Geometric Issues with a line, a plane and a sphere...

In summary, the conversation discusses various ways in which different geometric shapes can divide space into different parts. The topics of circles and spheres with infinite radii are also brought up. The possibility of using stereographic projection and mobius transformations to map circles and lines to other shapes is mentioned. Ultimately, the conversation raises questions about the definitions and properties of these shapes.
  • #1
dom_quixote
44
9
I - A point divides a line into two parts;
II - A line divides a plane into two parts;
III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion?

Note that the first two statements, the question of infinity must be considered.

For the third statement, is the division of three-dimensional space into two parts correct?

IV - Could the straight line be an arc formed by a circle of infinite radius?
 
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  • #2
dom_quixote said:
I - A point divides a line into two parts;
II - A line divides a plane into two parts;
Okay
dom_quixote said:
III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion?
Why not?
dom_quixote said:
For the third statement, is the division of three-dimensional space into two parts correct?
If I understand the idea, you have three parts: the volume inside the smaller sphere; the volume (annulus) between the spheres; the volume outside the larger sphere.
dom_quixote said:
IV - Could the straight line be an arc formed by a circle of infinite radius?
Infinite radius makes no sense mathematically. A radius is, by definition, a finite number.
 
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  • #3
  • #4
PeroK said:
Infinite radius makes no sense mathematically. A radius is, by definition, a finite number.

Thinking of a line as an infinite radius circle is actually kind of useful for some applications, but I think you need a much more technical understanding of certain topics than the OP question reflects.
 
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  • #5
Office_Shredder said:
Thinking of a line as an infinite radius circle is actually kind of useful for some applications, but I think you need a much more technical understanding of certain topics than the OP question reflects.
If we take the x-axis as a sphere of infinite radius, then the line ##y = 1## must be another sphere of infinite radius. In what way are these two spheres different? Different centre? And/or different infinite radius?

In what way are they circles?
 
  • #6
PeroK said:
If we take the x-axis as a sphere of infinite radius, then the line ##y = 1## must be another sphere of infinite radius. In what way are these two spheres different? Different centre? And/or different infinite radius?

In what way are they circles?
For example, stereographic projection sends circles to circles, except ones that pass through the pole become lines. Relatedly, mobius transformations send circles and lines to circles and lines.
 
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  • #7
Office_Shredder said:
For example, stereographic projection sends circles to circles, except ones that pass through the pole become lines. Relatedly, mobius transformations send circles and lines to circles and lines.
You can map a circle to the half open interval ##[0,1)##, but that doesn't make the interval itself a circle.
 
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  • #8
dom_quixote said:
I - A point divides a line into two parts;
II - A line divides a plane into two parts;
III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion?

Note that the first two statements, the question of infinity must be considered.
I'm not sure that I agree. If I say that I have a path that connects the two parts without crossing the dividing line, do I have to consider infinity to determine if that is correct?
dom_quixote said:
For the third statement, is the division of three-dimensional space into two parts correct?
Are the spheres concentric? Do you want to restrict their relative position?
dom_quixote said:
IV - Could the straight line be an arc formed by a circle of infinite radius?
You might be able to develop this idea. In the right situation, with the right development, it might make good sense.
 
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  • #9
PeroK said:
You can map a circle to the half open interval ##[0,1)##, but that doesn't make the interval itself a circle.
https://en.m.wikipedia.org/wiki/Generalised_circle

If you can find me an equivalent version for circles and half intervals, I will concede the point!
 
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  • #10
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  • #11
Yes, I imagined the problem with two concentric spheres. I think it is possible to equalize the volume of the two spheres, but one of them will be solid and the other will be hollow.
FactChecker said:
Are the spheres concentric? Do you want to restrict their relative position?
 

1. What is the difference between a line, a plane, and a sphere?

A line is a one-dimensional geometric figure that extends infinitely in both directions. A plane is a two-dimensional surface that extends infinitely in all directions. A sphere is a three-dimensional shape with a curved surface that is equidistant from its center point.

2. How do you find the equation of a line, plane, or sphere?

The equation of a line can be found using the slope-intercept form (y = mx + b) or the point-slope form (y - y1 = m(x - x1)), where m is the slope and (x1, y1) is a point on the line. The equation of a plane can be found using the general form (Ax + By + Cz + D = 0), where A, B, and C are the coefficients of the x, y, and z variables, and D is a constant. The equation of a sphere can be found using the standard form ((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2), where (h, k, l) is the center point and r is the radius.

3. What is the relationship between a line and a plane?

A line and a plane can intersect in three different ways: they can be parallel, they can intersect at a single point, or they can intersect at infinitely many points. The intersection of a line and a plane is a line if they are not parallel, and a point if they are parallel.

4. How do you determine if a point lies on a line, plane, or sphere?

To determine if a point lies on a line, you can plug in the coordinates of the point into the equation of the line and see if it satisfies the equation. To determine if a point lies on a plane, you can plug in the coordinates of the point into the equation of the plane and see if it satisfies the equation. To determine if a point lies on a sphere, you can use the distance formula to calculate the distance between the point and the center of the sphere. If the distance is equal to the radius, then the point lies on the sphere.

5. How do you calculate the distance between a point and a line, plane, or sphere?

The distance between a point and a line can be found using the formula d = |Ax0 + By0 + C| / √(A^2 + B^2), where (x0, y0) is the coordinates of the point and A, B, and C are the coefficients of the line's equation. The distance between a point and a plane can be found using the formula d = |Ax0 + By0 + Cz0 + D| / √(A^2 + B^2 + C^2), where (x0, y0, z0) is the coordinates of the point and A, B, C, and D are the coefficients of the plane's equation. The distance between a point and a sphere can be found using the distance formula d = √((x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2), where (x1, y1, z1) is the coordinates of the point and (x2, y2, z2) is the center point of the sphere.

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