Introduction to Computational Geometry Hackson Leung@HW311.

Introduction to Computational GeometryHackson Leung@HW311

Agenda• Dot, Line and Plane

• Cartesian Coordinate System

• Straight Line and Segment

• Distance – how to measure?

• Cartesian Coordinate Geometry

• Vector Geometry

• Intersections

• Polygons

DOT, LINE AND PLANETo begin with

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!)

Geometry

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…?

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’’

CARTESIAN COORDINATESYSTEM

You must have learnt it in Mathematics

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



x-axis

A straight line with negative slope, passing through origin



x-axis

A regular pentagon

Cartesian Coordinate System

y-axis

x-axis

• You may not know this

Two straight lines intersecting at one point

Cartesian Coordinate System

y-axis

x-axis

A straight line intersects the pentagon at two points

• You may not know this

Cartesian Coordinate System• So how to deal with…

• Learn more and you will know

y-axis

x-axis


y-axis

x-axis

A straight line intersects the pentagon at two points

LINE AND SEGMENTDiversify

Line and Segment• Line: infinitely long

• Segment: Finite “region” within the line

• Line is a more general representation of any line segment within itself

DISTANCEWay to describe how far we are

Distance• Problem: Given two points, how far are they?

• Use ruler to measure…• In computer, no way!• Not precise enough!

Length l P2

(x2, y2)

P1

(x1, y1)


• 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 P2

(x2, y2)

P1

(x1, y1)

2121 yyxxl


• 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

P2 (x2, y2)

P1

(x1, y1)a = | x2 – x1 |

b = | y2 – y1 |

221

221 )()( yyxxl

Distance• Problem: Given a line segment, how long is it?

• Treat the two ends as points• Go back to last slide

Distance• Problem: Given two line segments, which one

is the longest?• Easy

• Euclidean Distance• Apply Pythagoras’s Theorem• Compare the length

• …Really that easy?

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?)

• LA > LB LA2 > LB

2 when LA and LB are non-negative

CARTESIAN COORDINATEGEOMETRY

x, y and z

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

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-y1) = m(x-x1)

• Always possible to represent?

• Another choice: two-point form• y - y1 / x – x1 = y1 – y2 / x1 – x2

• If x1 = x2, then?

VECTOR GEOMETRYArrow’s World

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

Vector Geometry• Properties

• Length of vector =

• Addition: (x1, y1) + (x2, y2) = (x3, y3)

• Subtraction: Reverse addition only

22 yx

y-axis

x-axis

(x1, y1) (x2, y2)

(x3, y3)

Vector Geometry• Properties

• Multiplication• Dot Product: (x1, y1) (x‧ 2, y2) = x1*x2 - y1*y2

• Projection of (x1, y1) under (x2, y2) with product of their length

• Wedge Product: (x1, y1)^(x2, y2) = x1*y2 - y1*x2

• Dot Product on itself?• Wedge Product on itself?

Vector Geometry• Straight line equation

• Describe any vectors r that end on a line u, which passes through r0=(a, b) and parallel to r1=(c, d)

• u: r = r0 + tr1 , t is any real number

INTERSECTIONSWhen lines hit

Intersections• Recall the previous problem

• How to find the intersection point, if any?

y-axis

x-axis


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?


• Cartesian coordinate geometry• Given 2 straight line equations, find all distinct

intersection point(s)• HKOI1998 Junior Q1 – Simultaneous Equations• How about K lines?


• 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, r0 = OA, r1 = AB


• Vector geometry• Given 2 straight line equations, find all distinct

intersection point(s)• Two vectors are formed, let them be AB and CD

CDAB

CDAC

CDAB

CDOAOCu

CDOAOCCDABu

CDOCCDABuCDOA

vuCDvOCABuOA

)(

)(

,,


• 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?

CDAB

CDAC

CDAB

CDOAOCu

)(

ABuOACDAB

POLYGONOptional topic

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

Practice Problems• HKOJ2071 Little Stage

• HKOJ2074 Storage Centre

• HKOJ2980 Simultaneous Equation

Q&A• Remember: TFT is coming!

Introduction to Computational Geometry Hackson Leung@HW311.

Documents

Transcript of Introduction to Computational Geometry Hackson Leung@HW311.