## Sunday, March 27, 2011

### Polygon Offset Revisited

I had previously written a post concerning polygon offset that was incomplete in its scope.  Specifically, it dodged the problem of horizontal and vertical lines in polygons.

Since then I've had to confront this problem again.  Shown below is the code I'm using for the more complete solution.  It's a bit verbose, but it works.

import math

def calcoffsetpoint(pt1, pt2, offset):
"""
Get a point offset from the line
segment pt1-pt2 distance "offset".

Return two tuple of coordinates.
"""
theta = math.atan2(pt2 - pt1,
pt2 - pt1)
theta += math.pi/2.0
return (pt1 - math.cos(theta) * offset,
pt1 - math.sin(theta) * offset)

def getoffsetintercept(pt1, pt2, m, offset):
"""
From points pt1 and pt2 defining a line
in the Cartesian plane, the slope of the
line m, and an offset distance,
calculates the y intercept of
the new line offset from the original.
"""
x, y = calcoffsetpoint(pt1, pt2, offset)
return y - m * x

def getpt(pt1, pt2, pt3, offset):
"""
Gets intersection point of the two
lines defined by pt1, pt2, and pt3;
offset is the distance to offset
the point from the polygon.

Valid for lines with slopes other
than zero or infinity.

Returns a two tuple of coordinates.
"""
# get first offset intercept
m = (pt2 - pt1)/(pt2 - pt1)
boffset = getoffsetintercept(pt1, pt2, m, offset)
# get second offset intercept
mprime = (pt3 - pt2)/(pt3 - pt2)
boffsetprime = getoffsetintercept(pt2, pt3, mprime, offset)
# get intersection of two offset lines
newx = (boffsetprime - boffset)/(m - mprime)
newy = m * newx + boffset
return newx, newy

def getslopeandintercept(pt1, pt2, offset):
"""
Gets the slope and the intercept of the
offset line.
Result returned as a two tuple.
"""
m = (pt2 - pt1)/(pt2 - pt1)
b = getoffsetintercept(pt1, pt2, m, offset)
return m, b

def getoffsetcornerpoint(pt1, pt2, pt3, offset):
"""
Gets intersection point of the two
lines defined by pt1, pt2, and pt3;
offset is the distance to offset
the point from the polygon.

Returns a two tuple of coordinates.
"""
# starting out with horizontal line
if (pt2 - pt1) == 0.0:
ycoord = pt1 - math.cos(math.atan2(0.0, pt2 - pt1)) * offset
# a vertical line follows
if (pt3 - pt2) == 0.0:
xcoord = pt2 + math.sin(math.atan2(pt3 - pt2, 0.0)) * offset
# a sloped line follows
else:
m, offsetintercept = getslopeandintercept(pt2, pt3, offset)
# calculate for x with ycoord
xcoord = (ycoord - offsetintercept)/m
# starting out with a vertical line
if (pt2 - pt1) == 0.0:
xcoord = pt1 + math.sin(math.atan2(pt2 - pt1, 0.0)) * offset
# a horizontal line follows
if (pt3 - pt2) == 0.0:
ycoord = pt2 - math.cos(math.atan2(0.0, pt3 - pt2)) * offset
# a sloped line follows
else:
m, offsetintercept = getslopeandintercept(pt2, pt3, offset)
# calculate for y with xcoord
ycoord = m * xcoord + offsetintercept
# starting out with sloped line
if (pt2 - pt1) != 0.0 and (pt2 - pt1) != 0.0:
# if second line is horizontal
if (pt3 - pt2) == 0.0:
ycoord = pt2 - math.cos(math.atan2(0.0, pt3 - pt2)) * offset
m, offsetintercept = getslopeandintercept(pt1, pt2, offset)
# calculate for x with y coord
xcoord = (ycoord - offsetintercept)/m
# if second line is vertical
elif (pt3 - pt2) == 0.0:
xcoord = pt2 + math.sin(math.atan2(pt3 - pt2, 0.0)) * offset
m, offsetintercept = getslopeandintercept(pt1, pt2, offset)
# solve for y with x coordinate
ycoord = m * xcoord + offsetintercept
# if both lines are sloped
else:
xcoord, ycoord = getpt(pt1, pt2, pt3, offset)
return xcoord, ycoord

def offsetpolygon(polyx, offset):
"""
Offsets a clockwise list of coordinates
polyx distance offset to the inside of
the polygon.
Returns list of offset points.
"""
polyy = []
# need three points at a time
for counter in range(0, len(polyx) - 3):
# get first offset intercept
pt = getoffsetcornerpoint(polyx[counter],
polyx[counter + 1],
polyx[counter + 2],
offset)
# append new point to polyy
polyy.append(pt)
# last three points
pt = getoffsetcornerpoint(polyx[-3], polyx[-2], polyx[-1], offset)
polyy.append(pt)
pt = getoffsetcornerpoint(polyx[-2], polyx[-1], polyx, offset)
polyy.append(pt)
pt = getoffsetcornerpoint(polyx[-1], polyx, polyx, offset)
polyy.append(pt)
return polyy

I used this polygon to test it.

This is a fairly simple geometric problem, but it can be confusing.  Hopefully this will save someone the trouble of working through a solution.

Notes:

1) If you're using Python 2.x, feed your coordinates in as floats.  This will ensure the absence of integer division, which often yields zeros, which in turn yields ZeroDivision errors.

2) The polygon fed to the offsetpolygon function should not have a closing point.  The closing point will cause ZeroDivision errors.  Instead, append the point before drawing the polygon in gnuplot or some other plotting tool.

## Thursday, March 24, 2011

### jython + joda-time

joda-time is a date-time library written in Java.  It has advantages over Java's built in time utilities in terms of power and ease of use.  It has some functionality that Python's own datetime library does not.  Here is a sampling from within jython:

[OpenJDK Client VM (Sun Microsystems Inc.)] on java1.7.0-internal
>>> import sys

>>> # make sure the jar is in the CLASSPATH
>>> from org.joda import time as joda
>>> rightnow = joda.DateTime()
>>> rightnow
2011-03-24T19:54:45.544-06:00
>>> rightnow.getZone()
America/Edmonton

>>> # I'm actually in New Mexico - close enough
>>> rightnow.dayOfWeek().getAsText()
u'Thursday'
>>> rightnow.monthOfYear().getAsText()
u'March'
>>> rightnow.getDayOfMonth()
24

>>> # days are one indexed along with months

>>> # a date from the past
>>> declofindependence = joda.DateTime(1776, 7, 4, 0, 0, 0, 0)

The Period object allows you to count down to dates in the future of count from dates in the past in terms of years, months, and days:

>>> americasage = joda.Period(declofindependence, rightnow)
>>> americasage.getYears()
234
>>> americasage.getMonths()
8
>>> americasage.getWeeks()
2
>>> americasage.getDays()
6

joda-time can handle dates far into the future as well as those in the far distant past:

>>> farfuture = joda.DateTime(10000, 1, 1, 1, 0, 0, 0)
>>> farfuture.getYear()
10000
>>> farfuture
10000-01-01T01:00:00.000-07:00
>>>

Like most things Java, joda-time can be a bit more verbose than its Python equivalent.  Still, the ability to get dates beyond 9999 and some of the functionality may make it well worth the trouble.

## Saturday, March 19, 2011

### WKT (Well Known Text) + JTS Topology + jython

The last few posts have dealt with the JTS Topology library.  This one will wrap up that topic with a method for converting JTS's Geometry objects to and from text, WKT.

WKT is a standard of the Open Geospatial Consortium (OGC).  Objects expressed in WKT are not much different than their string representations in JTS.  Below is a code example of working back and forth between JTS and WKT.

Jython 2.5.2 (Release_2_5_2:7206, Mar 2 2011, 23:12:06)
[OpenJDK Client VM (Sun Microsystems Inc.)] on java1.7.0-internal
>>> import sys
>>> from com.vividsolutions.jts.geom import GeometryFactory
>>> from com.vividsolutions.jts.geom import Coordinate
>>> from com.vividsolutions.jts.geom import LinearRing
>>> from com.vividsolutions.jts.io import WKTWriter
>>> from com.vividsolutions.jts.io import WKTReader
>>> coord1 = Coordinate(0, 0)
>>> coord2 = Coordinate(0, 1)
>>> coord3 = Coordinate(1, 1)
>>> coord4 = Coordinate(1, 0)
>>> coord5 = Coordinate(0, 0)
>>> coords = [coord1, coord2, coord3, coord4, coord5]
>>> from com.vividsolutions.jts.geom.impl import CoordinateArraySequence
>>> cas = CoordinateArraySequence(coords)
>>> gf = GeometryFactory()
>>> lr = LinearRing(cas, gf)
>>> asquare = gf.createPolygon(lr, None)
>>> f = open('wkttest.xml', 'w')
>>> wktwriter = WKTWriter()

>>> # this is just my own personal preference
>>> # it is inefficient for storage, but
>>> # sometimes easier on the eyes
>>> wktwriter.setMaxCoordinatesPerLine(1)
>>> text = wktwriter.writeFormatted(asquare)
>>> f.write(text)
>>> f.close()
>>> print text
POLYGON ((0 0,
0 1,
1 1,
1 0,
0 0))
>>> f = open('wkttest.xml', 'r')
>>> text = f.read()
>>> asquare2
POLYGON ((0 0, 0 1, 1 1, 1 0, 0 0))

It's fairly easy to store JTS Geometry objects in human readable (WKT) format and reuse them.

There is a good bit more to WKT than what's shown here.  This was just a simple example for getting started.

WKB (Well Known Binary) is also available in JTS - this is used for efficient storage of Geometry objects, mainly in databases.

JTS also has a WKTFileReader class which can read in multiple geometries from a file.

## Friday, March 18, 2011

### Polygon Buffering with JTS Topology and jython

Previously, I had done a post regarding polygon offset.  It turns out that JTS Topology has polygon buffering capability.  I wanted to try this out and compare it to my results from the polygon offset post.

As with most things with a library like JTS, the software does most of the work for you:

import sys
# path to jar into CLASSPATH

from com.vividsolutions.jts.geom import Coordinate
from com.vividsolutions.jts.geom import GeometryFactory
# LinearRing is for the creation of the Polygon
from com.vividsolutions.jts.geom import LinearRing
# CoordinateSequenceArray is for the

# creation of the LinearRing
from com.vividsolutions.jts.geom.impl import CoordinateArraySequence
# CAP_ROUND is for the Polygon.buffer operation
from com.vividsolutions.jts.operation.buffer.BufferOp import CAP_ROUND

# coordinates
# PT 1
MONASTERY = [(1.1, 0.75),
# PT 2
(1.2, 1.95),
# PT 3
(1.9, 1.96),
.

.
.
# PT 21
(1.1, 0.75)]

gf = GeometryFactory()
coords = [Coordinate(*coord) for coord in MONASTERY]
cas = CoordinateArraySequence(coords)
lr = LinearRing(cas, gf)
polyx = gf.createPolygon(lr, None)

# offset of 0.15 inward
# rounded edges
# 10 points per quarter circle of rounded edges
polyinset = polyx.buffer(-0.15, 10, CAP_ROUND)

The result is shown below:

The rounded edges give an entirely different look to the shape. My version is shown below:

I'm glad I did the egg the way I did it (it looks more realistic).  Nonetheless, the polygon buffering capability of JTS is a useful tool, particularly for calculating offsets and distances for scientific or geographic purposes.

Notes:  Shapely is a Python package (CPython) with much of the same functionality as JTS.

## Thursday, March 17, 2011

### Clipping Voronoi Polygons with JTS Topology and jython

Following up on the last post on Voronoi diagrams and the JTS topology library, I will show clipping of the Voronoi polygons with another polygon (a polygonal bounding region) in this post.

The plot below is what we have currently, a Voronoi diagram (with points) clipped along a rectangular boundary:

Shown below is what I'd like to accomplish - clipping the red polygons with the blue surrounding one:

Here is the code, continued from the last post:

# because JTS can handle holes and "donuts", it
# has a LinearRing structure for constructing
# polygons
from com.vividsolutions.jts.geom import LinearRing
# the LinearRing requires a CoordinateArraySequence
# (it will not accept a list)
from com.vividsolutions.jts.geom.impl import CoordinateArraySequence

boundingcoords = [(10, 0), (2, 0),
(-2, 4), (2, 12),
(10, 6), (10, 0)]
boundingcoords = [Coordinate(*coord) for coord in boundingcoords]
casboundingcoords = CoordinateArraySequence(boundingcoords)
lr = LinearRing(casboundingcoords, geomfactx)

# None here fills in the spot where any inner rings
# would be in the polygon
boundingpoly = geomfactx.createPolygon(lr, None)

newpolygons = []
numpolygons = diagramx.getNumGeometries()
for numx in range(numpolygons):

newpolygons.append(diagramx.getGeometryN(
numx).intersection(boundingpoly))

And the result:

Note:  it is possible to get more than one polygon as the result of each intersection.  Through visual inspection I determined there would only be one polygon per intersection with the bounding area and structured my list (newpolygons) accordingly.

## Tuesday, March 15, 2011

### jython, JTS Topology, and Voronoi Diagrams

Last time I barely scratched the surface of the JTS library.  As it turns out, through ignorance I made a few things harder than they needed to be.  The GeometryFactory class is quite versatile and has a number of methods for creating objects from simple Coordinate objects.  In the example below, I've managed to create a clipped Voronoi diagram without too much code or difficulty.

import sys
# this line will be different depending on where
# the jts jar is stored and may be unnecessary if
# the jar is in your CLASSPATH

from com.vividsolutions.jts.geom import Coordinate
from com.vividsolutions.jts.geom import GeometryFactory
from com.vividsolutions.jts.triangulate import VoronoiDiagramBuilder
from com.vividsolutions.jts.geom import Envelope

# first some coordinates
coord1 = Coordinate(5, 7)
coord2 = Coordinate(3, 6)
coord3 = Coordinate(1, 4)
coord4 = Coordinate(2, 5)
coord5 = Coordinate(4, 2)
coord6 = Coordinate(7, 2)
coord7 = Coordinate(5, 5)
coord8 = Coordinate(2, 6)
coord5 = Coordinate(4, 2)
coords = [coord1, coord2,
coord3, coord4,
coord5, coord6,
coord7, coord8]

# the handy GeometryFactory
geomfactx = GeometryFactory()

# a MultiPoint Geometry object
mp = geomfactx.createMultiPoint(coords)

# the diagram builder class
vdb = VoronoiDiagramBuilder()

# it would be nice to clip the
# diagram with a rectangle;
# this is what the Envelope
# object is for
lowerleft = Coordinate(0, 0)
upperright = Coordinate(8, 8)
env = Envelope(lowerleft, upperright)
vdb.setClipEnvelope(env)

# load our sites (points) from the
# MultiPoint object
vdb.setSites(mp)

# this is where the builder object clips the
# the polygons with the Envelope rectangle
diagramx = vdb.getDiagram(geomfactx)

print 'Number of polygons = %d' % diagramx.getNumGeometries()
polygonx = diagramx.getGeometryN(0)
print 'One polygon:'
for coordsetx in polygonx.getCoordinates():
print coordsetx

Output:

Number of polygons = 8
One polygon:
(-5.0, -4.0, NaN)
(-5.0, 8.25, NaN)
(0.5, 5.5, NaN)
(2.7, 3.3, NaN)
(-2.166666666666666, -4.0, NaN)
(-5.0, -4.0, NaN)

There's more that can be done here (for instance, clipping the polygons with a non-rectangular shape).  As was true of the last post, this is just a small fraction of what the JTS Topology library can do.

## Sunday, March 13, 2011

### jython + the JTS geometry library

My favorite library for dealing with two dimensional points, lines, and polygons is Polygon by Jörg Rädler.  In a Java/jython environment, though, that's not available.  This led me to try my hand at JTS.

Constructs a GeometryFactory that generates Geometries having a floating PrecisionModel and a spatial-reference ID of 0."  For what I'm doing this is fine.

2) The preferred constructor for the Point object takes a CoordinateArraySequence of length one instead of a single Coordinate object.  This is analogous to a character in Python being a string of length one.

We don't know yet what our values are for our Point's coordinates.  Let's have a look.

>>> ptx
POINT (0 0)

OK, zero-zero.  Let's assign something to those coordinates away from the origin.

>>> ptx.getCoordinates().x = 44
>>> ptx.getCoordinates().y = 22
>>> ptx
POINT (44 22)
Even though the Point has only one coordinate, we still need to specify that with the index 0.

Well, that was a lot of work for just one point.  Let's try something more efficient and fun.

>>> cas2 = CoordinateArraySequence(6)
>>> somecoordinates = [(1, 5), (7, 14), (22, 44), (36, 12), (19, 1), (12, 4)]
>>> for numx in range(len(somecoordinates)):
...     cas2.getCoordinate(numx).x = somecoordinates[numx]
...     cas2.getCoordinate(numx).y = somecoordinates[numx]
...
>>> cas2
((1.0, 5.0, NaN), (7.0, 14.0, NaN), (22.0, 44.0, NaN), (36.0, 12.0, NaN), (19.0, 1.0, NaN), (12.0, 4.0, NaN))
>>> from com.vividsolutions.jts.geom import LineString
>>> ls = LineString(cas2, factoryx)
>>> ch = ls.convexHull()
>>> ch
POLYGON ((19 1, 1 5, 22 44, 36 12, 19 1))
>>>

LineString is one of the classes that can be instantiated from a CoordinateArraySequence.  I went with that mainly for ease of use.

There is a great deal more to JTS.  This post was mainly for getting started with the library.

## Wednesday, March 2, 2011

### jython + Google Guava's base.CaseFormat

I was trying to research Google Collections for use in jython and stumbled upon Solid Craft's blog and his description of Guava base's CaseFormat utility.  Where I work we're a Java shop, and use of camelCase infiltrates all code, even jython code.  CaseFormat provides functionality for changing these variable names to a number of different formats.

\$ /usr/local/jdk-1.7.0/bin/java -jar jython.jar
Jython 2.5.1 (Release_2_5_1:6813, Sep 26 2009, 13:47:54)
[OpenJDK Client VM (Sun Microsystems Inc.)] on java1.7.0-internal
>>> import sys
>>> # get downloaded jar into sys.path
>>> from com.google.common.base import CaseFormat as cf
>>> dir(cf)
['LOWER_CAMEL', 'LOWER_HYPHEN', 'LOWER_UNDERSCORE', 'UPPER_CAMEL', . . .]
>>> varname = 'someVariable'
>>> cf.LOWER_CAMEL.to(cf.LOWER_UNDERSCORE, varname)
u'some_variable'
>>> cf.LOWER_CAMEL.to(cf.UPPER_UNDERSCORE, varname)
u'SOME_VARIABLE'
>>> cf.LOWER_CAMEL.to(cf.UPPER_CAMEL, varname)
u'SomeVariable'
>>>

For cycling through a file full of camel case or underscored variable names, even by hand in the interpreter, this could save some time and typing.