Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

1. Ladders, Couches, and Envelopes
An old technique
gives a new approach
to an old problem
Dan Kalman
American University
Fall 2007

2. The Ladder Problem:
How long a
ladder can you
carry around a
corner?

3. The Traditional Approach
• Reverse the question
• Instead of the longest ladder that will go
around the corner …
• Find the shortest ladder that will not

5. A Direct Approach
• Why is this reversal necessary?
• Look for a direct approach: find the longest
ladder that fits
• Conservative approach: slide the ladder
along the walls as far as possible
• Let’s look at a mathwright simulation

10. About the Boundary Curve
• Called the envelope of the family of lines
• Nice calculus technique to find its equation
3/23/23/2
Lyx =+
• Technique used to be standard topic
• Well known curve (astroid, etc.)
• Gives an immediate solution to the ladder
problem

11. Solution to Ladder Problem
• Ladder will fit if (a,b) is
outside the region Ω
• Ladder will not fit if
(a,b) is inside the region
• Longest L occurs when
(a,b) is on the curve:
3/23/23/2
Lba =+
2/33/23/2
)( baL +=

12. A famous curve
Hypocycloid: point on a circle rolling within a
larger circle
Astroid: larger radius four times larger than
smaller radius
Animated graphic from Mathworld.com

13. Trammel of Archimedes

14. Alternate View
• Ellipse Model: slide a line with its ends on
the axes, let a fixed point on the line trace a
curve
• The length of the line is the sum of the semi
major and minor axes

17. • x = a cos θ
• y = b sin θ
12
2
2
2
=+ b
y
a
x

21. Family of Ellipses
 Paint an ellipse with every point of the
ladder
 Family of ellipses with sum of major and
minor axes equal to length L of ladder
 These ellipses sweep out the same region as
the moving line
 Same envelope

22. Animated graphic from Mathworld.com

23. Finding the Envelope
• Family of curves given by F(x,y,α) = 0
• For each α the equation defines a curve
• Take the partial derivative with respect to α
• Use the equations of F and Fα to eliminate
the parameter α
• Resulting equation in x and y is the
envelope

24. Parameterize Lines
• L is the length of ladder
• Parameter is angle α
• Note x and y intercepts
1sincos =+ αα L
y
L
x
Lyx
=+ αα sincos

25. Find Envelope

26. Find Envelope

27. Another sample family of curves
and its envelope

30. Find parametric equations for the envelope:

32. Plot those parametric equations:

34. Double Parameterization
• Parameterize line for each α:
x(t) = L cos(α)(1-t)
y(t) = L sin(α) t
• This defines mapping R2
→ R2
F(α,t) = (L cos(α)(1-t), L sin(α) t)
• Fixed α ⇒ line in family of lines
• Fixed t ⇒
ellipse in family of ellipses
• Envelope points are on boundary
of image: Jacobian F = 0

35. Mapping R2
→ R2
• Jacobian F vanishes when t = sin2
α
• Envelope curve parameterized by
( x , y ) = F (α , sin2
α) = ( L cos3
α, L sin3
α)

36. History of Envelopes
• In 1940’s and 1950’s, some authors claimed
envelopes were standard topic in calculus
• Nice treatment in Courant’s 1949 Calculus text
• Some later appearances in advanced calculus
and theory of equations books
• No instance in current calculus books I checked
• Not included in Thomas (1st
ed.)
• Still mentioned in context of differential eqns
• What happened to envelopes?

37. Another Approach
• Already saw two approaches
• Intersection Approach: intersect the curves
for parameter values α and α + h
• Take limit as h goes to 0
• Envelope is locus of intersections of
neighboring curves
• Neat idea, but …

38. Example: No intersections
• Start with given ellipse
• At each point construct the osculating circle
(radius = radius of curvature)
• Original ellipse is the envelope of this
family of circles
• Neighboring ellipses are disjoint!

42. More Pictures:
Family of Osculating Circles
for an Ellipse

45. Variations on the Ladder
Problem

47. Longest ladder has an envelope curve
that is on or below both points.

49. Longest ladder has an envelope curve
that is tangent to curve C.

50. The Couch Problem
• Real ladders not
one dimensional
• Couches and
desks
• Generalize to:
move a rectangle
around the corner

56. Couch Problem Results
• Lower edge of couch follows same path as
the ladder
• Upper edge traces a parallel curve C
(Not a translate)
• At maximum, corner point is on C
• Theorem: Envelope of parallels of curves is
the parallel of the envelope of the curves
• Theorem: At max length, circle centered at
corner point is tangent to original envelope
E (the astroid)

57. Good News / Bad News
• Cannot solve couch problem symbolically
• Requires solving a 6th
degree polynomial
• It is possible to parameterize an infinite set
of problems (corner location, width) with
exact rational solutions
• Example: Point (7, 3.5); Width 1.
Maximum length is 12.5

58. More
• Math behind envelope algorithm is
interesting
• Different formulations of envelope:
boundary curve? Tangent to every curve in
family? Neighboring curve intersections?
• Ladder problem is related to Lagrange
Multipliers and Duality
• See my paper on the subject