Midpoint polygon

Abstract: B o o l e a n p r o c e d u r e P O I N T I N POLYGON (n, x, y, xO, y0); v a l u e n, x0, y0; i n t e g e r n ; a r r a y x, y; r e a l x0, y0; c o m m e n t if t h e p o i n t s (x[i], y[i]) (i = 1, 2 , . . , n) a r e i n t h i s cyclic o r d e r t h e ve r t i c e s of a s imp le c losed p o l y g o n a n d (x0, y0) is a p o i n t n o t on a n y s ide of t h e po lygon , t h e n t h e procedure d e t e r m i n e s , b y s e t t i n g " p o i n t in p o l y g o n " to t r u e , w h e t h e r (x0, y0) l ies in t h e i n t e r i o r of t h e p o l y g o n ; b e g i n i n t e g e r i ; B o o l e a n b; x [ n + l ] : = x [ 1 ] ; y [ n + l ] : = y [ 1 ] ; b : = t r u e ; f o r i i= 1 s t e p 1 u n t i l n d o i f ( y < y [ i ] ~ y > y [ i + l ] ) h xO -x[i] -(yO -y[i]) X (x[i + 1] -z[i])/(y[i + 1] -y[i]) < 0 t h e n b := --1 b; P O I N T I N POLYGON := ~ b; e n d I~OINT I N P O L Y G O N

Abstract: We propose a very simple ray-shooting algorithm, whose only data structure is a triangulation. The query algorithm, after locating the triangle containing the origin of the ray, walks along the ray, advancing from one triangle to a neighboring one until the polygon boundary is reached. The key result of the paper is a Steiner triangulation of a simple polygon with the property that a ray can intersect at most O(log n) triangles before reaching the polygon boundary. We are able to compute such a triangulation in linear sequential time, or in O(log n) parallel time using O(n/log n) processors. This gives a simple, yet optimal, ray-shooting algorithm for a simple polygon. Using a well-known technique, we can extend our triangulation procedure to a multiconnected polygon with k components and n vertices, so that a ray intersects at most O(√κ log n) triangles. © 1995 Academic Press, Inc.

Abstract: Two polygon fill algorithms are presented for filling polygons on a graphics display. The first polygon fill algorithm fills polygons that are strictly convex. The second polygon fill algorithm fills a larger class of polygons than the first polygon fill algorithm which includes polygons being concave in the x direction, and polygons having crossing lines. The first polygon fill algorithm tests the polygon for strict convexity by testing for a consistent turning direction, and by testing for once around in the y direction. The first polygon fill algorithm then stores the maximum and minimum value of the pel selected by the Bresenham algorithm for each scan line of the polygon. The fill line is drawn from the pel having the minimum value to the pel having the maximum value for each scan line of the polygon. The second polygon fill algorithm tests the polygon to ensure that it can be filled with one unique fill line for each scan line of the polygon. The second polygon fill algorithm stores both a minimum value and maximum value for each scan line of the polygon for each line of the polygon. A fill line is then drawn from the least minimum value to the greatest maximum value for each scan line of the polygon.

Abstract: The geodesic center of a simple polygon is a point inside the polygon which minimizes the maximum internal distance to any point in the polygon. We present an algorithm which calculates the geodesic center of a simple polygon withn vertices in timeO(n logn).