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

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Presentation on theme: "Ladders, Couches, and Envelopes An old technique gives a new approach to an old problem Dan Kalman American University Fall 2007."— Presentation transcript:

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

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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

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10 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

11 Solution to Ladder Problem Ladder will fit if (a,b) is outside the region  Ladder will not fit if (a,b) is inside the region Longest L occurs when (a,b) is on the curve:

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

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17 x = a cos  y = b sin 

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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,  ) = 0 For each  the equation defines a curve Take the partial derivative with respect to  Use the equations of F and F  to eliminate the parameter  Resulting equation in x and y is the envelope

24 Parameterize Lines L is the length of ladder Parameter is angle  Note x and y intercepts

25 Find Envelope

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27 Another sample family of curves and its envelope

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30 Find parametric equations for the envelope:

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32 Plot those parametric equations:

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34 Double Parameterization Parameterize line for each  : x(t) = L cos(  )(1-t) y(t) = L sin(  ) t This defines mapping R 2 → R 2 F( ,t) = (L cos(  )(1-t), L sin(  ) t) Fixed   line in family of lines Fixed t  ellipse in family of ellipses Envelope points are on boundary of image: Jacobian F = 0

35 Mapping R 2 → R 2 Jacobian F vanishes when t = sin 2  Envelope curve parameterized by ( x, y ) = F ( , sin 2  ) = ( L cos 3  L sin 3  )

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 (1 st 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  and  + 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!

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42 More Pictures: Family of Osculating Circles for an Ellipse

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45 Variations on the Ladder Problem

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

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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

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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 6 th 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


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