Lecture 13 Geometry and Computational Geometry

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1 Lecture 13 Geometry and Computational Geometry Euiseong Seo 1

2 Geometry a Branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space We know geometry We don t know how to represent geometry in programming languages 2

3 Lines Lines are the shortest distance between any two points Lines are of infinite length in both directions Cf. line segments Line representations Two points: (x1, y1) and (x2, y2) A single point and a slope: y = mx+b m = (y1-y2) / (x1-x2) what if (x1-x2) = 0? General case ax + by + c = 0 These representations can be coverted to any other 3

4 Lines General case representation typedef struct { } line ax + by + c = 0 double a; double b; // default value is 1 doublec; 4

5 Line Problems Find an intersection of two lines check if they are parallel else find Whatelse? Angles of two lines Closest point on a line Rays half line with an origin 5

6 Triangles and Trigonometry Angle measurement Radians more general Degrees Terminologies Right triangle Perpendicular lines Internal/external angle Equilateral triangle 6

7 Triangle Area Signed triangle area positive b negative c a c a b 7

8 Circles Representation typedef struct { } circle; Formula point c; /* center */ double r; /* radius */ Problems Tangent line Intersection points 8

9 Faster Than a Speeding Bullet Superman flies from s to t He can see through the bullets He can t go through the bullets Find the distance he must travel to reach t 9

10 Line Segments A portion of a line typedef struct { point p1, p2; } segment; Are two segmentsintersect? Deal with degeneracy cases parallel total overlap 10

11 Polygons Closed chains of non-intersecting line segments typedef struct { int n; /* number of vertices */ point p[maxpoly]; /* vertices are ordered? */ } polygon; Convex polygons A polygon P is convex if any line segment defined by two points within P lies entirely within P Counter clock wise predicate 11

Convex Hull The smallest polygon containing a

12 Convex Hull The smallest polygon containing a set of points Graham scan algorithm 1. Find an extreme point 2. Sort the points in order of increasing angle about the pivot 3. Build the hull by adding edges when we make a left turn, and back-tracking when we make a right turn 12

13 Finding Area of a Polygon Triangulation A polygon can be broken down into triangles Van Gogh s Algorithm Area of a polygon is the sum of the triangles 13

14 Jordan Curve Theorem 14

15 Lattice Polygon Pick s Theorem 15

16 Rope Crisis in Ropeland! Rope-pulling (also known as tug of war) is a very popular game in Ropeland, just like cricket is in Bangladesh. Two groups of players hold different ends of a rope and pull. The group that snatches the rope from the other group is declared winner. Due to a rope shortage, the king of the country has declared that groups will not be allowed to buy longer ropes than they require. Rope-pulling takes place in a large room, which contains a large round pillar of a certain radius. If two groups are on the opposite side of the pillar, their pulled rope cannot be a straight line. Given the position of the two groups, find out the minimum length of rope required to start rope-pulling. You can assume that a point represents the position of each group. 16

17 Rope Crisis in Ropeland! Input The first line of the input file contains an integer N giving the number of input cases. Then follow N lines, each containing five numbers X 1, Y 1, X 2, Y 2, and R, where (X 1,Y 1 ) and (X 2,Y 2 ) are the coordinates of the two groups and R>0 is the radius of the pillar. The center of the pillar is always at the origin, and you may assume that neither team starts in the circle. All input values except for N are floating point numbers, and all have absolute value 10,000. Output For each input set, output a floating point number on a new line rounded to the third digit after the decimal point denoting the minimum length of rope required. Sample Input Sample Output

18 Nice Milk Little Tomy likes to cover his bread with milk. He does this by dipping it so that its bottom side touches the bottom of the cup, as in the picture below: Since the amount of milk in the cup is limited, only the area between the surface of the milk and the bottom side of the bread is covered. Note that the depth of the milk is always h and remains unchanged with repeated dippings. Tomy wants to cover this bread with largest possible area of milk in this way, but doesn t want to dip more than k times. Can you help him out? You may assume that the cup is wider than any side of the bread, so it is possible to cover any side completely. 18

19 Nice Milk Input Each test case begins with a line containing three integers n, k, and h (3 n 20, 0 k 8, 0 h 10). A piece of bread is guaranteed to be a convex polygon of n vertices. Each of the following n lines contains two integers x i and y i (0 x i,y i 1,000) representing the Cartesian coordinates of the ith vertex. The vertices are numbered in counterclockwise order. The test case n =0,k =0,h = 0 terminates the input. Output Output (to two decimal places) the area of the largest possible bread region which can be covered with milk using k dips. The result for test case should appear on its own line. Sample Input Sample Output

Grids Geometry Computational Geometry

Grids Geometry Computational Geometry Grids You will encounter many problems whose solutions need grid representation map VLSI layout the most natural way to carve space into regions regular pattern rectangular