R3 – Multiplication Links to Rotation and Rotication – Part 2
R3 Multiplication Link between Formulae: The R3 multiplication and rotication formulae also demonstrate how multiplication links to rotication. The post on multiplication use of trigonometry gives the formula for multiplication as: Part 1 of this post shows how the rotication formula can be rewritten in terms of Opposite Value types as: The multiplication calculations below […]
R3 – Multiplication Links to Rotation and Rotication – Part 1
Introduction: This post starts with a description of long multiplication in R3. It then describes the R3 multiplication links to rotation and rotication and how the different operations move a point. Long Multiplication Calculate (5^ + 7iv + 2j^)*(4v + 6iv + 4jv) = (20^ + 30iv + 66jv) by long multiplication and using the R3 multiplication table. For example: 4v […]
R3 – Quaternion Comparison
Comparison of Wave Number and Quaternion Multiplication Tables: An earlier post makes a detailed comparison of the Wave Number Multiplication table and the Quaternion Multiplication table. The only difference occurs when Quaternions are multiplied together, as in i*i, = -1, but in R3, 1^*1^ = 1^. In R3, multiplying 1^ and 1^ = 1^ and multiplying 1^ and 1v = 1v, leaving the operand […]
R3 – Return and Orthogonal Rules
Rotation: In this post, we delve into the Return and Orthogonal rules of rotation within the three-dimensional space R3. These principles are foundational to understanding the universal rules governing rotations, especially in contexts like Wave Numbers, where rotation operations play a crucial role. A rotation in R3 involves moving a point from one position to […]
R3 – Rotation Ball
Use a Tennis Ball as a Rotation Prop: A physical prop helps to understand 3d rotation. There are many digital 3d spheres on-line. They limit the user experience as they are only in 2d. It is difficult to find a physical 3d sphere to help with understanding rotation. A major requirement is to be able […]
R3 – Rotation – Roticvision
The R3 roticvision operation is the inverse of rotication. As seen in an earlier post, rotication requires an amount of rotation, an axis and a starting point in order to calculate the new location of a point. In contrast, roticvision requires a current point, an axis and an original starting point that are all Opposite […]
R3 – Rotation – Rotvision
The rotvision operation is the inverse of rotation in R3. Rotation requires an amount of rotation, an axis and a starting point to calculate the new location of a point. In contrast, rotvision requires a current point, an axis and an original starting point. Note that these are all Opposite Values. The rotvision operation returns […]
R3 – Complex Rotations – Comparing Wave Numbers and Quaternion Efficiency
Introduction This comparison of Wave Numbers and Quaternions evaluates their efficiency based on the rotation operation due to its computational complexity. Computer games and other high quality graphics frequently use rotation operations to implement the physics of movement, making this an important consideration for performance. Unit of Rotation The magnitude and unit rotation of a […]
Wave Number Calculator – Guide
Introduction This is the online Wave Number calculator guide. The calculator supports the R1, R2, and R3 dimensions. Chose from these using the top menu bar of the calculator. Wave Numbers use Opposite Values. Each Opposite Value consists of 3 parts. The first is the Counter. This gives the magnitude. The second part is […]
R3 – Complex Rotations – Example 2
This post gives a second example of a complex R3 rotation. It uses the Wave Number Rotation formula adapted from Rodrigues’ formula and shows the Quaternion equivalent. Wave Number Rotation Formula Example 2 Take for example the rotation of 60^o ↺(4v+ 5iv + 6j^) (1v + 2i^ + 3jv). Here θ = 60^o and v = (1v + 2i^ + 3jv). ur is […]