Rotations of 3+D implicit equations

Higher-dimensional geometry (previously "Polyshapes").

Rotations of 3+D implicit equations

Postby 개구리 » Mon Sep 14, 2020 3:52 am

For a 2D implicit function f(X,Y), the graph can be rotated by θ by substituting:
X = x*cos(θ) - y*sin(θ)
Y = x*sin(θ) + y*cos(θ)
Is there such a substitution for f(X,Y,Z) and f(X,Y,Z,W), adding the new angle variables φ and ϑ? I've been trying to make a generalized substitution algorithm based on spherical and glomic coordinates but it's quite difficult.
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Re: Rotations of 3+D implicit equations

Postby wendy » Mon Sep 14, 2020 8:50 am

For four dimensions, there is left and right quarterions. This product is not communitive, but associative.

For higher dimensions, the problem becomes quite difficult. The rotation-space (where each possible rotation, including speed, occupies a point), goes as n(n-1)/2, for n. This means for a 2d thing, we just have a speed scale. For 3d, you imagine a sphere, and the radial distance is again speed, but the direction is given by the axis through the north pole. For four dimensions, you have two 3d axies in 6d, by the quarterions.

There is no general algebra above 4d, except 8d, where the octonions serve. But other than that...

In all even dimensions, one can suppose swirlybobs, which arise out of that in analytical euclidean geometry, you draw a single line through two points, holds equally true for complex numbers. But the slope of the line does not change if you introduce a rotation w (applied to each axis), which rotates every line in a derived direction, while keeping the same slope (of the line). This is because a complex-euclidean line is a real 2-space, and adding w=cis(vt), where v = velocity and t = time, rotates every point around the origin in a fixed way (eg for a line Z=aX = bY, you have wZ=awX = bwY, which in CE3, is the equation of a 6d swirlybon.
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