Computational Geometry

The systematic study of algorithms and data structures for geometric objects, with a focus on exact algorithms that are asymptotically fast.

Two key ingredients of a good algorithmic solution:

Thorough understanding of the problem geometry.

Proper application of algorithmic techniques and data structures.

Example 1 Proximity

Closest café on campus?

Voronoi diagram

Delaunay triangulation

Example 2 Path Planning

Robot

How can a robot find a short route to the destination that avoids all obstacles?

Applications in Computer Graphics

Creating scene images on a display device.

Intersect geometric primitives (lines, polygons, polyhedra, etc.)

Determine primitives lying in a region.

Hidden surface removal – determine the visible part of a 3D scene while discard the occluded part from a view point.

Create realistic-looking scenes – taking into account lighting and computing shadows.

Deal with moving objects and detect collisions.

Applications in Robotics

How the robot perceives, understands, and acts upon its environment,

Motion planning.

Grasping.

Parts orienting.

Optimal placement.

Applications in GIS

Storage of geographical data (contours of countries, height of mountains, course of rivers, population, roads, electricity lines, etc.)

Large amount of data – requiring efficient algorithms.

Geographic data storage (e.g., map of roads for car positioning or computer display).

Interpolation between nearby sample data points

Overlay of multiple maps.

Applications in CAD/CAM

Design of Products with a computer.

Intersection, union, and decomposition of objects.

Testing on product specifications.

Design for assembly – modeling and simulation of assembly.

Testing design for feasibility.

Line Segments & Vectors

x

y

Dot (Inner) Product

Cross (Vector) Product

Turning of Consecutive Segments

Case of Collinearity

No change of direction Direction reversal

Intersection of Two Lines

p

1l

2l

0: 1111 cybxal

0: 2222 cybxal

Rewrite the above equations in terms ofdot products:

0)1,,(),,( 111 yxcba

0)1,,(),,( 222 yxcba

Hence is perpendicular to both and . )1,,( yx ),,( 111 cba ),,( 222 cba

)1,,( yx must be parallel to their cross product!

Homogeneous Coordinates)1,,( yxRepresent every point as

homogeneous coordinates of ),( yx

).,,(),,(),,( 122112211221222111 babaacaccbcbcbacba

)1,,( yx Intersection is parallel to the cross product vector:

Normalize the z-coordinate of the cross product to 1.

).1,,(1221

1221

1221

1221

baba

acac

baba

cbcb

Hence the intersection point: ).,(1221

1221

1221

1221

baba

acac

baba

cbcb

homogeneous coordinates of intersection

),( yx

Intersection Example

Find the intersect point of the lines:

087 yx 0143 yx

Take the cross product of the two coefficient vectors:

)17,23,25()1,4,3()8,7,1(

Intersection point: )17

23,

17

25(

Two Parallel Lines0: 1111 cybxal

0: 2222 cybxal

).,,(),,(),,( 122112211221222111 babaacaccbcbcbacba

02121 abba

).0,,( 12211221 acaccbcb

point at infinity

Two parallel lines “intersect” at infinity.

Line through Two Points

),( 11 yx

),( 22 yx

?0 cbyax

0)1,,(),,( 11 yxcba

0)1,,(),,( 22 yxcba

011 cbyax022 cbyax

)1,,()1,,(),,( 2211 yxyxcba Scaling yields the same line!

Ex. The line through (3, 1) and (-4, 5) has coefficients (i.e., homogeneous coordinates)

)19,7,4()1,5,4()1,1,3( and equation 01974 yx

Advantages of Homogeneous Coordinates

Scaling independent:

Uniform handling of all cases (horizontal, perpendicular, parallel lines, etc.)

No need to solve equations – just take a cross product.

)17,23,25( )34,46,50( )1,17

23,

17

25(

all represent the same point )17

23,

17

25(

Lines in the Space

For more, see http://web.cs.iastate.edu/~cs577/handouts/plucker-coordinates.pdf

Compute the distance between two lines.

Find their common perpendicular.

Find intersections with their common perpendicular.

(When the lines intersect, the two intersections coincide.)

Compute the distance between two lines.