How to Compute r in Stereographic Projection from R^4 to R^3?

[whattttt]

I am computing a stereographic projection in R^4 and i think i am correct in setting
x=rcos(x)sin(y)
y=rsin(x)sin(y)
z=rcos(y)
but can't see how to compute r as I do not know to visualise it graphically as was possible in R^3, any help would be greatly appreciated

 

[Tinyboss]

That doesn't look like stereographic projection to me...do you mean spherical?

 

[whattttt]

No. In a previous example it was in R^3 and polar coordinates were used with x=rcosx and y=rsinx and r was computed from the diagram and equaled something like 2tan(pi-theta) which could be computed from the diagram by projection onto the x-axis but I'm not sure if such a formula exists in R^4

 

[lavinia]

I am computing a stereographic projection in R^4 and i think i am correct in setting
x=rcos(x)sin(y)
y=rsin(x)sin(y)
z=rcos(y)
but can't see how to compute r as I do not know to visualize it graphically as was possible in R^3, any help would be greatly appreciated

Stereographic projection is the same in all dimensions. Draw straight lines from the north pole through the points of the sphere and calculate the intersection of these lines with the hyperplane that is perpendicular to the direction of the north pole. You do not need polar coordinates for this.

 

[blue_raver22]

Stereographic projection is a commonly used technique in mathematics and science to map points from a higher-dimensional space onto a lower-dimensional space. In this case, you are attempting to project points from a four-dimensional space (R^4) onto a three-dimensional space.

Your equations for x, y, and z are correct, as they follow the standard formula for stereographic projection in R^4. However, the challenge lies in determining the value of r, which represents the distance between the point in R^4 and the projection plane in R^3.

In order to determine r, you can use the Pythagorean theorem in combination with the given equations. This will allow you to calculate the distance between the point and the projection plane, and therefore determine the value of r.

Alternatively, you can also consider the projection plane as a three-dimensional subspace within R^4, and use linear algebra techniques to determine the distance between the point and the projection plane.

Overall, visualizing stereographic projection in higher dimensions can be challenging, but with the help of mathematical tools and techniques, you can accurately compute the projection. I recommend consulting with a mathematician or using online resources for further assistance in understanding and visualizing this concept.