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Unwrapping Involutes

As you pull the paper off of a grocery store tube of cinnamon rolls, several things happen at the same time: the seam of the wrapped portion of the paper forms a helix on the cylindrical tube; the line tangent to the helix follows the edge of the unwrapped paper to its corner; and the corner of the paper traces the involute of the base of the cylinder. One can similarly obtain the involute of any other base curve. This Demonstration illustrates a few such cases. It also shows the relationship of the construction to the geometry of the tangent developable surface of the seam.

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Snapshots 1, 2, and 3: The slider unwraps the right cylinder based on the selected plane curve and along a helical seam of slope 1/2 inscribed in the cylinder. We call the seam helical because, like the helix, its height above the base is proportional to the distance traversed along the base. The intersection of the tangent line of the seam and the plane containing the base traces the involute of the base curve.
Snapshots 4, 5, and 6: Another view of the involute of the base is obtained by considering the surface swept out by the movement of the tangent along the seam. The surface is called the tangent developable. The involute is the intersection of the tangent developable with the base plane. A tangent developable of a curve is the envelope of its osculating planes. The slider moves the tangent, normal, and osculating plane of the seam and illustrates how the tangent developable envelopes the osculating planes.
The tangent developable of a circular helix is also called a developable helicoid. A developable helicoid should not be confused with a right or rectangular helicoid, which is the surface swept out by a straight line meeting a helix to its axis orthogonally at both ends. The involutes of the cycloid and cardioid are an equal cycloid and similar cardioid, respectively, given that the starting point in both cases is at a vertex. The involute of a logarithmic spiral is not a logarithmic spiral, except in the limit, when the starting point is the limit point at the center of the spiral, which is not the case in this Demonstration.

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