One-point reflection

On the Euler Reflection Point

The Euler reflection point E of a triangle is known in literature as the common point of the reflections of its Euler line OH in each of its sidelines, where O and H are the circumcenter and the orthocenter of the triangle, respectively. In this note we prove that E lies on six circles associated with the triangles of Napoleon.

Affine connections, midpoint formation, and point reflection

It is a striking fact that differential calculus exists not only in analysis (based on the real numbers R and limits therein), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the “affine line” R; they act as infinitesimals. (Recall that an element x in a ring R is called nilpotent if xk = 0 for suit...

“Reflection and Choice”: A One-Time Experience?

One-Point Geometric Crossover

Uniform crossover for binary strings has a natural geometric interpretation that allows us to generalize it rigorously to any search space endowed with a notion of distance and any representation [6]. In this paper, we present an analogous characterization for one-point crossover and explicitly derive formally specific one-point crossovers for a number of well-known representations.

One-point Order-compactifications

We classify all one-point order-compactifications of a noncompact locally compact order-Hausdorff ordered topological space X. We give a necessary and sufficient condition for a one-point order-compactification of X to be a Priestley space. We show that although among the one-point order-compactifications of X there may not be a least one, there always is a largest one, which coincides with the...