1 11. Polygons

2 Polygons 2D polygons ( 다각형 ) –Polygon sides are all straight lines lying in the same plane 3D polyhedra ( 다면체 )  chap. 12 –Polyhedra faces are planar and edges are straight lines Topics –The types of polygons –Geometric properties –Convex hulls –Construction regular polygons –Symmetry –containment

3 Definitions Polygon –A many sided 2D figure bounded by a closed circuit of straight-line segments joining successive pairs of points Edges  line segments Vertices  points Type of polygon –Plane polygon: The vertices all lie in the same plane –Skew polygon: Otherwise –Curved surfaces Polygon edges are curve segment Polyline : it is not closed (Figure 11.1)

4 Definitions Plan polygon –Convex If the straight lines that are prolongations of the bounding edges of a polygon do not penetrate the interior If its edges intersect only at vertices –Concave : otherwise –Stellar (= star polygon) If the edges of the polygon intersect at points in addition to the vertex points –Regular polygon (Fig. 11.3) Equilateral : Has straight line edge all of equal length  직사각형 Equiangular: equal vertex angles  마름모꼴 Can be inscribed in a circle with which it shares a common geometric center Ex) 정삼각형, 정사각형, 정육각형

5 Properties Plane polygon properties –V – E = 0  V: vertices #, E: edges # A plane polygon must have at least 3 edges to enclose a finite area The number of vertices equals the number of edges –Triangulation (fig 11.5) Divide any plane polygon into a set of triangles (Fig. 11.5) T = V – 2  T: the minimum number of triangles –The sum of the exterior angles of a plane polygon is –The sum of the interior angles of a plane polygon (Fig 11.6) –The average angle of a polygon

6 Properties –The perimeter of a regular polygon Perimeter = EL  L: the length of an edge –The geometric center of a convex polygon –Area of a convex polygon (fig. 11.7) 1. divide the n-sided polygon into n triangles using geometric center 2. compute the length of each edge using Pythagorean theorem 3. calculate the area of each triangle using the length of its three sides 4. Sum the area of all the triangles –The perimeter of a convex or concave polygon

7 Properties –Detect a stellar polygon (fig. 11.8) Check each edge for the possibility of its intersection with other edges Intersection point: –The area of a polygon (vector geometry) The triangle A plane polygon with n vertices (n>=4)

8 The Convex Hull of a polygon The convex hull of any polygon (Fig. 11.9) –The convex polygon that is formed if we imagine stretching a rubber band over the vertex points –Convex hull of a convex polygon  indetification –Convex hull of a concave polygon Has fewer edges, encloses a larger area 3D convex Hulls –Polyhedra and sets of points in space

9 Construction of Regular Polygons Regular Polygons –Equal edge and interior angles –Their vertices lie on a circle Construction of Regular Polygons –Dividing the circumference of a circle into n equal parts –Gauss’s Theorem We can construct a regular convex polygon of n sides with a compass and straight edge if and only if 1. n = 2 k, where k is any integer, or 2. n =2 k  (2 r +1)  (2 s +1)  (2 t +1)  …, where (2 r +1), (2 s +1), (2 t +1), … are different Fermat prime numbers Ex) 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24 (can construct) 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25 (cannot construct)

10 Symmetry of polygons Revisited Regular polygons (fig ) –Three kinds of symmetry Reflection, rotation, and inversion A regular polygon with n edges –n rotational and n reflection symmetry transformations –n is an even number  an inversion symmetry transformation Ex) an equilateral triangle –3 rotations (120, 240, 360) –3 reflections (fig ) –No inversion (fig. 4.3) Ex) square –4 rotations(90, 180, 270, 360) –4 reflection (fig ) –1 inversion

11 Containment Determine if it is inside, outside, or on the boundary (fig ) Min-max Box Test (work for convex, concave) –The test point (x t, y t ) is not inside the min-max box It is not inside the polygon –The test point is inside the min-max box It still may not be inside the polygon Compute the intersections of y=y t with the edges of the polygon –There is always an even number of intersections (a vertex as 2) –Pair the x coordinates of the intersections in ascending order Ex) (x 1, x 2 ), (x 3, x 4 ) If x t falls inside an interval, it is inside the polygon If x t is identically equal to one of the interval limits, it is on the boundary Otherwise, it is outside the polygon

12 Containment Point-containment test (convex only) (Fig ) –Establish a reference point, (x ref, y ref ), that we know to be inside the polygon –Write the implicit equation of each edge Ex) f 1,2 (x, y), f 2,3 (x, y), f 3,4 (x, y), … –For given test point (x t, y t ) and For each and every edge Compare f i,j (x t, y t ), f i,j (x ref, y ref ) –Same sign  inside the polygon –Any one f i,j (x t, y t )=0 and the preceding condition obtain  on the boundary of the polygon –Any two sequential f i,j both equal zero  on the common vertex –Otherwise  outside the polygon

13 Containment Polygon-Polygon containment (Fig ) –Given two convex polygons If all vertex of one are contained in the other, then the first polygon is inside the second –Concave polygon ??