基于管内通光间隙形状的管道弯曲半径估算问询

Great question—this is a neat application of geometric optics and torus geometry! Let’s break down the problem and walk through the relationships you’re looking for.

First, let’s formalize the key parameters to avoid confusion:

• Let ( R ) = bend radius of the pipe (the radius of the circular arc followed by the pipe’s centerline)
• Let ( r ) = inner radius of the straight pipe (we’ll assume uniform wall thickness and negligible thickness for simplicity)
• Your crescent shape parameters: ( r_1 ) (radius of the inner bend’s arc, smaller value), ( r_2 ) (radius of the outer bend’s arc, larger value), ( d ) (distance between the centers of these two arcs)

Core Assumptions #

We’ll assume the pipe is bent into a perfect toroidal arc (constant bend radius), the bend is slight enough that ( R \gg r ) (so we can use small-angle approximations), and the light source is collimated (parallel rays passing through the pipe).

Deriving the Bend Radius Relationship #

For a toroidally bent pipe, the crescent-shaped clear aperture arises because rays near the inner bend side have less clearance (they’re closer to the pipe’s walls along the entire bend), while rays near the outer side have more.

Through geometric analysis of the torus’s projection onto the viewing plane (the end of the pipe), we can derive a simple relation between ( R ) and your crescent parameters:
[
R = \frac{r(r_1 + r_2)}{r_2 - r_1}
]

Additionally, the distance ( d ) between the crescent’s arc centers should satisfy this consistency check:
[
d = \frac{2r(r_2 - r_1)}{r_1 + r_2}
]

Let’s verify with a straight pipe case: if ( R \to \infty ) (no bend), then ( r_2 = r_1 = r ), so ( R ) becomes infinite as expected, and ( d = 0 ) (the crescent collapses to a perfect circle—correct!).

Is This "Simple Geometry"? #

Yes, fundamentally this relies on projecting the torus’s inner surface onto the viewing plane and finding the envelope of all rays that pass through the pipe without touching the walls. The small-angle approximation simplifies the math to basic algebraic relations, so it’s accessible with a solid grasp of 2D/3D geometry and projections.

References (No External Links) #

This kind of analysis is standard in optical inspection of bent pipes, and you’ll find similar derivations in textbooks focused on:

• Geometric optics for industrial inspection
• Differential geometry of surfaces (specifically tori and their projections)
• Pipe flow visualization techniques

备注：内容来源于stack exchange，提问作者Kepler