Ladders, Couches, and Envelopes

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Ladders, Couches, and Envelopes

• An old technique gives a new approach to an old
  problemDan Kalman
• American UniversityFall 2007

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The Ladder Problem

• How long a ladder can you carry around a
  corner?

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The Traditional Approach

• Reverse the question
• Instead of the longest ladder that will go around
  the corner
• Find the shortest ladder that will not

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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
• Lets look at a mathwright simulation

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About the Boundary Curve

• Called the envelope of the family of lines
• Nice calculus technique to find its equation

• Technique used to be standard topic
• Well known curve (astroid, etc.)
• Gives an immediate solution to the ladder problem

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Solution to Ladder Problem

• Ladder will fit if (a,b) is outside the region W
• Ladder will not fit if (a,b) is inside the region
• Longest L occurs when (a,b) is on the curve

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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
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Trammel of Archimedes
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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

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• x a cos q
• y b sin q

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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

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Animated graphic from Mathworld.com
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Finding the Envelope

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

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Parameterize Lines

• L is the length of ladder
• Parameter is angle a
• Note x and y intercepts

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Find Envelope
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Find Envelope
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Another sample family of curves and its envelope
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Find parametric equations for the envelope
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Plot those parametric equations
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Double Parameterization

• Parameterize line for each a x(t) L
  cos(a)(1-t) y(t) L sin(a) t
• This defines mapping R2 ? R2 F(a,t) (L
  cos(a)(1-t), L sin(a) t)
• Fixed a ? line in family of lines
• Fixed t ? ellipse in family of
  ellipses
• Envelope points are on boundary of image
  Jacobian F 0

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Mapping R2 ? R2

• Jacobian F vanishes when t sin2a
• Envelope curve parameterized by( x , y ) F (a
  , sin2a) ( L cos3a, L sin3a)

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History of Envelopes

• In 1940s and 1950s, some authors claimed
  envelopes were standard topic in calculus
• Nice treatment in Courants 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?

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Another Approach

• Already saw two approaches
• Intersection Approach intersect the curves for
  parameter values a and a h
• Take limit as h goes to 0
• Envelope is locus of intersections of neighboring
  curves
• Neat idea, but

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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!

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More PicturesFamily of Osculating Circlesfor
an Ellipse
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Variations on the Ladder Problem
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Longest ladder has an envelope curve that is on
or below both points.
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Longest ladder has an envelope curve that is
tangent to curve C.
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The Couch Problem

• Real ladders not one dimensional
• Couches and desks
• Generalize to move a rectangle around the corner

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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)

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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

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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