1 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Lecture 13 Geometry and

Computational Geometry

Euiseong Seo

(euiseong@skku.edu)

2 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

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

3 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

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

4 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Lines

General case representation typedef struct {

double a;

double b; // default value is 1

doublec;

} line

• ax + by + c = 0

5 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Line Problems

Find an intersection of two lines

• check if they are parallel

• else find

What else?

• Angles of two lines

• Closest point on a line

• Rays – half line with an origin

6 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Triangles and Trigonometry

Angle measurement • Radians – more general

• Degrees

Terminologies • Right triangle

• Perpendicular lines

• Internal/external angle

• Equilateral triangle

7 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Triangle Area

Signed triangle area

c

a

b

positive

b

a

c

negative

8 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Circles

Representation typedef struct {

point c; /* center */

double r; /* radius */

} circle;

Formula

Problems

• Tangent line

• Intersection points

9 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

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

10 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Line Segments

A portion of a line typedef struct {

point p1, p2;

} segment;

Are two segments intersect?

• Deal with degeneracy cases – parallel

– total overlap

11 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

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

12 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

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

13 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

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

14 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Jordan Curve Theorem

15 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Lattice Polygon

Pick’s Theorem

16 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Rope Crisis in Ropeland!

17 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Rope Crisis in Ropeland!

18 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Nice Milk

19 SWE2004: Principles in Programming | Spring 2015 | Euiseong Seo (euiseong@skku.edu)

Nice Milk