## Random TrianglesJanuary 26, 2017 The triangle is the simplest polygon, and triangles are technologically very useful. Many computer-generated images are built from triangles, and the finite element method decomposes a surface into triangular elements for analysis of such diverse properties as thermal conduction and mechanical stress.
Pneumatica. This steam engine, although primitive, predates James Watt by seventeen centuries.
A of an arbitrary triangle from the length of its sides appears in his Metrica (c. 60 AD). Using a, b, and c as the length of the sides, the formula is as follows:
Where the parameter, s, known as the semiperimeter, is half the perimeter; viz., If you want to eliminate the semiperimeter from the formula, the formula can be written in terms of the sides, only, as It's easy to verify this using a 3-4-5 right triangle, which will have an area of (1/2)(base)(height) = (1/2)(4)(3) = 6; viz, Eugen J. Ionascu of the Department of Mathematics, Columbus State University (Columbus, Georgia), has recently published his solution of the interesting problem of how often a fixed point will be interior to a random triangle when these are constrained to be drawn inside a variety of planar regions such as circles and regular polygons.[1] While the equations are dense for most of these planar regions, there's a simple case of triangles drawn inside a unit square in which numerical estimates are given for certain points. I became interested in this problem as a means to practice my computer simulation skills. The primary problem of such a simulation is the method of determining whether the point is inside or outside of the random triangle. An intuitive method is to draw vectors from the point to each vertex of the triangle and sum the angles between them. If these add to 360°, then the point is inside the triangle. This method is computationally intensive, so it's not ideal for a simulation. Another method is to define the location of the point in terms of vectors drawn between two of the vertices using what's called a barycentric coordinate system (see figure). A moment's reflection will reveal that defining the location of the point as a linear combination of these two vectors gives an easy way to determine whether or not the point is interior to the triangle. In reference to the figure, this will involve limits on the parameters, u and v and their sum.[2]
## References:- Eugen J. Ionascu, "Random triangles in planar regions containing a fixed point," arXiv, December 15, 2016.
- Barycentric coordinate system, Totologic Blog.
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