Name: Irene

Who is asking: Student
Level: All

Question: How can I prove that the average of two polygons will give me another one?

Hi Irene,

There are several possible meanings for 'average'.

The classical mathematical 'average' would be the Minkowski sum. This involves taking EVERY POINT in one polygon and adding it to EVERY POINT in the other, and plotting ALL the results. This produces a new polygon (convex from convex) which typically has new sides for EACH of the sides of the original pair - so the sum of two triangles can be a hexagon.

If this does not yet feel like an average, divide every vector by 2. Same kind of shape, just scaled down a bit.

This type of operation is often used in drafting and design to 'add' a circle to a polygon and create an 'offset' which is bigger and has rounded corners (coming from arcs of the circle).

Perhaps you have in mind a DIFFERENT average in which you have two labeled polygons of the same number of vertices: AB.... F, A'B' .... F'. You average each of these pairs.

The catch here is that the average of two convex quadrilaterals may NOT be convex. It will still have four vertices, and four sides (when you join the vertices). However, the 'average' of a triangle and its mirror image will actually be a set of collinear points - on the mirror.

Do you consider three collinear points and the connecting lines to be a triangle? I do - but I have actually had to work with the 'averaging' of polygons (and other shapes) to figure out a nice geometric theory for some problems in Computer Aided Design, and structural engineering!

So the question is good - and it raises even BIGGER questions. That is what mathematics is like. I saw a quote today saying that those who ask good questions sometimes contribute MORE to mathematics than those who answer them ;-/

Walter Whiteley
York University

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