$$x_1^2 + x_1i(x_2-x_4) + x_2x_4 = \fraccos(\phi),sin(\phi),sin(\phi)\biggr), \phi \in [0,2\pi[$$īut after a long while of calculation it ended up being wrong, because I've got nonsense. Now I'll include the above mentioned calculation: Note 2: As far as I understand, this is a circle that's been rotated 45°'s in the $x_3$, and 45°'s in the $x_4$ direction. When we intersect the sphere with a plane, we apply this definition again, so we must get a circle. Note: The intersection of a 4-dimensional sphere and a plane can only give you a 2-dimensional circle, since by definition a 4D sphere is the collection of points equal distance from the origin. I'm including the calculation of this intersetion, just to give you a deeper understanding of the problem, however my question only extends to how to obtain the parametric equation of the above circle. is a plane in 4 dimensions, with the correspondance of To show that this IS in fact a circle, you can solve the following system of equations to obtain their intersection: I need its 4-dimensional parameterization, using a $\phi \in [0,2\pi[$ value. The following rotated circle is given in 4 dimensions: What we need to do now is determine the equation of the tangent plane. 4D speed invariance postulate states: In a stationary coordinate system all objects move with the same 4D. I'm attending a differential geometry course, and I'm stuck at one part of a question that we've been asked. This postulate is directly derived from Formula (9).
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