Parallel Transport and Triangle Excess Angle

This can also be expressed as:Rotation = 2π - (180 degrees + Area(T)/R^2)where R is the radius of the sphere and Area(T) is the area of the triangle. This shows that the amount of rotation is dependent on the area of the triangle and the radius of the sphere.
  • #1
jimbo007
41
2
hi,
i am trying to show that the amount by which a vector is rotated by parallel transport around a triangle whose sides are arcs of great circles equals the excess of the sum of the angles over 180 degrees.

this is what i have found out so far
call the angles of the triangle (assuming locally flat space) a, b and c.
then
[tex]a+b+c=\pi +\frac{1}{R^2} Area(T)[/tex]

where Area(T) is the area of the triangle.

this is as far as i can get, i have looked up a few places on parallel transport but the notation used to explain it is very nasty...which is a bit over kill for which i believe to be a much easier problem than it looks.

pls help
 
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  • #2
, thanks.The amount by which a vector is rotated by parallel transport around a triangle whose sides are arcs of great circles is given by the equation:Rotation = 2π - (a + b + c)where a, b, and c are the angles of the triangle. From the equation above, it is clear that the amount of rotation equals the excess of the sum of the angles over 180 degrees.
 
  • #3


Hi there, it seems like you are on the right track! Let's break down the concept of parallel transport and triangle excess angle to make it easier to understand.

Parallel transport is a concept in differential geometry that describes how a vector is transported along a curve without changing its direction. In other words, imagine a vector attached to a moving object, as the object moves along a curve, the vector also moves along the curve but maintains the same direction. This is known as parallel transport.

Now, let's apply this concept to a triangle whose sides are arcs of great circles. As the vector is parallel transported along each side of the triangle, it will rotate around the triangle. The amount of rotation is equal to the angle between each side of the triangle.

Now, let's consider the sum of the angles of the triangle. In a flat space, the sum of the angles of a triangle is always 180 degrees. However, in curved space, the sum of the angles will be greater than 180 degrees. This excess angle is known as the triangle excess angle.

In your equation, a+b+c=\pi +\frac{1}{R^2} Area(T), you have correctly identified that the sum of the angles is equal to 180 degrees plus the triangle excess angle. This is because the area of the triangle is related to the curvature of the space.

Therefore, the amount of rotation of the vector by parallel transport around the triangle will be equal to the triangle excess angle. This is a fundamental concept in differential geometry and has many real-world applications, such as in general relativity.

I hope this helps clarify the concept of parallel transport and triangle excess angle for you. Keep up the good work!
 

FAQ: Parallel Transport and Triangle Excess Angle

1. What is parallel transport?

Parallel transport is a concept in differential geometry that describes the movement of a vector along a given path in a curved space without changing its direction. It is used to understand the concept of curvature and is an important tool in understanding parallelism and geodesics in curved spaces.

2. How is parallel transport related to the triangle excess angle?

The triangle excess angle is a measure of the difference between the sum of the interior angles of a triangle in a curved space and the sum of the angles of a corresponding triangle in a flat space. Parallel transport is used to calculate this difference, as it allows us to compare the two triangles and see how much the curvature of the space affects the angles.

3. Why is parallel transport important in understanding curved spaces?

Parallel transport is important because it helps us understand the concept of curvature and how it affects geometric objects in a curved space. It also allows us to compare geometric properties of objects in curved spaces to those in flat spaces, helping us visualize and conceptualize the effects of curvature.

4. How is parallel transport calculated?

The calculation of parallel transport involves finding a path that preserves the direction of a given vector as it moves along the path. This is done by taking into account the curvature of the space and using differential equations to calculate the change in direction of the vector as it moves along the path.

5. What are the practical applications of parallel transport and triangle excess angle?

Parallel transport and triangle excess angle have practical applications in fields such as physics, engineering, and computer graphics. They are used in understanding the behavior of objects in a curved space, such as the motion of planets in orbit, the design of curved structures, and the rendering of 3D objects in computer graphics.

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