ACM WorkshopComputational Geometry

Given two lines L1: A(x1,y1) B(x2,y2)L2: C(x3,y3) D(x4,y4)

Q. Find if they are perpendicular:

Method1:Calculate m1=(B.y-A.y)/(B.x-A.x) m2=(D.y-C.y)/(D.x-C.x)

If(m1*m2==-1) then “Perpendicular”

Basics

Problem with this approach:1.Check for divisibility by zero2. Loss of precision in divisibilty

Method 2:Check if dotProduct is zero:

int dotProduct(Point A,Point B,Point C,Point D) { int x1,y1,x2,y2;

x1=B.x-A.x; y1=B.y-A.y; x2=D.x – C.x;

y2 = D.y – C.y;return (x1*x2+y1*y2);

}

Basics

Q. How to calculate cross product:

int crossProduct(Point A, Point B, Point C , Point D) { int x1,y1,x2,y2; x1=B.x-A.x; y1=B.y-A.y; x2=D.x – C.x;

y2 = D.y – C.y;return (x1*y2-x2*y1);

}

Basics

◦ Find the distance between a line AB and point C◦ A(x1,y1) , B(x2,y2), C(x3,y3)

Question

A

B

C

Area of parallelogram = AB x AC Area of triangle = (AB x AC)/2 (Base * Height)/2 = (AB x AC)/2 Height = (AB x AC)/|AB|

Solution

A

B

C

Distance between a line segment (AB) and a point (C).

double linePointDist(Point A, Point B, Point C, boolean isSegment){ double dist = cross(B-A,C-A) / distance(A,B);

if(isSegment) { int dot1 = dot(B-A,C-B); // AB . BC if(dot1 > 0) return distance(B,C); int dot2 = dot(A-B,C-A); // BA . AC if(dot2 > 0) return distance(A,C);

} return abs(dist);

}

Code

Try to use struct/class for a point and all calculations using p.x and p.y◦ struct Point {

int x; int y;

} Use templating for extending point with double/float coordinates.

Overload operators for cross/dot product, add, subtract.

Sometimes, a clever heuristic will save you lot of work and knowledge otherwise required to solve a problem.

While comparing two floating point numbers, we take a very small value(epsilon ~ 10^-6) for comparison.

Tips

Find the area of a polygon. We start triangulating the polygon by dividing into number of triangles. Pick a vertex and two adjacent points. Calculate area of triangle so formed Now the vertex picked is same but other two vertex

change. Area of the polygon shown =

◦ ABC(AB x AC) +ACD(AC x AD) + ADE(AD x AE)◦ If it were a Concave Polygon??????

Triangulation

CD

E

B

A

int area = 0; //We will triangulate the polygon into triangles with points p[0],p[i],p[i+1]

for(int i = 1; i+1<N; i++){ area += cross(p[i]-p[0], p[i+1]-p[0]); // p[0]p[i] x p[0]p[i+1]

} return abs(area/2.0);

Code

Test if a point is interior or exterior

Question

Drawing a ray from given point out to infinity in some direction. Each time the ray crossed the boundary of the polygon, it would cross

from the interior to the exterior, or vice versa Odd number of time crosses: Interior Even : Exterior To implement, pick multiple large random number for a far point and vote

for the best answer

Solution

String testPoint(verts, x, y) {int N = lengthof(verts); int cnt = 0; double x2 = random()*1000+1000; double y2 = random()*1000+1000; for(int i = 0; i<N; i++){

if(distPointToSegment(verts[i],verts[(i+1)%N],x,y) == 0)

return "BOUNDARY"; if(segmentsIntersect((verts[i],verts[(i+1)%N],{x,y},{x2,y2}))

cnt++; } if(cnt%2 == 0)

return "EXTERIOR"; else

return "INTERIOR"; }

Code

A = y2-y1

B = x1-x2

C = A*x1+B*y1

A1B2x + B1B2y = B2C1

A2B1x + B1B2y = B1C2

To solve◦ A1x + B1y = C1◦ A2x + B2y = C2

Multiply 1 by B2 and 1 by B1

So x = (B2C1 – B1C2) / (A1B2 – A2B1)

Line-Line Intersection

double det = A1*B2 - A2*B1 { if(det == 0){

//Lines are parallel }else{

double x = (B2*C1 - B1*C2)/det double y = (A1*C2 - A2*C1)/det

} }

For line segment, check x,y lies on it or not.min(x1,x2) ≤ x ≤ max(x1,x2)

Precision issues:What if we just want to check if two line segments intersect or not??

Code

// sign of BA x CAint ccw(Point A, Point B, Point C);return value 0 if BAxCA = 0 => A, B, C are collinear 1 if BAxCA > 0 => Triangle ABC is aligned

anti-clockwise -1 if BAxCA < 0 => Triangle ABC is aligned

clockwise

Application exampleif(ccw(a, c, d) == ccw(b, c, d)) // a, b lie on the same side of line cd

Counter ClockWise (ccw)

// Line intersection AB and CDif(ccw(A, B, C) == 0 && ccw(A, B, D) == 0){

// A, B, C, D are collinearif( (AB . BC > 0 and AB . BD > 0) || (BA . AC > 0 and BA . AD

> 0) ){// Doesn’t Intersect

else// Intersect

}if( ccw(A, C, D) != ccw(B, C, D) && ccw(C, A, B) != ccw(D, A,

B) )// Intersect

else// Doesn’t Intersect

Check Line intersection

Given N sets of points, find the closest distance between two points.

Line Sweep Algorithm

10

10

20

12

8

20

10

8

166

12

6

4

4

8

Sort the set according to x-coordinates

Sweep a vertical line across the set of points ,from left to right, performing certain actions every time a point is encountered during the plane sweep

Lets say till now the closest pair encountered has a distance of d and now the sweep line is at a new point p.

In order to get a distance < d, we need to consider only points within a strip of distance d to the left of the sweep line.

Along y axis, we need to consider points within distance +d and –d from the current point p. Thus the strip reduces to d x 2*d rectangle.

These points will be stored in an ordered set D (sorted by their y-coordinates)

Steps

Remove the points further than d to the left of p from the ordered set D that is storing the points in the strip.

Determine the point on the left of p that is closest to it in d x 2*d rectangle

If the distance between this point and p is less than d (the current minimum distance), then update the closest current pair and d.

Actions when a point is encountered

There can be many points in d x2*d rectangle for a p, so how is complexity O(nlgn)?

In the given rectangle, each point(except p) will be at distance > d from each other .

If their existed a pair with distance(dnew) less than d, then the minimum distance till now would have been dnew.

So what is the maximum number of points that can exist in d x 2*d rectangle such that any two points are at distance at least d ?

Why God Why!!!

Maximum number of points in d x 2*d can be 6

Imagine a circle of radius d around each point representing the area that we are not allowed to insert another point into

We can place points somewhere inside the light blue area or on the edges of the circles

Hence we put one more in the middle of the left side leaving the entire box covered except for the remaining two corners and the middle of the right side (all three of these points being right on the boundaries of the circles)

So a maximum of 6 points.

Maximum Number ofPoints to Consider ?

We are only removing points which are distance > d along x-axis not along y-axis. Thus giving the width of rectangle as d. But how to maintain it’s height 2*d?

We cannot remove all points at a distance > d along y axis because the next point we consider may be directly above the current point which may require those points

A set may even contain close to n points when all points are on a vertical line

In that set we consider only points with y coordinates (y-d to y+d), and find the minimum distance

Why Use Sets ?

double lineSweep(Point arr[]){ d = distance(arr[0],arr[1]); l=0; set.insert(arr[0]); set.insert(arr[1]); for(r=2;r<n;r++) { while(point[l].x < (point[r].x – d))

{set.erase(point[l]);l++;}for(it=set.lower_bound(point[r].x,point[r].y-d);it <= set.upper_bound(point[r].x,point[r].y+d);it++){if(d>distance(point[r].x,point[r].y,it->x,it->y))

d=dist(point[r].px,point[r].py,it->px,it->py);}set.insert(point[r]);}return d;

}

Code