Math IA - Forensics - Mathematics internal assessment on forensic investigations PDF

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Mathematics internal assessment on forensic investigations...

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Mathematics SL Internal Assessment How to Solve a Crime Using Mathematics and Blood Splatters An Investigation into the Ellipses Formed by Blood Splatters in a Crime Scene: Using Trigonometry to Find a Killer Pin Code: fqr810

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Table of Contents: Introduction……………………………………………………………………… 3 Calculating the area of an ellipse (Geometry / Algebra)..………………………. 3 – 5 Calculating correlation between area of ellipse and angle of creation (Statistics) ………..……………………………………………………………… 5 – 6 Calculating the origin of the bullet (Trigonometry)……………………………. 6 – 9 Conclusion………………………………………………………………………. 9 – 10 Works Cited…………………………………………………………………….. 11

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Introduction: Due to modern day advancements in the collection and analysis of forensic evidence, forensic experts are able to use mathematics in order to further investigate a crime scene. As someone who has always been interested in the way evidence is collected that can help solve a crime, I decided to investigate how forensic experts use blood splatters in order to determine the angle from which someone was shot. Without sufficient evidence, it is often hard to convict a suspect, thus making the role of these forensic experts highly important in our growing society. Through examining the area of ellipses and how they are affected by the angle from which they are created, one could better understand what occurred in a crime scene. Additionally, understanding the trigonometry used by bloodstain analysts will allow me to further understand how mathematics is used in a crime scene investigation. For example, forensic experts often use bloodstains to calculate the area of convergence to determine the height at which the crime occurred, and often measure individual bloodstains in order to determine the angle where the suspect would have fired. The area of convergence allows for a predicted, two-dimensional visualization of the origin of the bloodshed, which must first be found before analysts can determine the position of the suspect. The analysts then take piece of string and using a protractor map where that angle leads to in a room and therefore the area of origin becomes clear to investigators (Bloodstain Pattern Analysis). Through my investigation, I decided to determine whether or not the area of the ellipse and the angle that the droplet was dropped from, had any correlation. If there was a correlation, I could then examine what the pattern was and begin to develop a way for forensic experts to examine blood splatters in order to determine the position of a shooter. I also decided to use the trigonometric formulas that bloodstain analysts used in order to comprehend more of the mathematics used in investigations. To begin this exploration of the mathematics used in crime scenes, I decided to take drops of liquid and angle a piece of paper at a few key angles, and measure the ellipse formed by the droplet Doing this, will help me to develop a clear understanding of how blood splatters are used in a crime scene investigation in order to determine the positioning of both a shooter and the victim. The information gathered can therefore be used by forensic experts when investigating crimes, as a way to calculate the correlation between the position of a shooter and the area of an ellipse. This investigation was inspired by my interest in true crime stories and the way in which crimes that occurred in the past can now be solved in our present day due to the advancements in technology. A specific case triggered my interest in this topic and that was the murder of JonBenét Ramsey, a case that began in 1996 and still has yet to be solved. Therefore, I wanted to explore the way in which crimes can be solved without the use of extensive technology and the way in which simple math could allow for a clearer understanding of a crime scene. Hence I chose to conduct the experiment using simple equipment and little technology in order to investigate how blood splatters can be used in an examination of a crime scene.

Finding the Area of an Ellipse – The Experiment: To begin collecting data, I took a piece of paper and a protractor and used a dropper placed 20cm away to drop droplets of liquid onto a paper angled at different angles. I chose to angle the paper at 0°, 15°, 30°, 45°, 60°, 75°, and 90°. 3

Figure 1: An example set up, where the paper is at 0° and the dropper

is placed 20cm away from the paper.

Figure 2: Table of values showing the lengths of the major and minor axis and the area of the ellipse Angle Major Axis (cm) Minor Axis (cm) Area (cm2) 0° 1.5 1.5 1.77 15° 1.9 1.3 1.94 30° 2.2 1.2 2.07 45° 2.6 1.1 2.25 60° 3.3 0.7 1.81 75° 3.5 0.4 1.10 90° 4.0 0.2 0.63 Figure 3: Sketches of the different ellipses formed by the experiment, using the measurements above to sketch them. And the outline of the ellipse created by the dropper at a 45°

I chose to sketch the ellipses after having done the full experiment, in order to preserve the idea of the shapes created by the water droplets. Furthermore, it allows me to see that as the angle increases, the ellipse becomes more and more stretched horizontally.

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Figure 4: Diagram of the radiuses a and b on an ellipse

b

a

In order to calculate the area of the ellipse, I used the equation: A=πab where a=0.5(major axis) and b=0.5(minor axis). I did this, as I realized that a circle, was a specific type of ellipse where the formula for the area of it is A=πr2. In the case of an ellipse, however, both values of the radius are not equal as an ellipse is technically a circle that has been pulled outward on both sides and “squashed” down, and therefore we must use variables a and b to distinguish between the radius of the major axis and minor axis. For the purpose of this investigation I used a GDC when multiplying with π in order to get a rounded measurement that would allow me to better analyze my data and form a conclusion. I chose to round my answer to three significant figures, as it was accurate enough within the nature of this investigation. Below I have included an example of these calculations using the angle 30°: A=πab 2.2 1.2 a= and b= 2 2 2.2 1.2 × 2 ) 2 A=π ¿ A=π 0.66 A ≈ 2.07345 A ≈ 2.07 I then used the data gathered and graphed the areas, in order to better comprehend the relationship between these points and thus how the area of the ellipse is affected by the angle from which it was created. I had noticed that as the angle increased, the major axis increased and the minor axis decreased, however, the area first increased and then decreased. Therefore I needed to graph the points in order to better visualize the relationship.

Correlation Between Area of Ellipse and Angle of Creation: Figure 5: Graph showing the correlation between angle ellipse was formed and the area of the ellipse.

Area of an Ellipse When Formed at Different Angles 2.5

Area (cm2)

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f(x) = − 0 x² + 0.03 x + 1.71 R² = 0.96

1.5 1 0.5

Through using excel to plot the points, I was able to generate an equation that allowed me to understand the

0 0

10

20

30

40

50

Angle (°)

60

70

80

90 100

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pattern more clearly. I found that the equation was a quadratic and was: y=−0.0005 x 2 +0.0284 x +1.7129 which showed me that the coefficient before the x2 value was very small. This could be used to understand that the area of the ellipse increases up to 45° and then begins to decrease. Hence, through determining the area, and lengths of the major and minor axis the bloodstain analyst is able to discover the angle of impact. I also chose to use my GDC and find the correlation between the points, using the scatter graph above. I thus found the R-value of the data set to be 0.72 using the function “LinReg (ax + b)” on the GDC indicating that there is a strong negative correlation between the area of the ellipse and the angle of impact.

Calculating the Origin of the Bullet: Due to my research on the work of bloodstain analysis, I understood that a significant amount of trigonometry is used within the investigation using the angle of impact found by the bloodstain analyst. Trigonometry allows for an accurate prediction of the angle at which the suspect attacked from, thus allowing for a clearer prediction of what may have occurred during a crime. A formula and theory provided by Cornell University allows for a deeper understanding of how trigonometry is used in a crime scene investigation when looking at blood splatters. The following image, taken from Wikipedia, represents the way in which analysts use the elliptical shape of the blood droplets on a wall to determine the angle of origin, or the angle at which the suspect fired. Figure 6: Diagram depicting the three-dimensional plane in which the angles are measured on a wall The plumb line is used as a reference for blood splatter analysts and is a line drawn perpendicular to the ground. The key angles in an investigation are α, β, γ. Angle α represents the angle of impact, angle γ represents the angle created by a line through the major axis of the ellipse (bloodstain) and the plumb line. Lastly, angle β points investigators towards the position of the suspect at the time of the crime, or where the shot was fired. * The “Vx”, “Vy” and “Vz” values have to do with velocity and take into account the aspect of force involved in creating a blood splatter. This thus involves physics calculations to find the velocity of a shot fired, and hence will not be explored in this investigation. The angle of impact, or angle α, is similar to the experiment I conducted with the dropper where I created different ellipses from different angles. This means that as 6

the angle of impact increases, the area of the ellipse decreases. Furthermore, the major axis increases as the angle of impact increases, which in relation affects angle γ as angle γ is created by connecting the major axis to the plumb line through a straight line. Thus, using the research I gathered I created my own diagram of the crime scene, in order to better understand my specific scenario using the ellipses created by the blood splatters.

Figure 7: Diagram showing the angles measured on a wall from the blood splatter in a real room

α

γ

β

Point X would be where the shooter is standing which is where angle β points to (found via the relationship explained below)

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According to Cornell University, the angles satisfy the relation (Rajchgot): tan (α) tan (β)= sin (γ) Therefore, using the data gathered, we were able to understand how the area of an ellipse was impacted by the angle it was created, however, to find the actual angle of impact the following formula is used by bloodstain analysts: w sin ( α ) =( ) l w=minor axis , l=major axis In theory, using the data gathered, if I substitute in the major and minor axis of my ellipses into this formula, I should get back to the angle at which I created the droplets from at the beginning of my investigation. To check this I used the example of 15° and the example of 60° and used the measurements within the given formula. 1.3 sin ( 15° )= 1.9 1.3 =0.68 sin ( 15° )=0.26 and For sin(60°) I used 1.9 sin ( 60 ° )= sin ( 60 ° )=0.87

0.7 3.3

and

0.7 =0.21 3.3

the absolute value to analyze this as negative measurements do not exist.

The calculations for both angles were off which may be due to small errors in the experiment. Neither angle worked with the given formula, which may be due to the errors in the experiment, as often as the angle increased it was harder to get the droplets onto the paper properly due to the nature of gravity. However, in theory and in a perfect world, this should have gotten back to the original angle. Furthermore, the liquid often formed imperfect ellipses due to it running on the page and thus I would often have to estimate the measurements of the ellipse. To calculate the β angle within the hypothetical scenario for the different angles I used to create the ellipses, I would need to create a plumb line and measure the angle of γ using a protractor. Due to the limits of this investigation, I would have to generate a random plumb line and positioning of a specific blood splatter. To do this I drew an ellipse and a plumb line and then connected the two lines using a line through the major axis of the ellipse. Figure 8: My hypothetical example of the blood splatter on a wall and the angle created between the splatter and plumb line Through the use of a protractor, I measured the angle γ to be 24°, and thus used this to calculate

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angle β using the ratio provided for each of the angles I had created at the outset of my investigation. As seen in Figure 5, the line drawn runs through the major axis of the ellipse and intersects with the vertical plumb line. I used this as the hypothetical blood splatter plan (for one particular droplet) when calculating the β angle.

The calculations of the β angle, for example, using the different angles of impact (angle α ¿ , in this case 15°. I chose to use the angle I had dropped the w liquid from as opposed to the, sin ( α )= , calculation as this was the actual angle l of impact used, despite the measurements of the ellipse not always corresponding. I also used the absolute value of all values calculated, as there are no negative angles involved in this scenario: sin (¿ 24 °) tan (15 °) tan ( β ) = ¿ 0.86 tan ( β ) = 0.91 tan ( β ) =0.96 0.96=¿ 43.8 ° tan−1 ¿

( )

Figure 9: Table of values using the formula described above Angle α (°) Angle γ (°) Angle β (°) 0 24 0 15 24 43.8 30 24 54.8 45 24 67.9 60 24 76.8 75 24 83.8 90 24 Undefined * * Angle β is undefined when angle α is 90° as tangent is defined as the sin of an angle 1 divided by the cosine of an angle, thus when finding tan90°, you get the fraction 0 , which is undefined as anything over 0 is undefined.

Conclusion: In conclusion, the data gathered in this exploration is applicable in crime scene investigations and is used in order to calculate the position of a suspect within the crime scene at the time of the attack. In this exploration, however, I did not take into account the speed or force at which the blood was projected onto a vertical surface and instead relied on the gravity involved in a droplet being dropped onto a horizontal

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or angled piece of paper. Therefore, there were limitations to the data gathered and I had to rely on hypothetical scenarios for the plumb line and calculations. Additionally, there may have been issues with accuracy when creating the ellipses, as it was difficult to drop the liquid onto the piece of paper as the angle increased in size. This would explain why when I tried to calculate angle α using the given formula; I did not get back to the angle I had actually used. Therefore, due to the nature of the experiment and the limitations, the ellipses created were not always accurate; however, through performing this experiment I can better comprehend the way mathematics is used within a crime scene investigation. Despite issues with accuracy, the exploration was able to provide insight into the relationship between the angle of impact and the angle of origin (or position of suspect). For example, I was able to conclude that the relationship between the angle of impact and the area of an ellipse was quadratic. I was also able to conclude that the angle of origin increased as the angle of impact increased, providing me with insight into the way blood splatters can be used as evidence. A possible extension to this exploration would involve doing this experiment on a vertical plane and using paint to make a splatter and taking the different ellipses formed and trying to determine the angle at which the paint splatter was created from. This would, however, need to use more of the complex mathematics used by blood splatter analysts in order to be accurate and would be far beyond the scope of the mathematics SL course. The implications of this investigation, however, allow one to understand the advancements in the evidence collection in crime scenes and the way in which mathematics can be applied in unique ways to identify the positioning of a suspect.

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Works Cited “Bloodstain Pattern Analysis.” Bloodstain Pattern Analysis: How It’s Done, www.forensicsciencesimplified.org/blood/how.html. “Bloodstain Pattern Analysis.” Wikipedia, Wikimedia Foundation, 25 Nov. 2017, en.wikipedia.org/wiki/Bloodstain_pattern_analysis. Empty Room. oklahomavstcu.us/empty-room.py. Rajchgot, Jenna. “Blood Splatter Trigonometry.” Department of Mathematics, Cornell University, www.math.cornell.edu/~numb3rs/jrajchgot/506f.html.

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