1. Introduction to Computational Geometry Hackson Leung@HW311

2. Agenda Dot, Line and Plane Cartesian Coordinate System Straight Line and Segment Distance – how to measure? Cartesian Coordinate Geometry Vector Geometry Intersections Polygons

3. DOT, LINE AND PLANE To begin with

4. Geometry ``Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space.’’ – Wikipedia What to deal with? Lines Polygons Planes Objects (in N dimensional!)

5. Geometry ``Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space.’’ – Wikipedia What to deal with (Computationally)? Lines Polygons Planes Objects (in N dimensional!)

6. Simple geometry revisit 1-Dimensional Simply R a single point and line only e.g. Number Line 2-Dimensional 2-tuple (a.k.a. pair) Can represent point, line and plane 3-Dimensional 3-tuple (a.k.a. Triple) Can represent…?

7. Computational Geometry Adopt algorithmic approach to solve problems in terms of geometry e.g. List out all possible distinct intersection points, given several lines In this session, we only focus on 2-D geometry only Further info can be found at ``Advanced Computational Geometry’’

8. CARTESIAN COORDINATE SYSTEM You must have learnt it in Mathematics

9. Cartesian Coordinate System Terminologies Don’t tell me that you don’t know them! y-axis x-axis A point with +ve x and y coordinates

10. Cartesian Coordinate System Terminologies Don’t tell me that you don’t know them! y-axis x-axis A straight line with negative slope, passing through origin

11. Cartesian Coordinate System Terminologies Don’t tell me that you don’t know them! y-axis x-axis A regular pentagon

12. Cartesian Coordinate System y-axis x-axis You may not know this Two straight lines intersecting at one point

13. Cartesian Coordinate System y-axis x-axis A straight line intersects the pentagon at two points You may not know this

14. Cartesian Coordinate System So how to deal with… Learn more and you will know y-axis x-axis Two straight lines intersecting at one point y-axis x-axis A straight line intersects the pentagon at two points

15. LINE AND SEGMENT Diversify

16. Line and Segment Line: infinitely long Segment: Finite “region” within the line Line is a more general representation of any line segment within itself

17. DISTANCE Way to describe how far we are

18. Distance Problem: Given two points, how far are they? Use ruler to measure… In computer, no way! Not precise enough! Length l P 2 (x 2, y 2 ) P 1 (x 1, y 1 )

19. Distance Problem: Given two points, how far are they? If we do not care about the actual distance… Think about that the world is full of grids… And you can only walk on their sides The famous Manhattan Distance Length l P 2 (x 2, y 2 ) P 1 (x 1, y 1 )

20. Distance Problem: Given two points, how far are they? If we do care about the actual distance… Manhattan distance gives us a brief idea to calculate the actual distances If you have learnt Pythagoras’s Theorem… Euclidean Distance Length l P 2 (x 2, y 2 ) P 1 (x 1, y 1 ) a = | x 2 – x 1 | b = | y 2 – y 1 |

21. Distance Problem: Given a line segment, how long is it? Treat the two ends as points Go back to last slide

22. Distance Problem: Given two line segments, which one is the longest? Easy Euclidean Distance Apply Pythagoras’s Theorem Compare the length …Really that easy?

23. Distance Problem: Given two line segments, which one is the longest? Difficulties Length of segment A = √4 = 2 Length of segment B = 1.9999999… In computer, it may turn out that A = B! Known as precision error Cure Use Manhattan Distance (does it work?) L A > L B  L A 2 > L B 2 when L A and L B are non-negative

24. CARTESIAN COORDINATE GEOMETRY x, y and z

25. Coordinate Geometry Basically from Cartesian Coordinate System How to describe: A point? (x, y) Coordinates for 2D A line? Straight line equation A line segment? Straight line equation, plus range of x and y Two points form a segment

26. Coordinate Geometry For straight line, we need to know its slope Given two points on the line slope = m = change of y value / change of x value What if x = 0? Common representation: point-slope form (y-y 1 ) = m(x-x 1 ) Always possible to represent? Another choice: two-point form y - y 1 / x – x 1 = y 1 – y 2 / x 1 – x 2 If x 1 = x 2, then?

27. VECTOR GEOMETRY Arrow’s World

28. Vector Geometry What is a vector? Geometric object which has both magnitude and direction (think of line segment) A notation of means a motion from A to B Notation: (x, y) means a point P from O (0, 0) to (x, y) (Note the terminology!) y-axis x-axis A vector from origin to a certain point

29. Vector Geometry Properties Length of vector = Addition: (x 1, y 1 ) + (x 2, y 2 ) = (x 3, y 3 ) Subtraction: Reverse addition only y-axis x-axis (x1, y1) (x2, y2) (x3, y3)

30. Vector Geometry Properties Multiplication Dot Product: (x 1, y 1 ) ‧ (x 2, y 2 ) = x 1 *x 2 - y 1 *y 2 Projection of (x 1, y 1 ) under (x 2, y 2 ) with product of their length Wedge Product: (x 1, y 1 )^(x 2, y 2 ) = x 1 *y 2 - y 1 *x 2 Dot Product on itself? Wedge Product on itself?

31. Vector Geometry Straight line equation Describe any vectors r that end on a line u, which passes through r 0 =(a, b) and parallel to r 1 =(c, d) u: r = r 0 + tr 1, t is any real number

32. INTERSECTIONS When lines hit

33. Intersections Recall the previous problem How to find the intersection point, if any? y-axis x-axis Two straight lines intersecting at one point

34. Intersections How to find the intersection point, if any? Cartesian coordinate geometry Given K straight line equations, find all distinct intersection point(s) Usually a line is defined by giving two arbitrary points which are on the line How to represent its equation? What if a line is a vertical line?

35. Intersections How to find the intersection point, if any? Cartesian coordinate geometry Given 2 straight line equations, find all distinct intersection point(s) HKOI1998 Junior Q1 – Simultaneous Equations How about K lines?

36. Intersections How to find the intersection point, if any? Vector geometry Given 2 straight line equations, find all distinct intersection point(s) Need to find their equations? Two vectors are formed, let them be AB and CD We let vector itself be a line In general, r 0 = OA, r 1 = AB

37. Intersections How to find the intersection point, if any? Vector geometry Given 2 straight line equations, find all distinct intersection point(s) Two vectors are formed, let them be AB and CD

38. Intersections How to find the intersection point, if any? Vector geometry Clearly, produces the intersection point What if is zero? Parallel but not the same line Parallel but they are on the same line How about the segment intersection?

39. POLYGON Optional topic

40. Polygon If time allows, we would discuss the following topics Polygon – Convex or Concave Is a point inside or outside a polygon? Given N points, can I find a minimum bounded polygon that includes those points? I have two convex polygons, tell me their intersection area I have two polygons, tell me if they can combine to a single polygon through rotation and translation

41. Practice Problems HKOJ2071 Little Stage HKOJ2074 Storage Centre HKOJ2980 Simultaneous Equation

42. Q&A Remember: TFT is coming!