Curved polygon

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An irregular pentagon - like all normals - stretches within the figure are also the diagonals of the same length.
Construction.

Arched polygons or arched polygons are a type of uniform thickness . They are based on a polygon , the sides of which are replaced by arcs between two adjacent corner points, the center of which is the opposite corner point. The underlying polygon must be convex and not overturned and have an odd number of corners.

The regular variant, which is based on a regular polygon , is known as the Reuleaux polygon . It is named after the German engineer and cinematographer Franz Reuleaux (1829–1905). The best-known sub-shape is the Reuleaux triangle .

construction

The polygon can be specified for the construction; this must meet the condition that all diagonals are of the same length. In this case the construction is quite simple. An arc of a circle must always be drawn around a corner point through the two opposite corner points.

But even without specifying a polygon, an arched polygon can be constructed with the compass alone. Here on the example of the pentagon arch ; however, the construction description can be adapted for any curved polygon:

  1. Define a point A and draw a circle i around it.
  2. Choose a point C on the circle i and, going further in a mathematically positive sense, a point D.
  3. Draw a circle j around point C through point A.
  4. Choose a point E on circle j, continuing from point A in a mathematically negative sense.
  5. Draw a circle k around point E through point C.
  6. Draw a circle l around point D through point A. The intersection of circles k and l, continuing from A in the mathematically positive sense, is point B.
  7. Draw a circle l around point B through points D and E.

The result is the pentagon ABCDE with the circular arcs AB, BC, CD, DE and EA.

Calculation of the scope

For regular pentagons, the circumference is calculated from the width b in the following way:

This calculation can be generalized to regular curved polygons of any, odd number of corners:

This shows that the circumference of any regular curved polygon is the same.

Comparison with the circle

The circle can be seen as a borderline case of a regular curved polygon, the number of corners of which approaches infinity. The width is the diameter of the circle.

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