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:
“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)
“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)