Stereographic projection in R^4

In summary, Stereographic projection can be used in any dimension, and involves drawing straight lines from the north pole to the points of the sphere and calculating their intersection with a perpendicular hyperplane. Polar coordinates are not necessary for this method.
  • #1
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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
 
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  • #2
That doesn't look like stereographic projection to me...do you mean spherical?
 
  • #3
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
 
  • #4
whattttt said:
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.
 
  • #5


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.
 

What is stereographic projection in R^4?

Stereographic projection in R^4 is a method of projecting points from a four-dimensional space (R^4) onto a three-dimensional space (R^3). It is commonly used in mathematics and physics to understand and visualize four-dimensional objects.

How does stereographic projection in R^4 work?

In stereographic projection, a four-dimensional point is projected onto a three-dimensional plane by drawing a line from the point to a fixed point on the plane (called the center of projection). This line intersects the plane at a single point, which represents the projection of the original point.

What are the applications of stereographic projection in R^4?

Stereographic projection in R^4 has various applications in mathematics, physics, and engineering. It is used to visualize four-dimensional objects, such as the four-dimensional sphere (known as a 3-sphere). It is also used in the study of four-dimensional geometry and in the visualization of complex data sets.

What are the advantages of using stereographic projection in R^4?

One of the main advantages of stereographic projection in R^4 is that it allows us to visualize four-dimensional objects in a three-dimensional space, which is easier for our brains to comprehend. It also helps us understand and analyze complex data sets in a more intuitive way.

Are there any limitations to stereographic projection in R^4?

While stereographic projection in R^4 is a useful tool for visualizing four-dimensional objects, it has its limitations. One of the main limitations is that it can distort the size and shape of objects, making it difficult to accurately measure or compare them. Additionally, it can only be used to project four-dimensional objects onto a three-dimensional space, not onto a two-dimensional space like a traditional map projection.

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