Midpoint Polygons

The Solution
   
  • Triangles: The Base Case
  •    
  • Convex Quadrilaterals
       
  • Non-Convex Quadrilaterals*
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  • Self-Intersecting Quadrilaterals*
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  • Convex Pentagons
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  • Convex Hexagons And Up
       
  • The General Case*
  • Pedagogy
       
  • Math 104
       
  • Original Discussion
  •    
  • Oriented Area Part 1
  •    
  • Oriented Area Part 2
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  • Vector Notation
  • Java Demonstrations
       
  • Area of Midpoint Triangles
  •    
  • Area of Midpoint Quadrilaterals
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  • Area Max for Convex Pentagons
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  • Infinite Area Ratios
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  • Star Pentagons
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  • Non-Midpoint Polygons
  • Bibliography

    Self-Intersecting Quadrilaterals

    Summary

    • The midpoint polygon is still a parallelogram.
    • The area ratio is still 1/2, but we have to be careful about orientation.

    Self-Intersection and Oriented Area

    Understanding the midpoint polygons of self-intersecting quadrilaterals requires use of the oriented, or algebraic, area. For a triangular region, this means that the area is positive if we order the points in such a way that we move counterclockwise around the region, and negative if we move clockwise. In this paper polygons are assumed to be oriented in the counterclockwise sense unless explicitly shown otherwise. Oriented area is discussed in more detail here.


    Figure 5. The triangle on the left is oriented in the counterclockwise direction and so has positive area; the triangle on the right has negative area.

    This actually gives us the oriented area for any planar polygon; we just subdivide it into triangles, calculate their areas, and sum them up. In fact, it doesn't even matter if parts of our triangles lie outside the region, as in the quadrilateral below:


    Figure 6. A self-intersecting quadrilateral. The quadrilateral has been subdivided into a counterclockwise triangle (123) and a clockwise triangle (134).

    If we traverse the quadrilateral in the order that the vertices are numbered, we go counterclockwise around the red region, so it has positive area. We go clockwise around the blue region, so it has negative area.

    By dividing the region into two triangles along the diagonal (which lies outside the original polygon), we can easily calculate the total area of the quadrilateral. The triangle 123 has positive area, and the triangle 134 has negative area. Thus the purple region outside the quadrilateral is counted once positively and once negatively, and so does not affect our calculation, leaving us with nothing but the red and blue regions, which have the correct sign.

    Once we have this decomposition, a simple adaptation of the proof for non-convex quadrilaterals shows us that the area ratio for self-intersecting quadrilaterals is also 1/2.

    Midpoint polygons for general quadrilaterals were discussed here and here.



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    Problem -- Solution -- Pedagogy -- Demos -- Bibliography