Internal Assessments & Written TasksMath SL

IB Math SL: Internal Assessment (IA) Example [Photography/Camera/Geometric Sequence]

Do you need help with your Math IA/Internal Assessment? In this post I will show you my IA that I submitted to IB! You can use this to see what a Math SL IA looks like and I hope this will inspire you to create your best Math SL IA to submit to your teachers! In this IA, I wrote about photography and its connection to Mathematics Standard Level subjects that we have studied. To view my Internal Assessment I submitted just open the PDF right below this! If the PDF does not show, you could read the text version  right below it (not recommended since it lacks formatting & images).



 

 

 

 

Math Internal Assessment

Investigating the Relationship between the Aperture of a Camera and a Geometric Sequence

 

1           Introduction

The aim of this investigation is to find the relationship between the aperture of a camera and a geometric sequence. As a photography enthusiast, I wondered about how a camera is able to turn a live scene and freeze it into an image. Photography used to be a time consuming and complicated hobby. However, the development of technology has enabled anyone to take their own pictures simply by using the devices they have in their pockets. As a camera becomes a more common tool/feature with the era of selfie, it seems that we do not actually understand how the camera itself works, especially how it is able to capture light and turn it into a digital image.

A camera is inseparable from light. An image’s exposure, meaning the amount of light per unit area allowed to reach the camera’s sensor during the process of taking a photograph, determines how bright or dark an image appears[1]. A factor that affects the exposure is called the aperture. This is a controllable circle shaped opening in the lens which directly affects the sum of light that gets in the camera sensor[2]. Aperture is measured in F-stop or f-number, with a larger number signifying a smaller opening in the lens resulting in lesser light received by the sensor; and a lower F-stop number creating a bigger diameter in the lens opening. It can be said that the relationship between the diameter and the F-stop number is inversely proportional to each other. Some examples of F-stop numbers and of the apertures can be seen in the picture below. The letter “ƒ/” signifies that it is an F-stop number.

Figure 1. Aperture and the inverse relationship of how the different f-stop looks like inside the lens  (modified image from: http://mitchmartinez.com/wp-content/uploads/2013/04/Exposure-Graphics-Aperture.gif.png )

Aperture can simply be represented as the pupil of a human eye which dilates and contracts depending on the amount of light that is allowed to enter the retina. F-stop numbers are the most widely used sequence or set of numbers used in photography. However, the unique sequence of numbers needs an explanation on how the abnormal sequence works. My main goal in this investigation was to apply mathematics to investigate the sequence that lies behind the F-stop numbers and to explore the relationship between F-stop numbers and the diameter and area of the circular aperture opening in the lens. Other than that, the relationship between F-stop number and lighting will be analyzed by looking at histograms. If these aims are fulfilled, this will help myself and others understand how cameras work and give credit to the mathematics that cameras deserve.

2           Aperture or F-Stop Numbers

In any camera, the standard F-stop numbers are given by this set of numbers (“ƒ/” signifies F-stop number). For example, for F-stop number 4, it will become ƒ/4.

ƒ/1       ƒ/1.4    ƒ/2       ƒ/2.8    ƒ/4       ƒ/5.6    ƒ/8       ƒ/11     ƒ/16     ƒ/22

To determine the sequence inside the numbers F-stop number, we can divide the numbers into two sets:

By dividing the numbers into two sets, it is visible that these numbers are actually mathematical patterns or geometric sequences which is basically the set of numbers that is obtained by multiplying the previous number by a fixed value called the common ratio (r). The sequence is able to be found by using the geometric sequence formula. The formula is broken down into a1 which is the first term of the sequence, r being the common ratio, and n being the number of the term to find.

We can use this formula to find the first and second set of number. In this example, the first and second value is used inside the formula.

First set of numbers Second set of numbers

 

The common ratio (r) is

Thus, the common ratio is proven:

The entire F-stop range can be found by multiplying by the common ratio (r) of 2. This geometric ratio signifies how the diameter of the aperture is doubled. Yet, by grouping these numbers into two sets, going to the next number in each set presents jumps in two F-stops.

In photography, the amount of light entering the lens depends on the area of the aperture hole which is shaped as a circle. When we decrease an F-stop number to the previous stop, the amount of light is doubled because the opening area of aperture is doubled. When the F-stop number is increased by one, the opposite happens, the amount of light is cut by half. However, if the diameter of a circle is doubled, then the area of the circle quadruples. The aperture becomes 4 times larger because the area varies in relation to the square of the radius. Thus, doubling the radius is identically equivalent to saying 2², which is 4. Therefore, to double the area of the circle, the diameter of the aperture must be multiplied by the square root of 2 (rounded to 1.4) instead. [3] On the other hand, to half the area of the circle, the diameter is divided by the square root of 2 (rounded to 0.7). To summarize, every time we multiply the diameter by the square root of two, the area of the aperture is doubled. If the diameter is multiplied by two, the area will be quadrupled.

From this information, we can create formulae to find the next and the previous F-stop number. In the formula  signifies the current f-stop number.[4]

We can find the next F-stop number by:

We are also able to find the previous F-stop number by:

For example, we can use this formula to find the next and the previous F-stop number of f/8.
When referring to the standard f-stop numbers, note that on the bottom row of the first formula of finding the aperture numberand not 11 and on the second formula
and not 5.6. However, in the photography, the number is made 11 and 5.6 to make the figures convenient to remember although the actual aperture value used by the lens is 11.313 and 5.657 as proven by the recent formula.

It is now observable how the F-stop numbers are basically multiples of a fixed number of .

Initial F-stop number

1 stop from

2 stops from

3 stops from

n stops from

Therefore, we can then use the formula to find the next aperture numbers.

ƒ/1 ƒ/1.4 ƒ/2 ƒ/2.8 ƒ/4 ƒ/5.6 ƒ/8 ƒ/11 ƒ/16 ƒ/22

 

 

Figure 2. Illustration showing the diameter of a lens and the focal length.[5]

To find the diameter value of the aperture, another factor that affects the size of the aperture is the focal distance called the focal length (ƒ) or the distance from the lens to the camera sensor in millimeters. In photography, the focal length is how much zoom the lens makes. The bigger the focal length, the more zoom the lens makes. A lens has several F-stops because there is a variable diaphragm inside the lens that is controllable. Therefore, ƒ/4 is essentially ƒ divided by 4. We can use this information to find the diameter of the lens aperture.

Let’s say that the focal length of the lens or the distance from the lens to the camera sensor is 50mm (the most common focal length in photography) and the F-stop number is ƒ/4. We can then find the diameter of the aperture by using the formula above.

 

 

Therefore, the F-stop value ƒ/4 means that the aperture diameter for a given lens has an effective maximum opening of one fourth of its focal length. With this information, we can also find the radius of the current aperture.

Since we know the radius, we are now able to find the area of the current aperture.

 

 

 

 

 

 

 

 

We can make a table for the calculations:

F-Stop Number Lens focal length (mm) Aperture Diameter (mm) Radius (mm) Area (mm2)
1 50 50.00 25.00 1962.50
1.4 50 35.71 17.86 1001.28
2 50 25.00 12.50 490.63
2.8 50 17.86 8.93 250.32
4 50 12.50 6.25 122.66
5.6 50 8.93 4.46 62.58
8 50 6.25 3.13 30.66
11 50 4.55 2.27 16.22
16 50 3.13 1.56 7.67
22 50 2.27 1.14 4.05

 

From the table, we can observe that even though the lens’ focal length to the camera sensor is the same, the aperture diameter is getting smaller when the F-stop is increased resulting in a smaller radius and smaller area of the circle. This is one of the reasons why a photo taken is too dark. It is because the diameter of the aperture is small and therefore it lets less light into the lens and to the sensor. Thus, in photography moving forward from one F-stop to the next one will cut the amount of light reaching the sensor in half because the diameter of the aperture opening is halved, letting in half as much light to the sensor. On the other hand, moving backward from one F-stop to the previous F-stop would mean that the amount of light reaching the sensor is doubled because the diameter and area of the aperture opening is doubled. The table clearly shows the inverse relationship between the F-stop number and the aperture diameter where the F-stop number is inversely proportional to the diameter of the aperture.

3           Histogram

Aperture and the various F-stop numbers allow a digital camera to control how much light goes into the camera sensor that affects the brightness or darkness of the image taken. However, how does a photographer determine how much he or she needs to adjust the F-stop number so that the image is perfectly exposed (neither too bright nor too dark)? It turns out, in photography the usage of histograms is very important in order to get a perfectly exposed image. A histogram is a type of graph that sanctions a photographer to discern how the values of some data are distributed. In photography, a histogram of the values of black and white is used to analyze how dark (black color) and how bright (white color) an image is. Most digital cameras will blissfully show a histogram of values for the image a photographer is about to take. If the camera is in automatic mode, it will automatically adjust the camera settings to take a perfectly exposed image.

The question is, what are actually these values that the histograms show us? In order to understand what the histogram means to the camera and photographer, we need to know how digital pictures are made. A digital image captured by a camera is fundamentally a two-dimensional grid of numbers, each of which belongs to a single pixel in the image. Every one of these pixels represents the brightness of a given spot in the image.[6] Computers store numbers in a binary format, meaning that all signals inside a computer have two (and only two) different therefore binary values: 0 and 1.[7] In computer terminology, one piece of information which can store either 0 or 1 is called a bit.[8] In a digital image, every single pixel of an image uses 8 bits of memory to store each value.

Therefore, we can calculate how many possible values of a shade of color (in the case of image brightness: black and white) for every pixel of an image:

Basically each pixel has one of 28 = 256 possible color values from 0 which is the darkest tone of black to 255 which is the brightest value of white. The closer the number is to 0 the darker the pixel is and the closer the value is to 255 the brighter the pixel is.

Figure 3. An example of a histogram with a visualization of the data presented. In this example, the image is said to be over-exposed (too bright)[9]

 

A histogram is a bar graph that is compacted together with no spaces between each bar. Since an image has millions of pixels (this is why cameras have “megapixels”), we are not able to see each individual bar in a photo histogram although it is actually there. In essence, the photo histogram shows how properly exposed an image is and tells the camera to automatically adjust the aperture and other settings so that the final image taken is not too dark nor too bright. In theory, a perfect image should have a bell curve or a normal distribution. We do not want an image to be too dark or too bright, but we want it to have perfect brightness where most of the image details are visible.

 

 

4       Proving The Theory/Application

 

Aperture: ƒ/2.0 Aperture: ƒ/2.8 Aperture: ƒ/4

 

From the example photos above, it is made possible to see the relationship between aperture and the histogram. From the series of pictures shown above, it can be concluded how the aperture affects the amount of light that enters the camera. The histogram proves how going from one F-stop number to the next cuts the quantity of light entering the camera in half by showing how the  axis, which represents the brightness of pixels, is shifting. The  axis which represents the number of pixels is also shown to keep a similar shape when the  axis changes.

5           Conclusion

This exploration has allowed me to explore the application of mathematics in our daily lives that too often are there without us realizing it. It turns out that the F-stop number that controls the aperture of a camera does not come from a random number, but is based on a geometric sequence. The standard F-stop scale is essentially a geometric sequence of numbers that corresponds to the sequence of the powers of the square root of 2 because an aperture is shaped as a circle. Just like a pupil, the aperture is in charge of controlling the area of the aperture or the opening of the lens that lets light in to the sensor controlled by the F-stop number. Decreasing the F-stop number by one stop will double the brightness of an image because it will increase the diameter of the aperture and in turn increase the area of the aperture letting double the amount of light into the lens and into the sensor. Increasing an F-stop number by one stop will half the brightness of an image because it will decrease the diameter of the aperture and in turn decrease the area of the aperture letting half as much light into the lens and into the sensor.

This relationship between the F-stop number and the area of the aperture hole is an inverse relationship. To prove how increasing the F-stop number halves the amount of light going into the sensor, a histogram of the values of black and white is used to analyze how dark (black color) and how bright (white color) an image is. The histogram maps out each pixels that has one of 28 = 256 possible color values from 0 which is the darkest tone of black to 255 which is the brightest value of white. The  axis represents the number of pixels. On the  axis, the closer the value is to 0, the darker the pixel is; and the closer the value is to 255, the brighter the pixel is. How increasing the F-stop number decreases the amount of light of an image by half is shown in a histogram by the  axis shift without the  axis shape changing significantly. In theory, a perfect image should have a bell curve or a normal distribution. We do not want an image to be too dark or too bright, but we want it to have perfect brightness where most of the image details are visible. Thanks to this mathematics exploration, I have learned a lot in order to take better photos and to determine whether a photo is too dark or bright. Mathematics do permeates our daily life.

 

 

Bibliography

Burden, Robert. “What Is an F-stop and Why Is It Important in Photography?” The Canadian Nature Photographer –. The Canadian Nature Photographer, 8 Aug. 2008. Web. 02 Apr. 2016. <http://www.canadiannaturephotographer.com/fstop.html>.

Cheney, Mack L., Robert J. Galla, and Tessa A. Hadlock. Facial Surgery: Plastic and Reconstructive. Boca Raton, Fla.: CRC, Taylor Et Francis, 2015. Google Books. Google Books, 2 Dec. 2014. Web. 10 Mar. 2016. <https://books.google.co.id/books?id=z07rBgAAQBAJ>.

“Exposure.” What Is ? Webopedia Definition. QuinStreet Inc. Web. 10 Mar. 2016. <http://www.webopedia.com/TERM/E/exposure.html>.

Hildebrandt, Darlene. “How to Read and Use Histograms – Digital Photography School.” Digital Photography School How to Read and Use Histograms Comments. Digital Photography School, 2012. Web. 08 Mar. 2016. <http://digital-photography-school.com/how-to-read-and-use-histograms/>.

Kumar, Sinu. “Relationship Between F-Stop Numbers and the Size of the Diaphragm Opening.” : Relationship Between F-Stop Numbers and the Size of the Diaphragm Opening Explained. School of Digital Photography, 27 Nov. 2013. Web. 8 Mar. 2016. <http://www.school-of-digital-photography.com/2013/11/relationship-between-fstop-numbers-and-the-size-of-the-diaphragm-opening-explained.html>.

Marshall, Jason. “How to Use Histograms to Take Better Pictures.” The Math Dude. Quick and Dirty Tips, 18 Dec. 2015. Web. 13 Mar. 2016. <http://www.quickanddirtytips.com/education/math/how-to-use-histograms-to-take-better-pictures?page=1>.

Saad, Tony. “Please Make A Note.” : The Mathematics of F/stop Aperture Numbers. Tony Saad, 4 Oct. 2010. Web. 11 Mar. 2016. <http://pleasemakeanote.blogspot.co.id/2010/10/mathematics-of-fstop-aperture-numbers.html>.

Watson, Gray. “The Story of 256.” The Story of 256. Gray Watson, 2002. Web. 16 Mar. 2016. <http://256.com/256.html>.

 

[1] “Exposure.” What Is ? Webopedia Definition. QuinStreet Inc. Web. 10 Mar. 2016. <http://www.webopedia.com/TERM/E/exposure.html>.

[2] Cheney, Mack L., Robert J. Galla, and Tessa A. Hadlock. Facial Surgery: Plastic and Reconstructive. Boca Raton, Fla.: CRC, Taylor Et Francis, 2015. Google Books. Google Books, 2 Dec. 2014. Web. 10 Mar. 2016. <https://books.google.co.id/books?id=z07rBgAAQBAJ>.

[3] Kumar, Sinu. “Relationship Between F-Stop Numbers and the Size of the Diaphragm Opening.” : Relationship Between F-Stop Numbers and the Size of the Diaphragm Opening Explained. School of Digital Photography, 27 Nov. 2013. Web. 8 Mar. 2016. <http://www.school-of-digital-photography.com/2013/11/relationship-between-fstop-numbers-and-the-size-of-the-diaphragm-opening-explained.html>.

[4] Saad, Tony. “Please Make A Note.” : The Mathematics of F/stop Aperture Numbers. Tony Saad, 4 Oct. 2010. Web. 11 Mar. 2016. <http://pleasemakeanote.blogspot.co.id/2010/10/mathematics-of-fstop-aperture-numbers.html>.

[5] Burden, Robert. “What Is an F-stop and Why Is It Important in Photography?” The Canadian Nature Photographer –. The Canadian Nature Photographer, 8 Aug. 2008. Web. 10 Apr. 2016. <http://www.canadiannaturephotographer.com/fstop.html>.

[6] Marshall, Jason. “How to Use Histograms to Take Better Pictures.” The Math Dude. Quick and Dirty Tips, 18 Dec. 2015. Web. 13 Mar. 2016. <http://www.quickanddirtytips.com/education/math/how-to-use-histograms-to-take-better-pictures?page=1>.

[7] Watson, Gray. “The Story of 256.” The Story of 256. Gray Watson, 2002. Web. 16 Mar. 2016. <http://256.com/256.html>.

[8] Ibid.

[9] Hildebrandt, Darlene. “How to Read and Use Histograms – Digital Photography School.” Digital Photography School How to Read and Use Histograms Comments. Digital Photography School, 2012. Web. 08 Mar. 2016. <http://digital-photography-school.com/how-to-read-and-use-histograms/>.

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