# Curved line shortest distance between two points?

• B
• ChrisisC
In summary, Einstein proposed the concept of geodesics, which states that a curved line may be shorter than a straight line in spaces with curvature. This applies to scenarios such as walking across a curved field or flying on intercontinental routes. This concept was not introduced by Einstein himself, but he did mention it in his theory of general relativity.
ChrisisC
Can someone explain how this is possible? it makes no sense to me. You wouldn't walk along the track surrounding a soccer field if you wanted to get to the other end... you would walk straight across the grass to reach your destination. Why did Einstein propose this?

ChrisisC said:
Can someone explain how this is possible? it makes no sense to me. You wouldn't walk along the track surrounding a soccer field if you wanted to get to the other end... you would walk straight across the grass to reach your destination. Why did Einstein propose this?
Imagine the field isn't flat, but 200 yards high. Would you still walk over the grass? This is basically what it is about in spaces with curvature: a giant heap in the middle. Or take the earth. There is no way directly through it. Masses bend spacetime and thus it has a curvature and the shortest way isn't a straight line anymore.

ChrisisC said:
Can someone explain how this is possible? it makes no sense to me.
Have you ever seen a map showing intercontinental airplane routes? Why do you think the airlines use curved routes if not because they are shorter and therefore save money.

Dale said:
Have you ever seen a map showing intercontinental airplane routes? Why do you think the airlines use curved routes if not because they are shorter and therefore save money.
Good idea! (black is the shortest way, grey the direct line - it shows at least the principle)

Dale
ChrisisC said:
Why did Einstein propose this?
What exactly are you referring here? The concept of geodesics was not introduced by Einstein.

A.T. said:
What exactly are you referring here? The concept of geodesics was not introduced by Einstein.

In the book "Hyperspace" by Kaku, Kaku says that Einstein said a curved line is less distance than a straight line. I'm assuming Kaku said this because it refers to GR. I could be wrong.

ChrisisC said:
In the book "Hyperspace" by Kaku, Kaku says that Einstein said a curved line is less distance than a straight line. I'm assuming Kaku said this because it refers to GR. I could be wrong.
Could also mean the pseudo-Euclidean line interval in Minkowski space time.

## 1. What is a curved line?

A curved line is a line that deviates from a straight path. It can be represented by a mathematical equation and can take on various shapes such as circles, ellipses, and parabolas.

## 2. Why is the shortest distance between two points on a curved line important?

The shortest distance between two points on a curved line is important because it allows us to find the most efficient path between those two points. This is useful in many fields including engineering, transportation, and navigation.

## 3. How is the shortest distance between two points on a curved line calculated?

The shortest distance between two points on a curved line can be calculated using the shortest distance formula, which takes into account the curvature of the line. This formula involves using calculus and can be solved using various methods such as differentiation or integration.

## 4. Can the shortest distance between two points on a curved line be shorter than the straight-line distance?

Yes, the shortest distance between two points on a curved line can be shorter than the straight-line distance. This is because the curved line may take a more direct path between the two points, even though it may appear to be longer when graphed on a two-dimensional plane.

## 5. Are there any real-life applications of calculating the shortest distance between two points on a curved line?

Yes, there are many real-life applications of calculating the shortest distance between two points on a curved line. Some examples include finding the most efficient flight path for airplanes, determining the optimal route for a train, and designing curved roads for transportation systems.

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