# Generalizing the Crease Length Problem

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Let be a convex region in the plane and a point. Thinking of as a piece of paper, fold a boundary point to to form a crease. Given a closed subset of the boundary of , the crease length problem is to determine the shortest and longest creases for .

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Contributed by: Roger B. Kirchner (May 2013)

Open content licensed under CC BY-NC-SA

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## Details

The endpoints of the crease formed by folding to are equidistant from and and therefore satisfy or , which is an equation for the perpendicular bisector of the segment joining and . The endpoints of the crease are found by determining the points where this line intersects the curves (lines for a polygon) defining the boundary of and choosing the ones on the boundary of .

Some problems are easy to solve "by hand". For example, suppose has and for boundary segments. Let and . Using the equation for the perpendicular bisector of and , the crease with endpoints and satisfies

, , .

When is a polygon and is on a linear segment of the boundary, the envelope of the creases is part of the parabola whose focus is and whose directrix is the line containing the linear segment.

When is a circle and is interior to the circle, the envelope of the creases is the ellipse with foci and . When is outside the circle, the envelope is part of the hyperbola with foci and .

Parametric equations of the perpendicular bisector of and are given by

, ,

where is the vector rotated counterclockwise 90°. Parametric equations for the envelope of the perpendicular bisectors therefore have the form

,

and is parallel to .

Thus, .

These are linear equations in and with matrix the transpose of , which can be solved for at nonsingular points of the envelope.

Let denote the ellipse whose center is and whose axes end at and . Then the ellipse has , , and . If is the flipping function, mapping a point to the point symmetric with respect to the perpendicular bisector of and , the folded part of the ellipse is mapped interior to ; is defined by

.

References

[1] L. W. Berman, "Folding Beauties," *The College Mathematics Journal*, 37(3), 2006 pp. 176–186.

[2] S. Ellermeyer, "A Closer Look at the Crease Length Problem," *Mathematics Magazine*, 81(2), 2008 pp. 138–145. www.jstor.org/discover/10.2307/27643095?uid=3739656&uid=2&uid=4&uid=3739256&sid=21102015257303.

[3] Mathematical Art Galleries. "Sharol Nau." (May 15, 2013) gallery.bridgesmathart.org/exhibitions/2012-joint-mathematics-meetings/sharol-nau.

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