Metrics on the plane and the sphere

In summary, plane is a Euclidean metric space while the sphere surface is a non-Euclidean metric space. Distance functions on plane can be applied to distance functions on sphere, except in the case of the circle, which is not an intersection.
  • #1
gianeshwar
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Are plane and surface of sphere different metric spaces?
Can distance function of plane be applied as distance function of surface of sphere?
Please correct my question if needed?
 
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  • #2
They certainly are different metric spaces! One is unbounded while the other is bounded. I presume that by "distance function of plane" you mean the "standard metric" in three dimension" [itex]d((x_0, y_0, z_0), (x_1, y_1, z_1))= \sqrt{(x_1- x_0)^2+ (y_1- y_0)^2+ (z_1- z_0)^2}[/itex] where (x, y, z) satisfy the equation of the plane.

Yes, with the provision that (x, y, z) satisfy the equation of the surface of the sphere instead of that of a plane, that same formula defines a metric on the surface of the sphere. But they are still different metric spaces. There is, after all, a difference between a metric formula and a metric space.
 
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  • #3
Like HIvy said, they are not homeomorphic, one, the sphere is compact, plane is not.
 
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  • #4
1.Plane is a Euclidean metric space and sphere surface ,I think is non Euclidean metric space.
2.How (x,y,z) of plane can satisfy surface of sphere,except in the condition when their intersection is the circle.?
3. Please bear with me as I am a learner.
 
  • #5
gianeshwar said:
1.Plane is a Euclidean metric space and sphere surface ,I think is non Euclidean metric space.
This doesn't mean anything until you have specified which metric you are using.

2.How (x,y,z) of plane can satisfy surface of sphere,except in the condition when their intersection is the circle.?
Yes, the intersection of a given plane with a given sphere is either
1) empty
2) a point
3) a circle

But that is not relevant to the original question:
Are plane and surface of sphere different metric spaces?
which is a question about homeomorphism of metric spaces, NOT a question about the intersection of two sets.

3. Please bear with me as I am a learner.
 
  • #6
gianeshwar said:
1.Plane is a Euclidean metric space and sphere surface ,I think is non Euclidean metric space.
2.How (x,y,z) of plane can satisfy surface of sphere,except in the condition when their intersection is the circle.?
3. Please bear with me as I am a learner.

Yes, a plane is Euclidean, in the sense that it is a manifold with a single chart , e.g., the identity, but the circle does not admit a global Euclidean structure as ## \mathbb R^n ## for any n (because, e.g., this would imply that ## \mathbb R^n ## is compact, which it is not ). For 2, see Ivy's reply.
 
  • #7
Thankyou very much dear friends! Your answers motivate me to study more.Will be back after more study.
 

1. What are metrics on the plane and the sphere?

Metrics on the plane and the sphere are mathematical tools used to measure distances, angles, and other geometric properties on two-dimensional and three-dimensional surfaces, respectively. They are essential for understanding and modeling the behavior of objects in these spaces.

2. Why do we need metrics on the plane and the sphere?

Metrics are important because they allow us to quantify the properties of objects on these surfaces and make accurate predictions about their behavior. For example, in navigation, metrics on the sphere are used to determine the shortest distance between two points on the Earth's surface.

3. How are metrics on the plane and the sphere calculated?

Metrics on the plane and the sphere are calculated using mathematical equations that take into account the curvature and other geometric properties of these surfaces. For example, the Pythagorean theorem can be used to calculate distances on the flat plane, while the Haversine formula is often used for distances on the sphere.

4. What are some common metrics used on the plane and the sphere?

Some common metrics used on the plane include the Euclidean metric, which measures the shortest distance between two points, and the Manhattan metric, which measures the distance between two points traveling only along the axes. On the sphere, the Haversine metric and the Vincenty formula are commonly used for distance calculations.

5. How do metrics on the plane and the sphere differ?

Metrics on the plane and the sphere differ in how they account for the curvature of these surfaces. On the plane, distances are measured using straight lines, whereas on the sphere, distances are measured along the surface of the sphere. Additionally, the equations used to calculate metrics on the sphere are more complex due to the non-Euclidean nature of the surface.

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