Properties
- The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals . Thus, d divides q.
- In the special case of d = 2, maximal arcs are known as hyperovals which can only exist if q is even.
- An arc K having one fewer point than a maximal arc can always be uniquely extended to a maximal arc by adding to K the point at which all the lines meeting K in d - 1 points meet.
- In PG(2,q) with q odd, no non-trivial maximal arcs exist.
- In PG(2,2h), maximal arcs for every degree 2t, 1 ≤ t ≤ h exist.
Read more about this topic: Maximal Arc
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)