Two conjectures regarding dense near polygons with three points on each line

Dense near polygons with two types of quads and three types of hexes

We show that the number of points at distance i from a given point x in a dense near polygon only depends on i and not on the point x. We give a number of easy corollaries of this result. Subsequently, we look to the case of dense near polygons S with an order in which there are two possibilities for tQ, where Q is a quad of S, and three possibilities for (tH , vH), where H is a hex of S. Using...

On Sets Which Meet Each Line in Exactly Two Points

ABSTRACT Using techniques from geometric measure theory and descriptive set theory we prove a general result concerning sets in the plane which meet each straight line in exactly two points As an application we show that no such two point set can be expressed as the union of countably many recti able sets together with a set with Hausdor measure zero Also as another corollary we show that no an...

On the Finiteness of Near Polygons with 3 Points on Every Line

Let S be a near polygon of order (s, t) with quads through every two points at distance 2. The near polygon S is called semifinite if exactly one of s and t is finite. We show that S cannot be semifinite if s = 2 and derive upper bounds for t .

Caging Polygons with Two and Three Fingers

We present algorithms for computing all placements of two and three fingers that cage a given polygonal object with n edges in the plane. A polygon is caged when it is impossible to take the polygon to infinity without penetrating one of the fingers. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging placements of two disc fingers in O(...

On the classification of dense near polygons with lines of size 3 Pieter