1. 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.

2. Example 1  Proximity Closest café on campus? Voronoi diagram
Delaunay triangulation
Voronoi diagram

3. Example 2  Path Planning
How can a robot find a short route to the destination that
avoids all obstacles?
Robot

4. 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.

5. Applications in Robotics
How the robot perceives, understands, and acts upon its
environment:
Motion planning.
Grasping.
Parts orienting.
Optimal placement.

6. 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.

7. Applications in CAD/CAM
Design of Products with a computer.
Intersection, union, and decomposition of objects.
Testing on product specifications.
Testing design for feasibility.
Design for assembly – modeling and simulation of assembly.

8. Line Segments & Vectorsy
𝑝 1 = ( 𝑥 1 , 𝑦 1 )
𝑝 1 − 𝑝 2 =
( 𝑥 1 − 𝑥 2 , 𝑦 1 − 𝑦 2 )
𝑝 2 = ( 𝑥 2 , 𝑦 2 )
𝑂 = (0, 0) x
Points (vectors): 𝑝 1 , 𝑝 2 , 𝑝 1 − 𝑝 2 = 𝑝 2 𝑝 1
Line segment: 𝑝 2 𝑝 1 = 𝑝 1 𝑝 2

9. Dot (Inner) Product 𝑦 𝑝 2 𝜑 𝑝 1 (0, 0) 𝑥
𝑝 1 ∙ 𝑝 2 = 𝑥 1 𝑥 2 + 𝑦 1 𝑦 2 = 𝑝 2 ∙ 𝑝 1 = | 𝑝 𝑝 2 | cos 𝜑

10. Cross (Vector) Product𝑦
𝑝 1 + 𝑝 2
𝑝 2
𝑝 1 × 𝑝 2 is the signed area
of the parallelogram. 𝜑
𝑝 1
(0, 0) 𝑥
𝑝 1 × 𝑝 2 = 𝑥 1 𝑦 2 − 𝑥 2 𝑦 1 =− 𝑝 2 × 𝑝 1 = | 𝑝 𝑝 2 | sin 𝜑
𝑝 1 and 𝑝 2 are collinear with the origin iff 𝑝 1 × 𝑝 2 = 0.

11. Turning of Consecutive Segments
Segments 𝑝 0 𝑝 1 and 𝑝 1 𝑝 Move from 𝑝 0 to 𝑝 1 then to 𝑝 2 .
𝑝 2
𝑝 1
𝑝 0
𝑝 0
𝑝 2
𝑝 1
𝑝 1
𝑝 2
𝑝 0
Counterclockwise Clockwise Turn of 0 or 𝜋
𝑝 0 𝑝 1 × 𝑝 1 𝑝 2 >0
𝑝 0 𝑝 1 × 𝑝 1 𝑝 2 < 0
𝑝 0 𝑝 1 × 𝑝 1 𝑝 2 = 0

12. Case of Collinearity 𝑝 0 𝑝 1 × 𝑝 1 𝑝 2 = 0
𝑝 0 𝑝 1 × 𝑝 1 𝑝 2 = 0
𝑝 1 , 𝑝 2 , 𝑝 3 are collinear.
𝑝 1
𝑝 2
𝑝 0
𝑝 2
𝑝 1
𝑝 0
No change of direction Direction reversal
𝑝 0 𝑝 1 ∙ 𝑝 1 𝑝 2 >0
𝑝 0 𝑝 1 ∙ 𝑝 1 𝑝 2 < 0

13. Intersection of Two Lines
𝑙 1 : 𝑎 1 𝑥+ 𝑏 1 𝑦+ 𝑐 1 =0
𝑙 1
𝑙 2 : 𝑎 2 𝑥+ 𝑏 2 𝑦+ 𝑐 2 =0 𝑝
Rewrite the above equations as dot products:
( 𝑎 1 , 𝑏 1 , 𝑐 1 ) ∙ (𝑥,𝑦,1) =0
𝑙 2
( 𝑎 2 , 𝑏 2 , 𝑐 2 ) ∙ (𝑥,𝑦,1) =0
Hence (𝑥,𝑦, 1) is perpendicular to both ( 𝑎 1 , 𝑏 1 , 𝑐 1 ) and ( 𝑎 2 , 𝑏 2 , 𝑐 2 )
(𝑥,𝑦, 1) must be parallel to their cross product!

14. Homogeneous Coordinates
Represent every point (𝑥, 𝑦) as 𝑥, 𝑦, 1 .
homogeneous coordinates of (𝑥, 𝑦)
Intersection (𝑥, 𝑦, 1) is parallel to the cross product vector:
𝑎 1 , 𝑏 1 , 𝑐 1 × 𝑎 2 , 𝑏 2 , 𝑐 2 =( 𝑏 1 𝑐 2 − 𝑏 2 𝑐 1 , 𝑐 1 𝑎 2 − 𝑐 2 𝑎 1 , 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 )
homogeneous coordinates of intersection
Normalize the 𝑧-coordinate of the cross product to 1.
𝑏 1 𝑐 2 − 𝑏 2 𝑐 1 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 , 𝑐 1 𝑎 2 − 𝑐 2 𝑎 1 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 ,1
Hence the intersection point:
𝑏 1 𝑐 2 − 𝑏 2 𝑐 1 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 , 𝑐 1 𝑎 2 − 𝑐 2 𝑎 1 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1

15. Intersection Example Find the intersect point of the lines:
Take the cross product of the two coefficient vectors:
Intersection point:

16. Two Parallel Lines 𝑎 1 𝑏 2 − 𝑏 1 𝑎 2 =0 𝑙 1 : 𝑎 1 𝑥+ 𝑏 1 𝑦+ 𝑐 1 =0
𝑙 1 : 𝑎 1 𝑥+ 𝑏 1 𝑦+ 𝑐 1 =0
𝑙 2 : 𝑎 2 𝑥+ 𝑏 2 𝑦+ 𝑐 2 =0
𝑎 1 𝑏 2 − 𝑏 1 𝑎 2 =0
𝑎 1 , 𝑏 1 , 𝑐 1 × 𝑎 2 , 𝑏 2 , 𝑐 2 =( 𝑏 1 𝑐 2 − 𝑏 2 𝑐 1 , 𝑐 1 𝑎 2 − 𝑐 2 𝑎 1 , 𝑎 1 𝑏 2 − 𝑎 2 𝑏 1 )
=( 𝑏 1 𝑐 2 − 𝑏 2 𝑐 1 , 𝑐 1 𝑎 2 − 𝑐 2 𝑎 1 ,𝟎)
point at infinity
Two parallel lines “intersect” at infinity.

17. Line through Two Points
𝑎𝑥+𝑏𝑦+𝑐=0?
𝑎 𝑥 1 +𝑏 𝑦 1 +𝑐=0
𝑎 𝑥 2 +𝑏 𝑦 2 +𝑐=0
( 𝑥 1 , 𝑦 1 )
( 𝑥 2 , 𝑦 2 )
𝑎, 𝑏, 𝑐 ∙ 𝑥 1 , 𝑦 1 ,1 =0
𝑎, 𝑏, 𝑐 ∙ 𝑥 2 , 𝑦 2 ,1 =0
𝑎, 𝑏, 𝑐 = 𝑥 1 , 𝑦 1 ,1 × 𝑥 2 , 𝑦 2 ,1
Scaling yields the same line!
Ex. The line through (3, 1) and (-4, 5) has coefficients (i.e., homogeneous coordinates)
and equation

18. Advantages of Homogeneous Coordinates
Scaling independent:
all represent the same point
Uniform handling of all cases (horizontal, perpendicular,
parallel lines, etc.)
No need to solve equations – just take a cross product.
For more, see

19. Lines in the Space Use Plucker coordinates: (𝑙, 𝑝×𝑙) 𝑙 𝑝 𝑂
Compute the distance between two lines. 𝑝
Find their common perpendicular.
Find intersections with their common
perpendicular. 𝑂
(When the lines intersect, the two
intersections coincide.)
For more, see