Introduction to Computational Geometry

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Introduction to Computational Geometry

• Hackson Leung_at_HW311

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Agenda

• Dot, Line and Plane
• Cartesian Coordinate System
• Straight Line and Segment
• Distance how to measure?
• Cartesian Coordinate Geometry
• Vector Geometry
• Intersections
• Polygons

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DOT, LINE AND PLANE

• To begin with

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Geometry

• Geometry (Greek ?e?µet??a 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!)

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Geometry

• Geometry (Greek ?e?µet??a 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!)

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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?

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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

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CARTESIAN COORDINATESYSTEM

• You must have learnt it in Mathematics

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Cartesian Coordinate System

• Terminologies
• Dont tell me that you dont know them!

y-axis
A point with ve x and y coordinates
x-axis
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Cartesian Coordinate System

• Terminologies
• Dont tell me that you dont know them!

y-axis
A straight line with negative slope, passing
through origin
x-axis
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Cartesian Coordinate System

• Terminologies
• Dont tell me that you dont know them!

y-axis
A regular pentagon
x-axis
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Cartesian Coordinate System

• You may not know this

y-axis
Two straight lines intersecting at one point
x-axis
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Cartesian Coordinate System

• You may not know this

y-axis
A straight line intersects the pentagon at two
points
x-axis
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Cartesian Coordinate System

• So how to deal with
• Learn more and you will know

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LINE AND SEGMENT

• Diversify

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Line and Segment

• Line infinitely long
• Segment Finite region within the line
• Line is a more general representation of any line
  segment within itself

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DISTANCE

• Way to describe how far we are

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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)
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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
P2 (x2, y2)
P1 (x1, y1)
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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 Pythagorass Theorem
• Euclidean Distance

P2 (x2, y2)
Length l
b y2 y1
P1 (x1, y1)
a x2 x1
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Distance

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

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Distance

• Problem Given two line segments, which one
• is the longest?
• Easy
• Euclidean Distance
• Apply Pythagorass Theorem
• Compare the length
• Really that easy?

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Distance

• Problem Given two line segments, which one
• is the longest?
• Difficulties
• Length of segment A v4 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 gt LB ? LA2 gt LB2 when LA and LB are
  non-negative

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CARTESIAN COORDINATEGEOMETRY

• x, y and z

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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

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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?

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VECTOR GEOMETRY

• Arrows World

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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!)

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Vector Geometry

• Properties
• Length of vector
• Addition (x1, y1) (x2, y2) (x3, y3)
• Subtraction Reverse addition only

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Vector Geometry

• Properties
• Multiplication
• Dot Product (x1, y1)? (x2, y2) x1x2 - y1y2
• Projection of (x1, y1) under (x2, y2) with
  product of their length
• Wedge Product (x1, y1)(x2, y2) x1y2 - y1x2
• Dot Product on itself?
• Wedge Product on itself?

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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

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INTERSECTIONS

• When lines hit

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Intersections

• Recall the previous problem
• How to find the intersection point, if any?

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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?

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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?

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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, r0 OA, r1 AB

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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

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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?

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POLYGON

• Optional topic

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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

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Practice Problems

• HKOJ2071 Little Stage
• HKOJ2074 Storage Centre
• HKOJ2980 Simultaneous Equation

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QA

• Remember TFT is coming!