1/13/2024 0 Comments Equation of 4d sphere![]() ![]() $$x_1^2 + x_1i(x_2-x_4) + x_2x_4 = \fraccos(\phi),sin(\phi),sin(\phi)\biggr), \phi \in [0,2\pi[$$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: dimensional polyhedra are called polytopes. ![]() The prefix 'hyper-' is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane, hypersphere. I need its 4-dimensional parameterization, using a $\phi \in [0,2\pi[$ value. Download Wolfram Notebook Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The following rotated circle is given in 4 dimensions: I'm attending a differential geometry course, and I'm stuck at one part of a question that we've been asked. ![]()
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