What Is the Exterior Angle at Infinity for a Semi-Infinite Strip?

In summary, the conversation discusses a shape in the complex plane that is the exterior of a semi-infinite strip bounded by 0 < y < 1 and x > 0. The rectangle formed by this shape has interior angles of 3*pi/2 and exterior angles of -pi/2. It is then stated that the exterior angle at infinity is 3pi, but the justification for this geometrically is unclear. The conversation also mentions determining non-overlapping triangles in the polygon, with each triangle having a sum of interior angles of pi radians. The concept of triangles on an infinite domain is also brought up, but it is unclear how this helps in understanding the shape.
  • #1
rsq_a
107
1
Suppose we have the following shape in the complex plane: The EXTERIOR of the semi-infinite strip bounded by 0 < y < 1 and x > 0. The two physical angles making up the rectangle have interior angles of 3*pi/2 and thus exterior angles of -pi/2.

Now, because the sum of the exterior angles of a polygon have to sum to 2pi, we can claim that the exterior angle at infinity is simply 2pi - (-pi/2 - pi/2) = 3pi.

However, I'm having trouble justifying this geometrically. I've looked at projecting the surface onto the Riemann sphere, but the exterior angle at infinity simply seems to be either pi or -pi. I see no reason why it has to do an additional orbit of 2pi.
 
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  • #2
How many non-overlapping triangles can you determine in the polygon? Each triangle contains sum of the interior angles being [tex]\pi[/tex] radians.
 
  • #3
symbolipoint said:
How many non-overlapping triangles can you determine in the polygon? Each triangle contains sum of the interior angles being [tex]\pi[/tex] radians.

How do you imagine triangles on an infinite domain? I don't see how this helps...
 
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FAQ: What Is the Exterior Angle at Infinity for a Semi-Infinite Strip?

1. What is the concept of "angle at infinity" in mathematics?

The angle at infinity is a mathematical concept used to describe the behavior of lines or curves as they approach infinity. It refers to the angle between two lines or curves that are parallel, but appear to intersect at a point at infinity.

2. How is the angle at infinity related to the concept of parallel lines?

The angle at infinity is closely related to the concept of parallel lines because it measures the angle between two lines that are parallel, but seem to intersect at infinity. In other words, the angle at infinity can be thought of as the "ultimate" or "limiting" angle between parallel lines.

3. Can the angle at infinity be measured or calculated?

No, the angle at infinity cannot be measured or calculated in the traditional sense, as it represents the behavior of lines or curves as they approach infinity. However, it can be conceptualized and used as a tool in mathematical proofs and constructions.

4. How is the angle at infinity used in projective geometry?

In projective geometry, the concept of the angle at infinity is fundamental. It allows for the extension of Euclidean geometry to include points at infinity, making it possible to study and describe properties of geometric figures that extend to infinity.

5. Is the angle at infinity the same for all geometric figures?

No, the angle at infinity can vary depending on the geometric figure being studied. For example, the angle at infinity for parallel lines is different from the angle at infinity for a parabola. The concept of the angle at infinity is used to describe the behavior of each figure as it approaches infinity.

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