CS 3250: Computational Geometry

Fall 2021: M, W 1:15 - 2:40 in Searles 126

Welcome to Computational Geometry, Fall 2021!

Computational geometry studies algorithms for problems that involve geometric data: (finite) sets of points, line segments and polygons. It is a young and growing field, driven by applications in graphics, robotics, autonomous vehicles, vision, image processing, and GIS. Some classical examples of geometric problems: Is a point inside or outside a polygon? Do two polygons intersect? Given a set of points in the plane, what is the closest pair of points? What is the smallest convex polygon that contains a set of points in the plane? What are all the points that fall in a given query range? What is visible from a point in a polygon? How many points are sufficient to guard any polygon? Given two sets of segments, what are their intersections? Given a set of points, what is their smallest enclosing circle? Given a set of polygonal obstacles, and a polygonal robot moving in 2D with translation and rotation, what is an efficient path between a start and end location? And many more. This class is an introduction to computational geometry and will cover some of the fundamental problems and techniques in the field.

Syllabus overview:

• Introduction and warmup
  • Finding collinear points
  • Finding the closest pair of points
• Geometric primitives
  • area of triangle, orientation, segment intersection.
• Convex hulls in 2D and 3D
  • naive, gift wrapping, quickhull, Graham scan, incremental, divide-and-conquer algorithm
  • lower bound
  • 3D hulls
• Segment intersection
  • Bentley-Ottman sweep
• Art gallery problem. Fisk's sufficiency proof.
• Polygon triangulation
  • quadratic, based on ear removal algorithm.
  • triangulation of monotone polygons.
  • polygon triangulation in O(n lg n) via trapezoidalization.
• Geometric searching
  • orthogonal range searching with kd-trees and range trees.
• Voronoi diagrams and Delaunay triangulations.
• Combinatorial motion planning in 2D
  • shortest path in a simple (non-convex) polygon with the funnel algorithm.
  • point robot among polygonal obstacles (trapezoid decomposition, visibility graph)
  • polygonal robot moving among polygonal obstacles with translation and rotation
  • C-space and C-obstacles; planning via graph search
• Heuristical motion planning (grid-based techniques, sampling, RRT)

Prerequisites: Data Structures (cs2100), Algorithms (cs2200) and Systems (2310). In other words, knowledge of:

• basic analysis: asymptotic notation, growth, solving recurrences.
• basic algorithm design techniques: divide-and-conquer, greedy.
• basic algorithms and data structures: searching, sorting, binary search trees, priority queues.
• basic programming in C/C++
• basic Unix, working from a terminal, compiling using a Makefile

CS curriculum: This is a 3000-level class that fullfills the Algorithms/Theory requirement, and also Projects requirement. Although it has lots of programming, it is not a projects class.

Discussion forum We will use Slack to facilitate communication and discussion outside of class. Please try to use this forum and avoid sending email ---- your classmates will benefit from seeing everyone else's questions and answers. You are encouraged to follow the posts on Slack and jump into the discussion. If you want to send me a private message, you can direct message (DM) me on Slack. Another nicer feature of Slack is that you'll be able to create your own channels for group work.

TAs:

• n/a (no alums in the pipeline)

Office hours:

• M, W: 3-4pm
• F: 12-2pm

Class webpage: http://tildesites.bowdoin.edu/~ltoma/teaching/cs3250-CompGeom/fall21/. This is hosted on my personal website at Bowdoin and will contain all class-related materials throughout the semester. You can bookmark this page or you can follow the link from my website. We will use the class Blackboard site very minimally.

Textbooks: There is no required textbook. The course is based on two classical textbooks, below. You can find them in Searles 224 (you're more than welcome to read them, but please do not take them out of 224).

• Computational geometry in C. J. O'Rourke.
• Computational geometry: algorithms and applications. Mark de Berg, Otfried Cheong, Mark van Kreveld, Mark Overmars.

Work and Grading: The work for the class will come from programming projects (approximately biweekly), with class engagement/participation serving as a tie breaker. The assignments are weighed equally, so each one is worth approx. 14.3% of the final grade.

Collaboration policy: You can work alone, or you can pair with a partner (pair-programming).

• Collaboration across teams is at level-1: that is, verbal collaboration without solution sharing. You are allowed and encouraged to discuss ideas with other class members, but the communication should be verbal and additionally it can include diagrams on board. Noone is allowed to take notes during the discussion (being able to recreate the solution later frommemory is proof that you actually understood it). Communication cannot include sharing pseudocode for the problem. Please read the department's collaboration policy.

• Pair programming policy: If you chose to work with a partner, you must work together, physically in the same place, for the entire project that you do together. The overall project must be a true joint effort, equally owned, created and understood by both partners. Specifically splitting the problem into parts and working on them separately is not allowed and violates the honor code for the class. If you start together as a team and are unable to complete the project together, then you will each inherit teh shared code, and split up to work individually on the remainder of the project. You would then submit individually, with a note on what happened.

  There are many advantages to working with a partner (e.g. more fun, more learning, good skill for the "real world"), and I encourage you to try. You may also consider alternating between working alone and with a partner, and changing partners.

  Remember that you are responsible for reading, understanding, and adhering to the policy. If you have any questions about any aspects of the policy, please do not hesitate to ask for clarification.