This document summarizes research applying curvature analysis to model crime patterns during protests in Baltimore in 2015. The researchers analyzed public transit data before and during the protests, imported location data into Mathematica, and calculated curvature metrics to determine if crime corresponded to areas of positive or negative curvature. Their results varied depending on the curvature algorithm and grid size used. The researchers concluded further analysis was needed but curvature analysis showed promise for understanding how infrastructure changes like shutting down transit affected criminal activity.

1. Application of Curvature Analysis as a
Methodology of Modeling Crime in
Baltimore
Abdallah Malik Naanaa, Robert Chen, Mark Branson Ph.D

2. April 27th, 2015
• On April 12th,
Freddie Gray was
arrested by police
and later died
from a spinal cord
injury.
• Protesting began
in wake of Freddie
Gray’s death by
officers.
• Began roughly
after 3 pm.
• Approximately
252 crimes were
reported to
police.
http://www.businessinsider.com/photos-of-riots-and-protests-in-baltimore-after-freddie-gray-death-2015-4

5. Why should we care?
• Public transportation is the primary manner
people move throughout the city of
Baltimore.
• There aren’t school buses in Baltimore City,
so kids have to ride public transportation.
• Rioters tended to be described as “teens”
or “kids”.
• Blockading roads and shutting down transit
systems essentially corralled younger
individuals into the Mondawmin area.
• A “purge” was planned by gangs to take out
officers, however most kids didn’t want to
take part.
• This impacted the city tremendously.
Billboard.com

6. Timeline of the Project
• Locating transit stop closures around the
Mondawmin neighborhood of Baltimore.
• Building graphs displaying transit times before
and during the riots.
• Generating plots of {latitude, longitude, time}
from this data and importing them into
Mathematica.
• Finding nearest neighbors of each data point and
sorting the group of points in a clockwise manner.
• Manipulating the data to achieve minimal slopes.
• Performing curvature calculations of the grouping
of points.
• Determining the curvature of the closest point
that correlates to each crime location.
It’s hypothesized that areas of positive curvature will
correspond to where the majority of crime occurred.

7. Deleting stops
• Routes and their
given diversions were
determined.
• Looking at
Mondawmin and the
bus routes, the stops
that weren’t serviced
were found.
• Each stop that was
shut down on the 27th
was then deleted
from our system.

9. Importing the Data
• Once graphs are built and
no more stops around
Mondawmin were present,
data was exported into a
“.csv” Excel file.
• Sets of 50x50 and 100x100
grids of data points were
used
• These sets of data can now
be imported into
Mathematica for analysis.
• The “Years” metric will be
discussed later on.

10. Nearest Neighbors
Four Neighbors
Utilization of Manhattan
Distance to find neighbors
Eight Neighbors
Utilization of Chessboard
Distance to find neighbors

11. https://en.wikipedia.org/wiki/Chebyshev_distance

12. Slope Data
• “Curvature analysis as a tool for subsidence-related risk zones identification in the
city of Tuzla (BiH)” revealed a slope problem with certain data sets.
• If slopes of data are 30° or more, 50% error will occur within the data.
• Numeric values must be minimized in order to achieve low slopes
• This is why time metric is in years instead of seconds.

13. Introduction into Curvature
• A measurement of
how a geometric
object deviates from
being flat or straight.
• Smooth curves exist
when there are no
sharp corners and the
change in arc length of
the object is not zero.
• The sign of the
curvature influences
angle measurements
of polygons on the
surface.

14. Angle
sums
greater
than 180°
Angle
sums
equal
180°
Angle
sums
fewer
than 180°

15. Gauss-Bonnet Scheme
• A method to formalize
angle measurements
from a triangle on a 3D
mesh.
• K is constant assumed
throughout the local
neighborhood.
• A is the accumulated
areas of the triangles.
• Gamma represents
exterior angles.

17. Taubin Algorithms
• P’s denote two principal
directions on a surface
• Mv denotes a symmetric matrix
that can be factorized into a
simpler form
• Principal curvatures can be
determined by the eigenvalues of
Mv
• Taubin I averages values based
upon the areas of triangles found
within the object.
• Taubin II averages the internal
angles of triangles found within
the object.

19. Data Visualization
• Once curvatures are
found, data can be
visualized by using the
“ListPointPlot3D”
command.
• Lists of all positive and all
negative curvature
values are created
separately.
• They are overlaid on a 3D
plot and visualized with a
specific color scheme.

20. Importing Crime Data
• Crime data was found in
the beginning of this
project.
• Each crime had it’s
latitude and longitude
imported into
Mathematica.
• A third coordinate of 0 is
needed for time in order
for determining nearest
points.

21. Results (Pos/Neg)
Paraboloid Gauss-Bonnet Watanabe-Belyaev I Watanabe-Belyaev II Taubin I Taubin II
Grid/Neighbors
50x50/4 155/97 155/97 131/115 155/97 n/a 252/0
50x50/8 131/121 169/83 193/59 76/176 149/103 146/106
100x100/4 81/156 60/101 167/85 112/140 n/a 252/0
100x100/8 174/78 95/157 118/134 123/129 117/135 130/122
Paraboloid Gauss-Bonnet Watanabe-Belyaev I Watanabe-Belyaev II Taubin I Taubin II
Grid/Neighbors
50x50/4 127/125 127/125 140/112 127/125 n/a 252/0
50x50/8 150/102 102/150 153/99 109/143 106/146 74/178
100x100/4 115/126 n/a 100/152 131/110 n/a 252/0
100x100/8 171/81 n/a 91/161 112/140 105/147 152/100
Paraboloid Gauss-Bonnet Watanabe-Belyaev I Watanabe-Belyaev II Taubin I Taubin II
Grid/Neighbors
50x50/4 146/106 153/99 140/104 153/99 n/a 150/102
50x50/8 127/125 139/113 176/76 92/160 110/142 131/121
100x100/4 145/107 n/a 101/151 142/110 n/a 229/23
100x100/8 141/111 n/a 121/131 98/154 101/151 152/100
Normal
Curvature
Protest
Curvature
Differences
Curvature

22. Conclusions
• Shutting down public transportation within a major city can have
detrimental effects, especially around areas of significant need.
• Crimes seemed to happen in more positively curved areas, the statistical
significance of this needs to be explored.
• Gauss-Bonnet might prove of use once the data is examined more
thoroughly.
• Taubin II is one of the best algorithms used in this study and needs to be
improved upon for future usage.
• Utilizing 100x100 grids show similar results to that of 50x50 grids.
Future Directions:
• Optimizing each curvature algorithm that gave results “n/a” and
rerunning data
• Formatting neighbors to include 6 point neighbors
• Taking into account multiple days of the occurrence
• Looking at the average curvature change for each method

23. Limitations
• More crimes occurred and weren’t reported to
authorities at the time.
• This was a multiple day event, not just April 27th.
• We believe that more transit stops were closed during
the day, but we have limited cooperation from the MTA.
• The computers took some time to process information
with Mathematica when it came to the 100x100 grids.
• The paper that specifically referenced the algorithms
said that they would work for any three-dimensional
mesh, but we see otherwise.

24. References
• http://www.motherjones.com/politics/2015/04/how-
baltimore-riots-began-mondawmin-purge
• http://mta.maryland.gov/local-bus
• http://www.businessinsider.com/photos-of-riots-and-protests-
in-baltimore-after-freddie-gray-death-2015-4
• https://www.google.com/fusiontables/DataSource?docid=1nh
w4JURMKNrwyp6NG0tWZv6iGSJuAKCx1fNkO6jq#rows:id=1
• “Curvature analysis as a tool for subsidence-related risk zones
identification in the city of Tuzla (BiH)”
• “A Comparison of Gaussian and Mean Curvature Estimation
Methods on Triangular Meshes of Range Image Data”
• http://www.wolfram.com/mathematica/

25. Paraboloid Gauss-Bonnet Watanabe-Belyaev I Watanabe-Belyaev II Taubin I Taubin II
Grid/Neighbors
50x50/4 0.6151 0.6151 0.5198 0.6151 - 1.0000
50x50/8 0.5198 0.6706 0.7659 0.3016 0.5913 0.5794
100x100/4 0.3214 0.3492 0.6627 0.4444 - 1.0000
100x100/8 0.6905 0.3770 0.4683 0.4881 0.4643 0.5159
Average 0.5367 0.5030 0.6042 0.4623 0.5278 0.7738
Standard Deviation 0.1382 0.1416 0.1174 0.1120 0.0635 0.2273
Paraboloid Gauss-Bonnet Watanabe-Belyaev I Watanabe-Belyaev II Taubin I Taubin II
Grid/Neighbors
50x50/4 0.5040 0.5040 0.5556 0.5040 - 1.0000
50x50/8 0.5952 0.4048 0.6071 0.4325 0.4206 0.2937
100x100/4 0.4563 0.3492 0.3968 0.5198 - 1.0000
100x100/8 0.6786 0.3690 0.3611 0.4444 0.4167 0.6032
Average 0.5585 0.4067 0.4802 0.4752 0.4187 0.7242
Standard Deviation 0.0854 0.0596 0.1036 0.0374 0.0020 0.2967
Paraboloid Gauss-Bonnet Watanabe-Belyaev I Watanabe-Belyaev II Taubin I Taubin II
Grid/Neighbors
50x50/4 0.5794 0.6071 0.5556 0.6071 - 0.5952
50x50/8 0.5040 1.2301 0.6984 0.3651 0.4365 0.5198
100x100/4 0.5754 0.5635 0.4008 0.5635 - 0.9087
100x100/8 0.5595 0.5397 0.9237 0.3889 0.4008 0.6032
Average 0.5546 0.7351 0.6446 0.4812 0.4187 0.6567
Standard Deviation 0.0301 0.2868 0.1924 0.1056 0.0179 0.1491