So long story short, what I'm trying to do is graph a four dimensional parametric curve using the website Desmos in the same way that this graph represents three dimensional parametrics.
What it's doing is representing 3D points by rotating them according to Euler angles and then projecting them down onto the 2D graph. That is for any (x,y,z), it will first rotate the point according to the slider "a" resulting in (cos(a)*x - sin(a)*y , cos(a)*y + sin(a)*x , z). The other two sliders will then perform those rotations in largely the same way, which leads to the transformations at the bottom of the linked graph.
This is where my issue comes in. The way I'm rationalizing these rotations in 3D space is that as the point is rotated, each rotation is using trig functions two switch between two variables smoothly. "a" switches between x and y, then "b" switches between z and y based on the y value after the "a" transform. These rotations work because they're all done about some axis.
My trouble comes from not knowing how rotations translate into higher dimensions. In two and three dimensions, a rotation is a process of switching between two variables with trig functions. At the same time though, 2D rotations are made around a point while 3D are around an axis. I've heard of 4D rotations going around a plane, but I'm not sure what that would mean or how that would fit into the framework I have set up. help
This is the best I could come up with for a 4D parametric graph. For points (x,y,z,w), "a" switches between x and z, "b" between y and w, and "c" between x and y. The transforms occur in that order. It looks like something, but I'm not sure if that something is in any way an accurate projection.
Any help appreciated.