Talk:Expansion (geometry)

Question

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A regular {p} polygon expands into a regular 2n-gon.

I think it must be

A regular {p} polygon expands into a regular 2p-gon.

Jumpow (talk) 18:26, 8 January 2017 (UTC)

See also – related concept that results in tangential continuity

What is the correct mathematical name for the similar concept that expands flat sides or faces in the same manner but also expands convex corners to arcs or, generally in n-dimensions, hyperspherical and hypercylindrical segments? The result is a shape whose points are equidistant to a nearest point of the original shape, though the converse is not necessarily true. Indeed it is the locus of points outside the shape that are the specified distance away from the nearest point on the original shape.
    In Inkscape, the operation is called ‘outset’, and its converse – to create a path that is (approximately) the locus of points inside the shape that are the specified distance away from the nearest point of (the outline of) the original shape – is called ‘inset’. The outset of a convex n-dimensional shape by a positive distance is an n-dimensional shape with the property of tangential continuity (i.e. G¹), even if the original shape did not have that property. Furthermore, an inset or outset of distance d₂ where d₂ > 0 of an outset or inset respectively by distance d₁, where d₁ > d₂, of any shape yields a shape that has the property of tangential continuity.
    This is because outset and inset are not inverse functions of each other, they are both functions that are not injective. The inset of an outset by the same distance of a convex shape yields (a Bézier curve that approximates) the original shape, but that of a concave shape or the outset of an inset by the same distance of any shape can both result in filleting, but only concave or convex filleting respectively, so may leave all or some corners and edges unfilleted. However, the inset by distance d₂ of the outset by distance d₁ + d₂ of the inset by distance d₁ of any shape yields a shape that both has the property of tangential continuity and is either the original shape or a filleting of the original shape. If the original shape does not have tangential continuity then the result of these 4 operations is a filleting. The convex parts with radius of curvature less than d₁ are filleted by radius d₁ and the concave parts with radius of curvature less than d₂ are filleted by radius d₂.
    Whatever be the mathematical names for the operation (similar to ‘expand’) and the result (similar to ‘expansion’), the concept should be mentioned in the See also section of this article. The noun similar to ‘expansion’ should fit in this example: “The outline of a 20 pence coin is an ⟨‘expansion’-like noun⟩ of a Reuleaux heptagon.”
—James Haigh (talk) 2017-06-30T00:34:15Z

The outset by distance d₁ of the inset by distance d₂ + d₁ of the outset by distance d₂ of any shape yields a shape that also both has the property of tangential continuity and is either the original shape or a filleting of the original shape, but for some cases where filleting occurs, this can be a different filleting to the inset-outset-inset version. Consider a comb shape where either outset-inset-outset fills the comb or inset-outset-inset removes its teeth. So not only are outset and inset not commutative, which is pretty obvious, but even the outset-inset and inset-outset pairs are not commutative in all cases. —James Haigh (talk) 2017-06-30T01:07:57Z