Created 2005-03-12 Modified 2007-01-01
Chelton Evans
Convexity and Polygon Representation with Algebra
Intro
What is Convexity?
Convexity and Polygon Representation in Algebra
Mathematical Representations of Polygons
Simple Polygon Examples
Complex Polygon Examples
Non-Unique Polygon Algebra
Maths Polygon Algorithm
Subtraction
C++ shape rep implementation
Complexity
Is a point inside or outside a polygon? This is part of a general problem called Inclusion in a Polygon.
An algorithm to determine this for a general polygon will be constructed. By using partitions a point can be tested if it is within the partition.
What goes for 2D is also true for 3D as well. Non-convex 2D and 3D shapes can be fully described by algebra operations of addition(+) and multiplication(.) .
Polygons can be represented generally as non-convex shapes. Convex shapes are closed regions where a line between any two points in the region is made up of other points in the region.
While this is the definition it does not tell the full story. Indeed its better to view convexity as multiplication and non-convexity as addition. Since this unfortunately not well known (although authors such as Addler have written layman's books on mathematics and group theory which are outstanding) the general communities have a limited view of these areas and so generally do the practitioners!!! AND is multiplication and OR is addition.
By far the simplest representation of how to compose the shapes is with the diagrams and algebra where a number corresponds with the line. Whats represented by the number is tested. eg given test point does it satisfy the numbers test. If so return numeric 1 else return numeric 0 corresponding to true or false.
Notice that the lines have the same anti-clockwise winding independent of whether the AND(.) operator is being used or the OR(+) operator is being used. This is good as it further simplifies describing the shape.
The maths formula + for or and . for convexity is relatively easily found for static geometry where the programmer determines the expression by looking at the geometry. However can this be done by an algorithm?
Here is an example of a section of the shape having its formula found. Both expressions are equivalent and were formed by transversal of the line from one end to the other from opposite ends.
The current line segment partitioned points as being inside or not. If they were inside they had to be in the nested algebra expression.
Each of the lines as a direction. Consider all the lines and subtract the lines which when partitioning the plane include all points. ie subtract all convex lines.
Here is what remains of the initial example.
Now transverse the line separating the line : convex regions and non convex regions which intersect no other lines, and other lines which do intersect other lines. The non-intersecting lines are expressed as algebra. If they are convex the lines are a product, if non-convex they are a sum. All these are summed. The non-intersecting lines are subtracted leaving the intersected ones remaining.
The subtracted initial convex regions are multiplied with the non-intersecting line sum.
Now what to do with the remaining lines is where the algorithm becomes recursive. Firstly in this simple example nothing remains and terminate. Assuming a more complicated shape, take the remaining points and apply the same algorithm to generate the algebra expression, then add it to the non-intersecting line sum. Repeat till no more points.
Thus the algorithm is complete. Although simple I have not implemented it because there is a lot to do. For example classifying the line segment into convex and non convex pieces is from a programming point of view some work. Thats why I cheated, creating the algorithm and placing in my notes without implementing it. Cheers.
A circle is always convex, and so is a triangle, and so is a sphere. But a donut is not. In maths it is a set of points where if we construct a straight line between the points, all the points are in the set too. In geometry if we take any two points in the shape then a line between these two points must also be in the shape for the shape to be convex.
Convex shapes provide a guarantee or completely define the shape. Algorithms that need to know if they are inside the shape often break down shapes into convex regions. Drawing and collision detection are two areas where these most work.
If an object is not convex then it is concave. A convex object is also a partition of two states: you are either inside it or not.
The mathematics of partitions defined by two operators does not express the idea of subtraction. This is because the partiton can be defined to include this operation. However for a maths system to be explict the not operator needs to be introduced. Then we can define subtraction.
Technically its probably complexity as the worst case would require computation of every half space. However the implementation can take advantage of boolean algebra so that if say a sum is being summed and a one is added stop summing because the result is true and return the result.
Further simplifying algebra reduces the computational cost by factoring like terms. Since this results in trees of expressions which may or may not need to be evaluated, the expected complexity may be dependent on the geometry and the sample points being tested.
Since this in part uses a layered approach (there is an outer layer and then inner layers for more detail) and the points must first pass the outer layer tests, the algorithm can achieve constant time for all points that are rejected by the outer layer.
A direct analogy is in graphics the use of LOD where from far away a much less detailed model is used for the graphics. So while linear complexity is not appealing, if this can be minimized so constant complexity dominates then the technique could be useful.
The Complex Polygon Examples give examples of the outer layer in 1.2.3.4 .
