计算由线段旋转生成的新线段的长度

Hey there, let's unpack this problem step by step—first, a quick clarification: pure rotation doesn't change the length of a line segment, so when you say rotating a segment of length a gives one of length b, I think what you're describing is a scenario where we have two segments (one length a, one b) positioned at a rotation angle θ relative to each other, and we need to find the length of the segment connecting their non-shared endpoints (that's c). Let's break this down:

First, a critical note: you can't find a unique value of c using only a and b—you also need to know the rotation angle θ (the angle between the original segment and the rotated segment). This makes intuitive sense: rotating the segment by 0° vs 180° will give drastically different lengths for c.

Core Formula: Law of Cosines #

The simplest and most common setup is when the two segments share a common endpoint (let's call this point O). So:

• Original segment: OA, length = a
• Rotated segment: OB, length = b
• The angle between OA and OB is the rotation angle θ
• Segment c connects points A and B

For this triangle OAB, we use the Law of Cosines, which relates the sides of any triangle to one of its internal angles:

c² = a² + b² - 2ab * cos(θ)

Solving for c gives:

c = √(a² + b² - 2ab * cos(θ))

Example Cases to Verify #

Let's test this with common rotation angles to make it concrete:

• 0° rotation (segments are colinear, same direction): cos(0°) = 1, so c = √(a² + b² - 2ab) = |a - b| (the gap between the overlapping segments)
• 90° rotation (right angle between segments): cos(90°) = 0, so c = √(a² + b²) (this is the Pythagorean theorem, a special case of the Law of Cosines)
• 180° rotation (segments are colinear, opposite directions): cos(180°) = -1, so c = √(a² + b² + 2ab) = a + b (the total length of the two segments end-to-end)

If Rotation Isn't Around a Shared Endpoint #

If you rotate the original segment around a point that's not one of its endpoints (like its midpoint), you'll need to use coordinate geometry:

1. Assign coordinates to the original segment (e.g., let it span from (0,0) to (a,0)).
2. Apply the rotation transformation (plus scaling if the length changes from a to b) to each endpoint using rotation matrices.
3. Use the distance formula between the corresponding endpoints to calculate c.

This method will still lead you back to the same core relationship once you compute the angle between the two segments.

内容的提问来源于stack exchange，提问作者Hanlon