CS133 Computational Geometry Computational Geometry Primitives 1 - PowerPoint PPT Presentation

CS133 Computational Geometry Computational Geometry Primitives 1

Point Representation A point in the 2D Cartesian space is represented as a vector from the origin to the point 𝑞 = 𝑦, 𝑧 p p y O x 2

Line Segment Representation A line segment is represented by its two end points Length= 𝑏 − 𝑐 Straight lines are usually represented by two points on it p1 p2 3

Application: CCW sort How to sort a list of points in a CCW order around the origin? Naïve solution: Calculate all the angles and sort Advantages: Easy and can reuse an existing sort algorithm as-is Disadvantages: arctan calculation is complex and might be inaccurate 4

CCW Sort What is the cross/dot product of the vectors of two points? 5

CCW Comparator 𝑅 2 𝑅 1 A point in 𝑅 2 or 𝑅 3 precedes a point in 𝑅 4 or 𝑅 1 Two points in the Two points with same half are the same angle ordered based on are assumed to their angle be equal 𝑅 3 𝑅 4 6

-1: 𝑞 1 precedes 𝑞 2 CCW Comparator +1: 𝑞 2 precedes 𝑞 1 0: On the same angle compare( 𝑞 1 = 𝑦 1 , 𝑧 1 , 𝑞 2 = (𝑦 2 , 𝑧 2 ) ) If ( 𝑦 1 < 0 and 𝑦 2 > 0 ) OR ( 𝑦 1 > 0 and 𝑦 2 < 0 ) Return 𝑦 1 < 0 ? -1 : +1 cp = 𝑞 1 × 𝑞 2 If 𝑑𝑞 < 0 Return -1 If 𝑑𝑞 > 0 Return +1 // cp = 0 If ( 𝑧1 < 0 and 𝑧 2 > 0 ) OR ( 𝑧1 > 0 and 𝑧2 < 0 ) Return 𝑧 1 < 0 ?-1 : +1 Return 0 7

CCW Sort How to sort a set of points around their geometric center? First, compute the geometric center Then, translate the points to make the origin at the center 8

Collinearity Given three points, check if they are on the same straight line 𝑞 2 Collinear( 𝑞 1 , 𝑞 2 , 𝑞 ) Return 𝑞 1 𝑞 2 × 𝑞 1 𝑞 == 0 𝑞 𝑞 𝑞 1 9

Missing Square Problem Hint: Test the collinearity of three points on the figure 10

Direction Given a straight line (ray) and a point, find whether the point is to the right or left of the line 𝑞 Direction( 𝑞 1 , 𝑞 2 , 𝑞 ) 𝑑𝑞 = 𝑞 1 𝑞 2 × 𝑞 1 𝑞 𝑞 2 𝑞 If 𝑑𝑞 < 0 Return “right” 𝑞 𝑞 1 If 𝑑𝑞 > 0 Return “left” Return “on -the- line” 11

Point-line Relationship Given a line segment and a point, find whether the point is on the line segment or not Coincident( 𝑞 1 (𝑦 1 , 𝑧 1 ) , 𝑞 2 (𝑦 2 , 𝑧 2 ) , 𝑞(𝑦, 𝑧) ) If NOT Collinear( 𝑞 1 , 𝑞 2 , 𝑞 ) Return false Return 𝑦 ∈ min 𝑦 1 , 𝑦 2 , max 𝑦 1 , 𝑦 2 AND ( 𝑧 ∈ min 𝑧 1 , 𝑧 2 , max 𝑧 1 , 𝑧 2 12

Line-line Relationship Given two straight lines, find whether they intersect or not IsIntersected( 𝑞 1 , 𝑞 2 , 𝑞 3 , 𝑞 4 ) If 𝑞 1 𝑞 2 × 𝑞 3 𝑞 4 ≠ 0 Return true // intersected in a point // The two lines are parallel If Collinear( 𝑞 1 , 𝑞 2 , 𝑞 3 ) Return true // Lines are coincident Return false // Parallel ant not coincident 13

Line-line Intersection Given two straight lines, find their intersection Solve the two linear equations that represent the two lines One solution ➔ The lines intersect in a point Infinite solutions ➔ The lines are coincident No solutions ➔ The lines are disjoint parallel 14

Line-line Intersection 𝑞 1 , 𝑞 2 : First line (input) 𝑞 3 , 𝑞 4 : Second line (input) 𝑞 0 : Intersection point (output) 𝑞 0 must be collinear with 𝑞 1 𝑞 2 and 𝑞 3 𝑞 4 𝑞 1 𝑞 2 × 𝑞 1 𝑞 0 = 0 𝑞 3 𝑞 4 × 𝑞 3 𝑞 0 = 0 𝑞 2 𝑞 3 𝑞 0 See the rest of the 𝑞 4 derivation in the notes 𝑞 1 15

Line Segment Intersection Given two line segments, find whether they intersect or not. If they intersect, find their intersection point (Homework) 16