R3 – Rotation – Hyperspherical Coordinates
This post covers the formula for hyperspherical coordinates in R3. Examples of rotation using the formula are given. Hyperspherical Coordinates Wikipedia defines the hyperspherical coordinate system as: ‘a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate r, and n […]
R3 – Rotation – Spherical Coordinates
This post covers the formula for spherical coordinates in R3. Examples of rotation using the formula are given. Cartesian Coordinates The Cartesian coordinates of a point are based on the intersection of perpendicular lines from the point with the x, y and z-axes. For example: the Cartesian coordinates at a point on the surface of […]
R3 – Rotation – Multiplication Table
Orthogonal Rule The layout of the R3 axes is defined as part of the Wave Number axioms. The Axioms of Rotation provide the Orthogonal rule which states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90o. This allows for the derivation of the R3 unitary multiplication table, shown below, from rotations […]
R3 – Rotation – Examples
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R3 – Rotation – Simple Examples
Tait-Bryan Angles This post looks at simple R3 rotations that consist of 90o rotations around a single axis on the unit sphere. A 90^o axis-angle rotation around the x-axis is a z-y’-x’’ rotation using the Tait-Bryan angles of φ = 90^o, θ = 0o and ψ= 0o. Rotation of 90^o around the x-axis moves the […]
Rotation Sphere
R3 – Rotation Definition and Quaternions What is Rotation in R3? The R3 rotation operation is a fundamental aspect of the Wave Number system. The facts expressed in its axioms support this. A rotation moves a point, represented by Opposite Values, to a new location, also represented by Opposite Values, through circular movement around an […]
R2 – Rotation and Multiplication – Examples
This post gives examples of R2 unitary rotations and the equivalent multiplications. A Double Rotation in R2 and its Equivalent Multiplication Take for example two rotations of 45^o around the z-axis from the point located at (1^, 0): A rotation of π^/4 ↺ 1^ moves a point at 1^ to (√2/2^, √2/2i^). See this in […]
R2 – Rotation – Multiplication Table
Orthogonal Rule used to Derive Multiplication Table The Axioms of Rotation provide the Orthogonal rule that states that multiplication by a unitary is the equivalent of rotation around the unitary’s single point axis by 90o. This allows the derivation of the R2 unitary multiplication table that follows. Multiplication by i^ Using the post-it note prop, it […]
R2 – Rotation – Cartesian & Euler
This post describes the role of Cartesian Coordinates and Euler’s formula in R2 Wave Number Rotation. Cartesian Coordinates and Euler’s Formula in R2 The Cartesian coordinates of a point are based on the intersection of perpendicular lines from the point with the x and y axes. For example: The Cartesian coordinates at π^/4 or 45^o […]
R2 – Rotation – Simple
This post gives some simple R2 examples of clockwise and counterclockwise rotations around the origin. Counterclockwise Examples Other R2 Rotations A similar set of clockwise R2 rotations can be done using v values of π. For example: πv/2 ↺ 1^ = iv: Other R2 rotation values of π^ or πv allow the movement of a […]