The phenomenology of triangle centers
Orbits and sweeps, and the dymamics of points
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I call a set of points that follow a certain path through the triangle "a sweep." It is one of a new group of triangle properties gained by experimental observation rather than by theory, although many of the observations are now explained by theory. There are two reasons that phenomenological geometry (based on direct observation) can be more interesting than synthetic geometry. The first is ETC, the grand compilation of triangle points compiled by Clark Kimberling. We can observe the properties of 100 points or 1000 points rather than only a few. The tool of observation is a computer with Geometer's Sketchpad using Yao Liu's documents that display the ETC points. Yao has now plotted the first 260 points. A big mathematics package such as Mathematica or Maple is also useful.
The second reason is that we are moving to a new theoretical basis in triangle geometry. The advances of the 1800's came from realizing that projective and affine techniques create much of what we call Euclidean geometry. In this triangle geometry reflected the general mathematical interest in projective geometry in the 1800's. But mathematicians have been busy for the last hundred years with an emphasis on more abstract structures. As these abstract structures are more likely to play out in the relations between points, phenomological observation may be a very good way to discover them and their consequences.
Below you will find lots of pictures and some movies.
The orbit of the incenter is the invariant curve of the projective transformation ABCG -> ABCIo. It is the orbit of Io, the incenter, in the group of triangle points under barycentric multiplication. Here is a picture of the orbit of the incenter.
Sweep: This is the favored region for centers in the triangle interior.
One way to understand a group is to create orbits by repeated operation by a particular group element. The group is then the direct product of all the orbits. The sweep is the direct product of several orbits.
The affine parallelogram is a basic organizing structure relating triangle centers. A large proportion of points lie in this parallelogram.
The incenter generates all the affine structure if the triangle, hence its overwhelming importance. The orbit of the incenter is obtained by repeated barycentric multiplication and division of G, the centroid, by Io, the incenter, which is the projective transformation ABCG -> ABCIo. This is the set of points ( : bn : ) for n an integer.
The orbit of an arbitrary point P = ( u : v : w ) is the set of points { ( un : vn : wn ) }.
The orbit concept as defined here is an affine invariant concept.
Figure: The orbit of the incenter is shown. The coordinates of the points are shown in red. The Kimberling X-numbers are shown in blue. AP is Conway notation for the preCevian vertex of P.
The sweep is a path through the triangle central region that moves from the vertex with the smallest angle to the one with the largest angle. The P-sweep emphasizes points originating from the direct product of the P-orbit and the Io orbit. The So-sweep is shown below.
If P lies close to the orbit of Io, the P-sweep will represent a broadening of the Io-orbit to a region adjacent to it.
Figure: The So-sweep. This shows the points on the cartesian product of the Io and So orbits and is a wonderful example of the favored region in the interior of the triangle. Red points are part of the Io orbit or the So orbit.
The affine parallelogram is a basic structure that measures and constrols the distribution of points affinely related to a given point. For the 3000 points listed in ETC it turns out to be a defining structure. The affine parallelogram is alway associated with a form of circumconic known as a Mineur conic as well as an hyperbola with perspector at infinity.
The following picture shows the affine parallelogram for Go, the Gergonne point. I often call this picture "the friends of the incenter," since it organizes so many of the points associated with and immediately derived from the incenter. The red hyperbola is the Feuerbach hyperbola.
In the phenomenological context, triangle points seem abnormally to find themselves inside the affine parallelogram of the orthocenter for which the associated conics are the Kiepert and Jerabek hyperbolas.
The wonderful thing about Yao Liu's Skechpad documents is that one can follow the motion of points a the shape of the triangle changes. I will call the motion of points their dynamics.
It has always been obvious that points are dynamically different. It is also obvious that the most significant structures of the triangle are the most rigid ones. The most significant points on the Euler line are O, G, N, H precisely because we always know where they are. This is a very rigid structure. The circumcenter of the tangential triangle is also on the Euler line, but as the triangle changes, its position varies wildly.
The movies below clearly show many examples of rigid and non-rigid behavior.
The following pictures show the distribution of the first 260 ETC centers as the triangle moves from almost isosceles to almost degenerate. Each picture clearly shows the sweep pattern and its association with the affine parallelogram of H and the Kiepert and Jerabek hyperbolas.
Red points are weak; blue strong; green fissile. The red hyperbola is Kiepert, the blue one is Jerabek. The Kiepert hyperbola always follows the sweep; the Jerebek one often does.
Figure: this shows the shape of the triangle that puts the maximum number of points in the sweep area. Note that a large number of points are inside the affine parallelogram of H which is light yellow.
Figure: Here as the triangle moves towards a right angle at C, points begin to accumulate at the right angle. These are mostly point whose C coordinate contains SC. The sweep path is associated with the Kiepert and Jerabek hyperbolas as well as the affine parallelogram of H. Points begin to leave the sweep area via well define paths as shown in the trail of points toward the upper left (following the Euler line) and lower right.
Figure: Here the triangle is approximately right showing the accumulation of points at the right angle. These are mostly point whose C coordinate contains SC. The sweep path is still associated with the Kiepert hyperbolas as well as the affine parallelogram of H. Points have left the sweep area to cluster near the midpoint of the c-edge and the C-vertex.
Figure: As the triangle becomes isosceles the asymptotes of the KH and JH begin to line up, forming pathways to infinity. Note that the affine parallelogram extens and spreads out, still containing a great many point.
Figure: The triangle is highly obtuse. Note that points still largely reside in the affine parallelogram of H and many are still clumped at the C vertex. The asymptotes of the Kiepert and Jerebek hyperbolas tend to eachother as the triangle gets obtuse.
The first move shows the strong points only. Shown are the circumcircle, the nine point circle, the (H, 2R) circle, the Kiepert and Jerabek hyperbolas, the Seiner ellipses, and the Euler and Broard lines.
The second movie shows all points.