Geometric proof: minimum angle to point in a line segment

In summary: SarahIn summary, Kyle shared a proof for finding the minimum angle theta between a nonzero vector v and any point on a line segment defined by endpoints r and s in R_3, with the assumption that the segment does not pass through the origin. He reduced the problem to two dimensions by projecting v onto the plane P defined by r, s, and the origin. His proof seems to be correct, but he can also consider the case where w is not parallel to r or s for a more accurate minimum angle. Sarah agrees with Kyle's approach and offers to review other parts of his proof if needed.
  • #1
kylecronan
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Hi,

I have a problem I need to solve for a piece of software I'm writing, and I think I've got it but it would be great if somebody could take a quick look at this proof and see if I've overlooked anything. Thanks in advance.

Here is the problem: We're working in R_3 here. Given a line segment L defined by endpoints r and s, and a nonzero vector v, what is the minimum angle theta between v and any point on the line segment? We can assume the segment does not pass through the origin.

And here's what I came up with: If the line defined by r and s passes through the origin (outside the segment), all angles will be the same and clearly theta = the angle between v and r (or s). Otherwise r, s and the origin define a plane P. We can reduce the problem to a two dimensional analysis as follows. Let w be the projection of v into P. If w is zero, then all the vectors in L will be orthogonal to v, and so theta = pi/2.

For nonzero w, check if w lies between r and s. That is, if angle wr and angle ws are both <= angle rs. If so, then there exists a vector to some point in L that is just a scalar multiple of w, so theta is simply the minimum angle between v and P, ie angle vw (which will be less than pi/2). Similarly, if -w lies between r and s, then theta is the angle between v and -w.

If ±w lie outside of the angle between r and s, then the angle with v will be monotonic while traversing the line segment (this is correct, right?). Therefore the minimum angle will occur at one of the endpoints, and so theta is the minimum of angle vr and angle vs (which could be greater than pi/2).

Where I have written angle xy I will be calculating according to the usual formula acos(x . y / |x| |y|). I'm interested in the direction of v but not its length.

Okay, if you're still with me and have any thoughts for me, thank you!

-Kyle
 
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  • #2


Hello Kyle,

Thank you for sharing your proof with us. It looks like you have a good understanding of the problem and your solution seems to be correct. I agree with your approach of reducing the problem to two dimensions by projecting v onto the plane P defined by r, s, and the origin.

One thing to consider is the case where w is not parallel to r or s. In this case, you can use the dot product to calculate the angle between v and the projection of w onto the line segment. This will give you a more accurate minimum angle theta.

Overall, your proof seems solid and I don't see any major oversights. Good job! Let me know if you have any other questions or if you would like me to review any other parts of your proof.

 

1. What is a geometric proof?

A geometric proof is a method of logically showing that a statement or theorem is true using the principles and rules of geometry. It involves using deductive reasoning to establish a sequence of logical steps that lead to the desired conclusion.

2. How is the minimum angle to a point in a line segment determined?

The minimum angle to a point in a line segment is determined by drawing a perpendicular line from the point to the line segment. The angle formed between the perpendicular line and the line segment is the minimum angle.

3. Why is the minimum angle important in geometry?

The minimum angle is important in geometry because it helps define the relationship between two lines or line segments. It can also be used to determine if two lines are parallel or perpendicular to each other.

4. Can the minimum angle be greater than 90 degrees?

No, the minimum angle cannot be greater than 90 degrees. This is because a line segment is the shortest distance between two points, and the angle formed by a perpendicular line is always 90 degrees.

5. How is a geometric proof different from other types of proofs?

A geometric proof is different from other types of proofs because it uses the principles and rules of geometry, such as theorems, postulates, and definitions, to establish the validity of a statement. It also involves visual representations, such as diagrams and drawings, to aid in the logical reasoning process.

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