Computational Geometry: Numbers & Structures for Intersecting Objects - Prof. Juan Cebral, Study notes of Computer Science

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Data Structures and Algorithms

More geometric algorithms

Searching and sorting

Representing numbers on a computer

Computational geometry

Representation of points

• Coordinate arrays

Representation of segments

• List of end points

Representation of polygons

• List of vertices

Representation of surfaces

• List of triangles / quads

Representation of solids

• List of tetrahedra / hexahedra

Segment intersection in 2D

int ccw(pointT p0,pointT p1,pointT p2) {

dx1=p1.x-p0.x; dy1=p1.y-p0.y; dx2=p2.x-p0.x; dy2=p2.y-p0.y; if( dx1dy2 > dy1dx2 ) return +1; if( dx1dy2 < dy1dx2 ) return -1; if( (dx1dx2<0) || (dy1dy2<0) ) return -1; if( (dx1dx1+dy1dy1)<(dx2dx2+dy2dy2)) return +1; return 0;

} int intersect(segT s1,segT s2) {

return ((ccw(s1.p1,s1.p2,s2.p1)

*ccw(s1.p1,s1,ps,s2.p2)) <=0 )

&& ((ccw(s2.p1,s2.p2,s1.p1)

*ccw(s2.p1,s2,ps,s1.p2)) <=0 );

}

Inclusion in a polygon

Problem: given a point and a polygon represented as anarray of points, determine whether the point is inside oroutside the polygon

Approach: cast a ray from the point to infinity and countthe number of intersections with polygon segments

Number is odd: inside, number is even: outside

Account for cases when the ray lies on a point or asegment

Unstructured grids

Mesh representation

• List of points + list of elements

Derived data structures

• List of elements surrounding each point – List of points connected to each point – List of neighbor elements to each element – List of boundary elements / points

These derived data structures are many times useful fordeveloping efficient algorithms on unstructured grids

Elements surrounding points

int esup1[npoin+1]; int esup2[max_store];

ip

e

1

e

2

e

3

e

4

e

5

ip

e

1

e

2

e

3

e

4

esup1 esup

point

point

point

Elements surrounding points: usage

for(i=0;

i<npoin;

i++)

loop

over

points

(i)

nel=esup1[i+1]-esup1[i];

num

elem

surr

i

lel=&esup2[esup1[i]];

list

elem

surr

i

for(j=0;

j<np;

j++)

loop

over

surr

elem

iel=lel[j];

get

next

elem

surr

I

do_work...

Points surrounding points

int psup1[npoin+1]; int psup2[max_store];

psup1 psup

point

point

point

ip

p

1

p

2

p

3

p

4

p

4

ip

p

1

p

2

p

3

p

4

p

5

p

6

p

7

p

8

Points surrounding points: usage

Do

ipoin=1,npoin

loop

over

points

jp1=psup1(ipoin)+

first

point

surr

ipoin

jp2=psup1(ipoin+1)

last

point

surr

ipoin

do

jp=jp1,jp

loop

over

pts

surr

ipoin

jpoin=psup2(jp)

gather

point

do_work...

enddo

enddo

Elements surrounding elements

int

esuel[nnode*nelem];

init

esuel[1:nnode*nelem]=-

loop

over

elements

(ie)

loop

over

nodes

of

ie

(a)

get

b=next

node

of

ie

get

c=next

node

of

ie

loop

over

elements

surrounding

b

(je

ie)

if(je

and

ie

have

nodes

in

common)

then

store

je

as

neighbor

opposite

to

a

endif endloop endloop endloop

a

b

c

e

a

e

c

e

b

e

Searching and Sorting

Searching •

Sequential search

Binary trees

Balanced trees

Hashing

Radix searching

External searching

Sorting •

Bubble sort

Selection sort

Insertion sort

Shel

l

sort

Quicksort

Radix sort

Merge sort

Heap sort

Sequential search

Suppose we are searching for an integer i

The simplest method for searching is to insert all theelements (integers) into an array and search sequentiallyfor the given number (i)

This method uses N+1 comparisons for unsuccessfulsearches and about N/2 comparisons for successfulsearches

Binary trees

A

tree

is a non-empty collection of

vertices

and

edges

A

vertex

or

node

is a simple object that can have a name

and carry other associated information

An

edge

is a connection between two vertices

A

path

in a tree is a list of distinct vertices in which

successive vertices are connected by edges in the tree

One node in the tree is designated as

root

: there is

exactly one path between the root and each of the othernodes in the tree

Binary trees

Each node has exactly one

parent

Each node has a number of

children

Nodes with no children are called

leaves

or

terminal

nodes

Nodes with children are called

internal

nodes

Any node is the root of a

subtree

A set of trees is called a

forest

The simplest tree is a

binary

tree:

nodes have no or exactly two children

A

B

C

D

E