现有一个90度的直角过道，过道宽度为一米，然后我现在需要把一个宽0.5米的矩形沙发推过这个过道，仅考...

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现有一个90度的直角过道，过道宽度为一米，然后我现在需要把一个宽0.5米的矩形沙发推过这个过道，仅考虑二维平面，请问这个家具的最大长度是多少？

Thought

Analyzing geometry for rectangle in corridor

I'm considering the path of a line through the inner corner (0,a) touching the bottom y=0 and right wall x=b. For a finite-width rectangle, one long side touches the inner corner, with the opposite side shifting down-right by width 'w'. The line is parallel to this side, offset by normal vector. I need to ensure it avoids walls.

Computing the geometry along the corridor

I’m calculating the geometry for a shifted line, taking into account the rectangle’s width ‘w’. The line’s equation involves slope and intercept adjustments. For projections onto the walls, I’ll compute the distance based on intersections with y=0 and x=b, focusing on distances between lower side intersections and the right wall.

Thought for 1m 21s

Answer

最大长度约为 1.83 米。

把沙发看成“宽度为 $t=0.5$ m 的矩形刚体”过一个两边宽度都为 $1$ m 的 90°直角走廊。对于这种二维问题，宽矩形通过直角弯的临界长度为：

L(\theta)=a\csc\theta+b\sec\theta-t(\cot\theta+\tan\theta)

其中 $a=b=1$ ， $t=0.5$ 。由于两条走廊等宽，最危险的位置出现在对称角度：

\theta=45^\circ

代入：

L_{\max} = \csc45^\circ+\sec45^\circ-0.5(\cot45^\circ+\tan45^\circ)

= \sqrt2+\sqrt2-0.5(1+1) =2\sqrt2-1

L_{\max}\approx 1.828\text{ m}

所以，这个 0.5 米宽的矩形沙发最大长度约为 1.83 米。

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