Write a program to recognize line patterns in a given set of points.

Computer vision involves analyzing patterns in visual images and
reconstructing the real world objects that produced them. The process
in often broken up into two phases: *feature detection* and
*pattern recognition*. Feature detection involves selecting
important features of the image; pattern recognition involves
discovering patterns in the features. We will investigate a
particularly clean pattern recognition problem involving points and
line segments. This kind of pattern recognition arises in many other
applications, for example statistical data analysis.

**The problem.**
Given a set of *N* points in the plane,
draw every line segment that connects 4 or more distinct points in the set.

**Brute force.**
Write a program `Brute.java` that examines 4
points at a time and checks if
they all lie on the same line segment, printing out any such line
segments to standard output and plotting them
using `StdDraw`.
To get started, you may use the data type
Point.java
and the client program
PointPlotter.java
which reads in a list of points from standard input
and plots them.
You will need to supply additional methods in `Point.java`
in order to support the brute force client, e.g., checking whether three
or four points lie on the same line.

**A sorting solution.**
Remarkably, it is possible to solve the problem much faster than the
brute force solution described above.
Given a point `p`, the following method determines whether `p`
participates in a set of 4 or more collinear points.

- Think of
`p`as the origin. - For each other point
`q`, determine the angle it makes with`p`. - Sort the points according to the angle each makes with
`p`. - Check if any 3 (or more) adjacent points in the sorted order have equal
angles with
`p`. If so, these points, together with`p`, are collinear.

Write a program `Fast.java` that implements this algorithm
using `Arrays.sort()` and a user-defined `Comparator`
for `Point` objects.

**Input format.**
The data file consists of an integer *N*, followed by *N*
pairs of integers (*x*, *y*) between 0 and 32,767.

%more input6.txt%more input8.txt6 8 19000 10000 10000 0 18000 10000 0 10000 32000 10000 3000 7000 21000 10000 7000 3000 1234 5678 20000 21000 14000 10000 3000 4000 14000 15000 6000 7000

**Output format.**
Print to standard output the line segments that your program discovers
in the format below (number of collinear points in the line segment, followed by the points).

Also, plot the points and the line segments using standard draw. Using the%java Brute < input8.txt4: (10000, 0) -> (7000, 3000) -> (3000, 7000) -> (0, 10000) 4: (3000, 4000) -> (6000, 7000) -> (14000, 15000) -> (20000, 21000) %java Fast < input6.txt5: (14000, 10000) -> (18000, 10000) -> (19000, 10000) -> (21000, 10000) -> (32000, 10000)

For full credit, `Fast.java` must print and plot a
*minimal representation*: that is, only print one representation of each
line segment and don't print subsegments.
It's ok if `Brute.java` does not produce a minimal representation.

**Analysis.**
Estimate (using tilde notation) the running time (in seconds) of your
two programs as a function of the number of points *N*.
Provide empirical and mathematical evidence to justify your hypotheses.

**Deliverables.**
Submit the files: `Brute.java`,
`Fast.java`, `Point.java`. Also submit any
other auxiliary files, if any, that your program needs
(excluding our standard libraries).
Finally, submit a
readme.txt file and answer the questions.

Copyright © 2005.