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 the amount of rotation required to get from the original starting point to the current point around the given axis.
For example, the rotation π^/4 ↺ 1^ = (√2/2^ + √2/2i^) gives the rotvision equation (√2/2^ + √2/2i^) /↺ 1^ = π^/4.
Here (√2/2^ + √2/2i^) is the current point and (0, 0, j^) is the default z-axis. The original starting point is 1^ and the result is π^/4.
Another example is the rotation π^/3 ↺(4^+ 5i^ + 6j^) (1^ + 2i^ + 3j^) = (1.627^ + 1.447i^ + 3.043j^). This gives the rotvision equation:
- (1.627^ + 1.447i^ + 3.043j^) /↺(4^+ 5i^ 6j^) (1^ + 2i^ + 3j^) = π^/3
Syntax of Rotvision in R3
Rotvision in R3 is expressed as:
(Current Point) /↺?, (Starting point) = ??o or aπ?.
In rotvision, /↺ is the Operation, the ? is the axis and together with the original starting point form the Operator. The default is the z-axis, described by the pair of Opposite Values (0, 0, 0) and (0, 0, j^). The current point is the Operand. The Operator and Operand must be Opposite Values. The result is expressed as an angle of rotation.
In the equation:
- (1.627^ + 1.447i^ + 3.043j^) /↺(4^+ 5i^ + 6j) (1^ + 2i^ + 3j^) = π^/3
The starting point (1^ + 2i^ + 3j^) and the axis (4^+ 5i^ + 6j) form the operator. The current point (1.627^ + 1.447i^ + 3.043j^) is the Operand.
The details of how rotvision is calculated in R3 have not been worked out. As a result it is not included in the online calculator.
Examples
- iv /↺z 1^ = πv/2
- jv /↺y 1^ = π^/2
- 2π^/3 ↺(1^+ 2i^ + 1j^) (1v + iv + j^) = (1.061^ + 1.207iv + .646jv)
- So, (1.061^ + 1.207iv + .646jv) /↺(1^+ 2i^ + 1j^) (1v + iv + j^) = 2π^/3
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