Intersection of a 4D line and a 3D polyhedron in 4D

In summary, the intersection of a 4D line segment and a 3D polyhedron in 4D is a point in 4D if they intersect. This can be proven by looking at the line segment in a coordinate system where the 3D object is aligned with three axes, and using the same approach as with a plane and a line in 3D.
  • #1
LCDF
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Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
 
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  • #2
If they are not in the same 3D space then the intersection will be a point. You can even use the full 3 D volume and a full line and it will still be a single point.

You can look at the line segment in a coordinate system where the 3D object is aligned with three axes, for example.
 
  • #3
Thanks. Is there a mathematical proof for your observation? A hint would also work.
 
  • #4
My second paragraph is a description how to get to a proof.

It's basically the same as with a plane and a line in 3D, just with one more coordinate.
 
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Related to Intersection of a 4D line and a 3D polyhedron in 4D

1. What is the concept of a 4D line and a 3D polyhedron in 4D?

A 4D line is a geometric figure with four dimensions, similar to a 3D line in three dimensions. A 3D polyhedron in 4D is a three-dimensional shape that exists in four-dimensional space, similar to a 2D polygon in three dimensions.

2. How do you determine the intersection of a 4D line and a 3D polyhedron in 4D?

The intersection of a 4D line and a 3D polyhedron in 4D can be determined by finding the points where the line and the polyhedron intersect in four-dimensional space. This can be done by solving a system of equations that represent the coordinates of the line and the polyhedron.

3. Is it possible for a 4D line and a 3D polyhedron to not intersect in 4D?

Yes, it is possible for a 4D line and a 3D polyhedron to not intersect in 4D. This can occur if the line is parallel to the polyhedron or if they are in different planes in four-dimensional space.

4. How does the intersection of a 4D line and a 3D polyhedron in 4D differ from the intersection in lower dimensions?

In lower dimensions, such as 3D and 2D, the intersection of a line and a polyhedron can result in a point, a line, or a polygon. In 4D, the intersection can result in a point, a line, a polygon, or a polyhedron, depending on the orientation and position of the line and the polyhedron in four-dimensional space.

5. What are some real-world applications of studying the intersection of a 4D line and a 3D polyhedron in 4D?

Studying the intersection of a 4D line and a 3D polyhedron in 4D can have applications in fields such as computer graphics, computer vision, and physics. It can also be useful in understanding and visualizing higher-dimensional spaces and objects.

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