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WOLFRAM|DEMONSTRATIONS PROJECT

The Ladder around the Corner Problem

​
test
​
move the ladder around the corner
This Demonstration visualizes a typical problem of constrained optimization: find the maximum length of a ladder that can be moved horizontally around the corner between two perpendicular corridors of different widths
a
and
b
.
The ladder touches the outer walls at the points
(x,0)
and
(0,y)
, and the corner at
(a,b)
. In order for the ladder not to intersect the gray area,
y(x)≤bx/(x-a)
throughout the ladder's range of motion. That is, the red point must stay below the hyperbola
y(x)=bx/(x-a)
(orange). Since the ladder has fixed length
ℓ
, the red point must be on the circle
ℓ(x)=
2
x
+
2
y(x)
(blue).
The blue circle touching the orange hyperbola at the green point represents the largest such circle with radius
ℓ
max
. The coordinates of the green point are
(x,y)=(a+
1/3
a
2/3
b
,
2/3
a
1/3
b
+b)
. The length of the ladder is then
ℓ
max
=
3
(
2/3
a
+
2/3
b
)
[1].
The Locator defines the corner at
(a,b)
. The slider lets you move the ladder around the corner to verify the solution visually. The "test" button sets the ladder to its maximum length at its critical point.
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