations. For example, take the cubic equation: As described in an earlier post, the square of the expression (a? + bi? + cj?)2 is (a2? + b2i2 + c2j2). This allows […]
R3 – Square Roots
Definition The square root of an expression in R3 is made of the roots of the individual Opposite Values with the signs remaining constant. Square of an Expression Given the generic expression (a? + bi? + cj?): Note that as a result of the Zero Rules of Rotation a??b?? + b??a?? = 0, so This […]
R3 – Roots – Definition
This post covers R3 roots including their derivation. Definition Wikipedia defines the square root as ‘a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x.’ Wave Numbers uses this definition and adapts it for roots of higher degrees. The root of an Opposite Value […]