# The Ladder around the Corner Problem

The Ladder around the Corner Problem

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 and .

a

b

The ladder touches the outer walls at the points and , and the corner at . In order for the ladder not to intersect the gray area, throughout the ladder's range of motion. That is, the red point must stay below the hyperbola (orange). Since the ladder has fixed length , the red point must be on the circle (blue).

(x,0)

(0,y)

(a,b)

y(x)≤bx/(x-a)

y(x)=bx/(x-a)

ℓ

ℓ(x)=

x+y(x)

2

2

The blue circle touching the orange hyperbola at the green point represents the largest such circle with radius . The coordinates of the green point are . The length of the ladder is then [1].

ℓ

max

(x,y)=a+ab,ab+b

1/3

2/3

2/3

1/3

ℓ=

max

a+b

2/3

2/3

3

The Locator defines the corner at . 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.

(a,b)