Mathematics of Photography

Mathematics of Photography 01

Mathematics is just one of my passions. As well as teaching Mathematics, MultiMedia, and ICT as my “Day Job”, I also work as a Photographer on weekends.

Modern Digital Photography involves quite a bit of behind the scenes Mathematics.

This Mathematics includes Geometric Sequences, Polyhedrons, Fractions, Geometry, Circle Areas, Angles, Algebra Formulas, and Symmetry.

An understanding of this Mathematics can definitely make you a better Photographer, even if it just means you know the best Viewpoints, Angles, and Geometric Composition to use when taking pictures with a Mobile Phone.

I have found that understanding the Mathematics of a DSLR Camera really helps me do fast effective problem solving on Photoshoots.

Digital Cameras have preset auto modes which work well in about 80% of situations, but there are other times when a Camera needs to be put into full Manual Mode and some Mathematical Problem Solving invoked.

 

Here at Passy’s World we have produced a comprehensive series of articles on the Mathematics of Photography, including plenty of real life examples as well as selected Videos which can be viewed.

We recommend working through these articles in the order in which they appear in the Overview Below.

The eight Mathematics of Photography articles are as follows:

– Photo Composition Rules

– Digital Camera Variables

– The ISO Variable

– The Aperture Variable

– The Shutter Speed Variable

– The White Balance Variable

– Combining Variables for Correct Exposure

– Flash Photograpy and Diffusers

Contained below is an outline of each article followed by a clickable link to the full article.

It is important to work through the articles in the order they are given here.

 
 

Mathematics of Photography 02

Like great Works of Art, the elements in a Photograph need to be arranged in a manner which is interesting and pleasing to the eye.

In addition, Photography is a two dimensional medium. Therefore we need to make use of the elements which are present in the photo to create a sense of depth and three dimensions.

Photos also need a main focal point, as well as leading our eyes on a journey through the picture.

Having these things makes for great photos, which are pleasing to the eye, and we call this “Composition”.

There are a Number of Composition Rules involving Percentage, Geometry, Symmetry, a Grid of Thirds, and even the “Golden Ratio” and its spiral.

To find out about these rules and their Mathematics, click the link below:

Photo Composition Rules

 
 

Mathematics of Photography 03

Most people use their camera on “Auto” mode, and get plenty of good pictures.

However, there are many items on a camera which can be adjusted manually to get even better pictures!

In this “How To” article we introduce several of these camera settings, (known as “Variables”), and show how they can be used to create quality photos.

An overview of the three main Variables: “ISO Light Sensitivity”, “Aperture”, and “Shutter Speed” is provided.

This provides the preliminary background to more detailed articles on each of the Variables.

It is therefore vitally important to read this article before proceeding forward to the specific detailed articles.

To read the Introductory Article about Camera Variables, click the link below.

Digital Camera Variables and Settings

 
 

Mathematics of Photography 04

ISO sets a Camera’s sensitivity to light

Effectively it works like a Brightness Control on your camera, and you can turn it up higher if you are in a dark room, or in a shady forest.

If you are in Bright Sunlight, then you need to turn the ISO down to a low value or else your photo will come out all white and washed out with too much brightness.

Camera ISO is one of the three key Variables of Photography, the other two being Aperture and Shutter Speed.

Every photographer needs to understand ISO in order to get bright and clear pictures from their equipment.

ISO values increase in a “Geometric Progression” or a “Geometric Sequence” (Factor of Two, which results in Doubling).

This ISO sequence is: 100, 200, 400, 800, 1600, 3200, 6400 and etc. For every step up in this scale, the brightness of your picture doubles.

To get the full story on ISO and how to set it correctly on Digital Cameras for a variety of lighting conditions, click the link below.

The ISO Variable

 
 

Mathematics of Photography 05

Aperture is one of the key variables in Digital Photography. The other two are ISO Light Sensitivity and Shutter Speed.

An understanding of Aperture is critically important to isolate subjects in portraits and get full detail in Landscape pictures.

The Aperture Scale consists of a seemingly odd set of decimal and whole numbers, and getting your head around these values is not at all easy for Beginner Photographers.

In this article we examine Aperture in detail, and mathematically explain where the Aperture “f-numbers” come from.

Click the following link to read the full article.

The Aperture Variable

 
 

Mathematics of Photography 06

Shutter Speed is mainly used for creating dramatic effects by either freezing action or blurring motion.

The Speed is expressed in a fraction of a second, and is how long we let light in through the camera lens to the sensor.

Think of the Shutter as like opening and closing a set of venetian blinds quickly to let a burst of light into a darkened room.

Shutter Speed forms a Geometric Progression of Fractions with a Common Ratio of 1/2.

Shutter Speed, (also called “Exposure Time”), becomes critically important in situations outside of the norm, such as the following:

1) Fast Moving Sports Action

2) Dimly lit indoor areas such as Bars and Clubs

3) Night Time Photography

4) Theatre and Musical Performances

5) Indoor Photos using Flash

6) Indoors Photos where we do not use Flash

7) Photos where we want motion blur for fast moving objects

8) Bird and Wildlife Photography

9) Photographing Young Children and Animals

In this article we will show you what Shutter Speed is, what the speed numbers mean, and how to set shutter speeds to values which should produce great photos.

Click the following link to read our article on Shutter Speed.

The Shutter Speed Variable

 
 

Mathematics of Photography 07

Different light sources produce light with slightly different colour tints, but our eyes do a great job correcting these variations.

Humans do not see a shift in colour as we move from a sunny garden into a shaded area, or go into a room and turn a light on.

Generally wherever we are, a piece of plain white paper always looks white.

However Digital Cameras DO detect light source differences, and sometimes create pictures with incorrect looking colours in them.

We need to know how to perform “white balance” adjustments, so that we can obtain pictures with realistic colours in them.

White Balance involves the Primary Colours of the Spectrum forming a Kelvin Temperature Scale from “cool” blue/violet through to warm “red” candle light.

What we have is a Mathematical Scale of color, and White Balance blends various colours together so that a white sheet of paper will look white, no matter what coloured lighting we are in.

You can read the full article on White Balance at the following link:

The White Balance Variable

 
 

Mathematics of Photography 08

In this article we look at Combining Variables for Correct “Exposure”.

We tie together the three key Digital Camera Variables: ISO, Shutter Speed, and Aperture, and discuss how to set these up to get good exposures in Camera Manual Mode.

When these three variables are optimised and unified together, we get a nice clear picture with great contrast and highlights. We say that the photo is correctly “Exposed”.

You can read the full article on Getting Correct Exposure Using Key Variables at the following link:

Combining Variables for Correct Exposure

 
 

Mathematics of Photography 09

In this article we discuss Flash Photography and how to use Light Diffusers to make Flash Photos look so much better.

Diffusers come in all types of Geometric Shapes and Sizes and create softened light which is more like Daylight.

The Intensity of Light from a point source like an on camera Flash follows an Inverse Square Law with the Distance the Flash is from the subject.

This is very important to have an intuitive idea about Inverse Square Law when changing the Power on a Flash, or when we move further away from the subject being photographed.

The full article about Flash and Diffusers can be read at the following link:

Flash Photograpy and Diffusers

 
 

Working through this series of articles will definitely make you a better photographer.

Keep in mind that I spent a couple of years working with Digital Photography to reach this level of understanding to a point where I could write about it to share my knowledge with others.

So if you are a complete beginner and feel a bit overwhelmed, then that is okay, these things just take practice and time.

It is best to work through all of our eight articles a bit at a time, try things out, and then come back to the articles to pick up a bit more of the finer detail.

These articles can also be great to refer to when a student says something like: “I don’t need to know Mathematics, I’m going to be a Photographer!”.

Well there is a lot of Mathematics involved with Photography, Cameras, and Lighting; even if you are only the model being put into poses and surrounded by intense studio lights.

The Maths is helpful, but the main thing about being a Photographer is to get out there taking lots of pictures!

Life is short and there can never be too many photos of its little magic moments!

Enjoy,
Passy

 
Photos By Passy

Mathematics of Photography 10

My Photography website can be found at the following link:

Photos By Passy Website

If you would like to check out a Portfolio of my Photography Work, then check out the Photo Albums on Flickr at the following link:

Photos By Passy Portfolio Albums

Finally if you are interested in reading more “How To” articles about Photography, then visit the “How To” section on my website at this link:

Photos By Passy How To Articles

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Free Exponents Posters

Exponents Rules on Wall Pic Two
Image Copyright 2013 by Passy’s World of Mathematics

We have a great set of 12 color Posters covering the Exponent Laws that is available for free to all subsribers to Passy’s World.

If you are not already a subscriber, then click the following link to find out how to become an email subscriber.

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If you would like to see a preview of the full set of posters, then view the following SlideShare presentation.

 
 

Exponents Laws on Wall Pic One
Image Copyright 2013 by Passy’s World of Mathematics

The posters are available as either a PDF or a PPT Presentation.

We suggest you then print them in color on standard A4 or Letter sized paper, and then use a Color Photocopy to enlarge them to Double Size or Larger.

If you are a subscriber you can get this Poster Set for free by emailing Passy’s World at the link shown below:

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Related Items

Why Exponents and Indices are Important

 
 

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These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.

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Exponential Population Growth

crowded china city
Image Source: http://img.gawkerassets.com

In this lesson we look at Exponential Growth of Populations.

Exponential growth involves increases starting off as reasonably small, and then dramatically increasing at a faster and faster rate.

The world’s accelerating population growth is a major concern in terms of how our planet can feed and provide fuel for the current 7.2 billion plus people who currently live in our world.

In addition to the explosion of humans on our own planet, Animals and Bacteria also increase exponentially.

The following video shows how Bacteria divide and multiply exponentially.

 
 

Human Population Growth

By the year 2000, there were around 10 times more people on Earth than there were just 300 years ago in 1700.

WORLD POPULATION

Year —– Population
1700 —– 600 000 000
1800 —– 900 000 000
1900 —- 1 500,000 000
2000 —- 6 000,000 000
2011 —- 7 000,000 000

On a graph, the increase looks like this:

Population Increase Graph
Image Source: http://www.pfaf.org

 

Here is an excellent two and a half minute video which shows the history of the world’s population increase:

 
 

Increasing Life Expectancy

pic of old people having fun
Image Source: http://elimfamilychurch-eastbourne.org.uk

The death rate has reduced, and people now live a lot longer and many more children are produced and live longer.

In 1960 the average age of death was 53 years old, but now it is around 75 years old.

The increase in health and life expectancy was historically unevenly spread throughout the world.

But over the last 200 years, Long Life and Good Health have generally increased throughout most of the world, and this has contributed to our exponentially huge population increase.

The following four minute Swedish video shows what has happened over the last 200 years.

 
 

Population Growth Equation

Population Growth Formula
Image Copyright 2013 by Passy’s World of Mathematics

Population Growth follows Mathematics which involves exponents.

Exponential increases start off slow, but then sharply increase to a massive explosion in size, just like the power of a rocket engine igniting at take off.

An exponents formula, similar to the one used on compound growth for superannuation and interest bearing investments, can be used to estimate the Populations of Humans, Animals, and Bacteria.

Full Population Formula
Image Copyright 2013 by Passy’s World of Mathematics

This Equation involves the exponents of Rate x Time, and this is why Exponential patterns of increase in Populations occur.

 

Find out more about this equation at the following link:

Click here for Population Growth Mathematical Equations

 

There is a set of Population Growth fully worked example Maths Problems at the following link:

Click here for Population Growth Example Maths Questions

 
 

World Population Growth

Although Total Population is dramatically increasing, the actual Percentage Rate of world’s population growth is slowing down.

Throughout the 1960s, the world’s population was growing at a rate of about 2% per year.

By 1990, that rate was down to 1.5%, and by the year 2015, it’s estimated that it will drop down to 1%.

Currently each second, 5 people are born, and 2 die, which means that each second of the day we get an extra three people on the planet.

Real Time World Population Meter

There is an excellent real time Population meter which ticks over continuously, at the following link:

http://www.worldometers.info/world-population/

 

Family planning initiatives, an ageing population, and the effects of epidemic diseases such as AIDS, are some of the factors behind this rate decrease.

Even at these very low rates of population growth, the population increase numbers are still staggering.

By 2015, despite a low expected 1% growth rate, experts estimated there would be 7 billion people on the planet.

We actually reached 7 billion people four years earlier than this in 2011.

By 2050, there may be as many as 10 billion people living on Planet Earth.

Can the planet support this population and when will we reach the limit of our resources?

 

Space on the planet is not the problem, as 7 billion people standing shoulder to shoulder would only occupy the area of the city of Los Angeles.

There are already 23 megacities around the world where people live very large numbers.

 

The main problem is not space, but an inbalance in food and fuel, with 5% of the earth’s population consuming 23% of the world’s energy.

Or in other words 1/20th of the world’s people using up 1/4 of the energy.

13% of the world does not have clean drinking water, and 40% do not have adequate sanitation and sewerage treatment.

The following video is about the science of overpopulation: how it has occurred and what it means to our future.

 
 

The following video shows that Population Growth is not the key part of our problems.

War and Poverty are far bigger problems, than our overpopulation.

 
 

The Future of Our Planet

The following one hour video documentary shows that extreme poverty has decreased, especially in Asia.

In fact, for the first time on our history, poverty could be totally eliminated.

Although it is one hour long, it is definitely worth watching.

 
 

Related Items

Why Exponents and Indices are Important

 
 

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Ocean Mathematics

Ocean mathematics into slide
Image Copyright 2013 by Passy’s World of Mathematics

Welcome to Ocean Mathematics!

Passy’s World is pleased to present a series of lessons all about the Mathematics related to the following aspects of the World’s Oceans:

Mathematics of Ocean Waves

Mathematics of Surfing

Surfboard Design and Geometry

Power Generation from Waves

Mathematics of Tsunamis

Sharks Mathematics

Mathematics of Ships at Sea

These lessons have been assembled as part of a Lecture Presentation at the Mathematics Association of Victoria 2013 Annual Conference in Melbourne Australia.

Links to each of these lessons are given below, as well as a link to the PowerPoint Presentations which were used during the lecture.

 
 

Mathematics of Ocean Waves and Surfing

Surfing Mathematics 1
Copyright Image Purchased by Passy’s World from Dreamstime.com

In this lesson we look at some of the Mathematics associated with Surfing.

We cover the Mathematics of Ocean Waves, as well as the mathematics and physics associated with catching and riding a surfable wave. People do not need to think about mathematics when actually surfing; however there are certain mathematical realities which are foolish to ignore!

Click the following Link to access this lesson:

Mathematics of Ocean Waves and Surfing

 

Surfboard Design and Geometry

kelly slater doing big cutback
Image Source: http://wikimedia.org

This lesson covers Modern Surfboard Design, and its associated mathematical geometry.

First we look at the main types of Surfboards, and then we examine the specific geometries of the component parts. We also look at computerised shaping and manufacture of surfboards, involving 3D Printing technology and Parametric Geometrical Equations.

Click the following Link to access this lesson:

Surfboard Geometry and Design

 
 

Power Generation from Waves

Oceanlinx power plant in water
Image Source: http://2.bp.blogspot.com

In this lesson we look at Wave Energy from the Oceans, and some of the Mathematics and Geometry associated with this “Blue Energy”.

Click the following Link to access this lesson:

Wave Power Mathematics

 
 

Mathematics of Tsunamis

Tropical Tsunami
Image Source: http://millicentandcarlafran.files.wordpress.com

This lesson examines the Mathematics associated with the incredible destructive power of Tsunami Waves. It’s all about Equations and Algebra Substitution!

Click the following Link to access this lesson:

http://passyworldofmathematics.com/tsunami-mathematics/

 
 

Sharks Mathematics

Shark Mathematics One
Image Copyright 2012 by Passy’s World of Mathematics

This lesson covers some mathematics associated with Sharks, such as Levy Walks, Biting Force, and Ratio and Proportion.

Click the following Link to access this lesson:

http://passyworldofmathematics.com/shark-mathematics/

 
 

Mathematics of Ships at Sea

Blue Marlin Sinkable Ship
Image Source: http://i.imgur.com

This lesson contains some Mathematics of Ships including Archimedes Principle, Stabiliser Wings, the Plimsoll Line, the Bulbous Bow, and a number of other aspects relating to ship design and mathematics.

Click the following Link to access this lesson:

Mathematics of Ships at Sea

 
 

MAV Conference PowerPoints

To download the PowerPoint Presentations from the MAV Conference, click the links below and save the files to your computer.

Mathematics of Ocean Waves and Surfing

Click the link below to download this 12MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/IntroWavesSurfPPTv4.pptx

 

Surfboard Design and Geometry

Click the link below to download this 11MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/SurfGeomDesignPPTv3.pptx

 

Power Generation from Waves

Click the link below to download this 3MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/WavePowerPPTv2.pptx

 

Mathematics of Tsunamis

Click the link below to download this 2MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/TsunamiMathPPTv1.pptx

 

Sharks Mathematics

Click the link below to download this 2MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/SharksMathPPTv1.pptx

 

Mathematics of Ships at Sea

Click the link below to download this 4MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/ShipsMathPPTv1.pptx

 
 

AMC Teacher Resources

AMC website Image

The Australian Maritime College has an excellent set of free resources for Mathematics Teachers.

Among these are some very good mathematical exercises related to Ocean Mathematics.

The AMC is planning to add more online interactive lessons, but as of late 2013 the current the interactive lessons are as follows:

A Study of Similar Vessels (an application of Curve Fitting) – 5mins

Wind Farm Feasibility Study (an application of Probability) – 12mins

Vessel Speeds in Waves (an application of Differentiation) – 5mins

Ship Hydrostatics (an application of Integration) – 12mins

Wave Refraction (an application of Trigonometry) – 12mins

Ocean Waves (an application of Superposition) – 5mins

Scaling Laws (an application of Algebra) – 8mins

 

Go to the following link to find their Resources page:

http://www.amc.edu.au/why-study-maths

To get the workbooks and other resources, it is necessary to register your school and complete an order form online.

These resources are free to Australian schools, and if you would like to see what some of the workbook exercises look like, then check out the following link:

Click here for AMC Online Maths Workbook PDF

 

The AMC can be contacted about this program at the following email address: whystudymaths@amc.edu.au

 
 

Surf Aid Mathematics Resources

Surfing Mathematics 20
Image Screen Captured from Surf Aid Website

Surf Aid is a not for profit organisation sponsored by Billabong with a kean interest in the preservation of Mentawai in Indonesia.

The Mentawai Islands are a chain of about seventy islands and islets off the western coast of Sumatra in Indonesia.

It is totally free to join the “Surfaid Schools Program”, and all that is required is submission of your email address.

The Surfaid website can be found at this link:

http://www.surfaidinternational.org/schoolsprogram

Downloadable Free Units which are in ZIP and PDF format which could be of use to Mathematics Teachers are the following:

AUS Maths 12-14

Mathematics – Connections through Surfing

Technology

Destination Mentawai Islands

The Economics Of Aid

Crossing The Divide – Primary

These are all free PDF and ZIP file downloads

A Typical example might be the following questions about traveling to surfing locations on various Islands:

“If the motor launch can manage an average speed of 15knots, calculate the journey time between each
location. 1knot is 1 nautical mile per hour. You can assume there is sufficient sailing staff to keep moving 24h
per day. Round up your answers to the nearest day and complete worksheet 5.3.

If you move 10º around the equator how many nautical miles have you travelled?

What angle around the equator (change in longitude) corresponds to 1 nautical mile?”

 

Another activity we looked at was all about “Planning an Overseas Surfing Trip”.

Eg. Costs, Savings Plan, Items needed and their Cost, and so on.

It was very surfing orientated and might alienate non-surfers and female students, but could easily be adapted to be a group of friends planning a Bali Holiday, and undertaking various tours and activites whilst in Bali.

We will be going through all of the Surfaid Materials, and seeing what could be incorporated into some middle school mathematics, even though the school Passy works at has a tiny minority of students who have ever been Surfing.

We suggest you could easily do the same for your classes.

 
 

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Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

 

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These include items of mathematical interest, funny math pictures and cartoons, as well as occasional glimpses into the personal life of “Passy”.

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While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.

 

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Enjoy,
Passy

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Mathematics of Ships at Sea

Blue Marlin Sinkable Ship
Image Source: http://i.imgur.com

This ship is the “Blue Marlin”, one of the most amazing vessels to ever to sail the seas.

The Marlin can carry an incredible 75,000 tonnes.

It reaches a top speed of a sedate 13 knots, powered by a gigantic 17,000 horse power diesel engines, with a crew of 24.

(Note 13 Knots = 13 x 1.852 = 24 km / hr).

Steering it is said to be like controlling a floating office block.

The barges pictured above – which weigh a total of 60,000 tonnes – were exported from Korea to be completed in Rotterdam.

No crane in the world is big enough to lift these types of cargoes onto the Marlin’s deck, and an ingenious method is used to load the cargo.

The deck is submersible and can disappear under 13 metres of water when the Marlin’s ballast tanks are pumped full of water.

Boats, oil rigs and ships can then be floated on before the deck is raised again by emptying the ballast tanks.

There is a great article about the Blue Marlin at the following link:

Click here for Daily Mail Blue Marlin Newspaper Article

 

The following three minute video shows a giant oil rig that was built in Korea being loaded onto the Blue Marlin for transport to the USA.

 

Obviously there is some very exact mathematics involved here, because ships are designed via Archimedes Mathematical Principle to float rather than sink!

 
 

In this lesson we look at some of the Mathematics of Ships.

This includes Archimedes Floating Principle, as well as the Geometry of Hulls and Stabilisers.

We also look at Ship Stability, the Plimsoll Line, and the Mathematics of Loading Ships with Cargo.

 
 

How Ships Float

big cruise liner
Image Source: http://twistedsifter.wordpress.com

Today’s colossal size cruise ships, such as the “Oasis of the Seas” and the “Allure of the Seas”, remain afloat thanks to buoyancy and displacement.

Ships are designed to displace the amount of water equivalent to their own mass.

The ocean meanwhile pushes up and keeps the ship afloat, or buoyant.

In other words, the pressure of the water pushing up on the bottom (or hull) of a cruise ship counteracts the downward force of the vessel’s gravity, thereby making it float on the water’s surface.

(Basically Positive and Negative Integer values cancelling themselves out to zero).

This basic idea is often referred to as “Archimedes’ Principle”.

According to this principle, an item floats when it displaces more than its own weight of the liquid that it’s immersed in.

The surrounding fluid pushes back with a force equal to that of the amount displaced, so that the two opposing forces are equal and the object floats.

Mathematics of Ships 1

 

A Ship will float as long as it weighs less than the volume of water that its hull displaces.

The following one minute video explains Archimedes Principle

 
 

Tall Cruise Ships

pic of hotel style cruise liner
Image Source: http://www.cruiselawnews.com

However, there are some concerns about cruise liners becoming too tall.

In the past, there was a reasonable and safe ratio between a vessel’s draft (below the waterline) and air draft (above the waterline).

These days cruise liners seem to be getting designed primarily as multi-storey Hotels, and secondly as actual ships.

These cruise ships have lost the traditional proportions between what’s below and above the waterline, making the vessels dependent on stabilizers not only to battle through rough weather, but also to stay upright with only slight to moderate breezes.

“Stablisers” are discussed in the next section.

The following link is to an article about ships getting too top heavy and too tall.

Click here for Cruise Liners Article

 

The following 46 minute documentarty viceo is about the world’s largest cruise shipe “The Oasis of the Seas”.

 
 

Ship Stabilisers

stabiliser sticking out of a ship
Image Source: http://sonnyv.smugmug.com

The purpose of stabilizers is to reduce the roll (i.e., the sideways motion) of a ship.

Today, all modern cruise ships have stabilizers. Most ships have two stabilizers, one on each side of the ship. Larger ships like Cunard Line’s Queen Mary 2 and Royal Caribbean’s Voyager, Freedom and Oasis class ships, have four stabilizers, two on each side.

The stabilizers are shaped like airplane wings and extend out from the side of the hull in a perpendicular fashion when in use. They can pivot up and down like the ailerons on an airplane’s wings. Consequently, as the water flows over a stabilizer it can be turned upwards or downwards to exert dive or lift.

When a sensor detects that a wave is pushing the ship one way, the ship’s systems automatically pivot the stabilizers so as to exert pressure in the opposite direction. Captain William Wright of Royal Caribbean reports: “they eliminate about 85% of the roll of the vessel.”

When the ship is in port or when the seas are calm, the stabilizers can be folded back hydraulically into compartments along the side of the hull. Stabilizers can be deployed independently and so the ship’s officers have the option of deploying the stabilizers on one side of the ship or on both sides to suit the sea condition.

Mathematics of Ships 2

Stabilizers do create some drag and thus can in theory reduce speed and fuel efficiency. However, any such loss has to be balanced against the savings in speed and fuel efficiency resulting from reducing the ship’s motion.

The following diagram shows how the lift works for various stabiliser positions.

ship stabilisers diagram
Image Source: http://www.uq.edu.au

 
 

Sea Sickness

sea sickness
Image Source: http://toohighortullo.wordpress.com

The purpose of cruise ship stabilizers is to reduce the side to side rocking motion of the ship.

They help a ship move more smoothly, which cuts down the chance of seasickness for passengers.

When there is a great deal of movement, it can cause a discrepancy between what a person sees and what her inner ear senses. This is what causes seasickness. The smoother the ride, the less chance for this to happen.

 
 

Stabilising Pool Tables

pool table on ship
Image Source: http://images.dailydawdle.com

How do pool tables stay level on cruise ships?

Ships, and even some large yachts, have pool tables that are kept level when the ship has a slight roll by specially made gimbals.

These keep the pool table from rolling with the ship or yacht in mild conditions.

This makes the pool table self-levelling.

The pool table rests on gimbals and not just table feet on a floor or deck.

Gimbals are also often used on the stoves and cooking tops on ships and yachts.

A Gimbal consists of three concentric rings – the following picture shows what a basic pool table gimbal looks like:

Gimbal
Image Source: http://productions.andrewgentle.com

The following web page has a video of a pool table that uses gyroscopes and computers to create perfectly accurate self levelling.

Click the Link below to watch a Video of a Self-Levelling Pool Table.

Click Here for Pool Table Video

 
 

Swimming Pools on Ships

pool inside pool on a ship
Image Source: http://i47.photobucket.com

What about swimming pools on ships ?

How do they endure stormy weather where the ship rolls side to side ?

Quite simple really – They have a Pool inside a Pool as shown in the following YouTube clip

 

However for passenger safety, they usually close the pool during storms on ships.

 
 

Geometry of Ship Hulls

ship hull diagram
Image Source: http://www.shipcruise.org

(Click to Enlarge the above Image on a new Screen)

 

A ship’s hull is the body of the ship, which sits below the main deck.

Through years of trial and error engineers have found that making the hull rounded, wide and deep, helps disperse the weight of the ship across the body of the ship.

Large cruise ship hulls are shaped like a squarish shaped letter “U”

This design allows water to flow away from the vessel, dissipates drag, and facilitates a smooth ride.

 

Hulls Inside Hulls

In addition to providing passengers with a fluid ride, the hull helps protect a cruise ship so that it doesn’t sink.

Icebergs, reefs, sandbars, and other large, sharp objects can rip apart a ship’s outer layers.

To prevent a major catastrophe shipbuilders typically use extra-strength steel and insert double hulls, or a hull inside a hull,
as an extra precaution.

Mega cruise ships also stay afloat thanks to bulkheads.

These vertical watertight dividers are installed throughout the interior of the hull and help keep damaged ships afloat by containing incoming water into compartments.

By containing the gushing water, the ship should not flood and sink.

 
 

Bulbous Bow

Most large ships have a bulb like protrusion under the water line which pushes out a bow wave to partially cancel the waves hitting the bow and provide “cutting through” ability.

Mathematics of Ships 4

However, recent design changes to save fuel consumption costs on large oil tankers has actually changed the traditional ship hull shape, and totally removed the Bulbous Bow.

The following video explains that the designers tried out over 40,000 different hull designs to come up with the final new design.

Prior to the use of computer modelling, it would have been impossible to manuallu do the necessary calculations for 40,000 different designs without it taking years and years of work!

 
 

Ballast Water

diagram about ship ballast
Image Source: http://www.netpeckers.co.in

Mathematically, there is no real limit to how big a ship can be, as long as it fits through the waterways it sails, or the harbors it enters.

The “Carnival Destiny”, for example, is too large to fit through the Panama Canal.

Of course, the larger a ship gets, the more passengers and cargo it can carry.

This additional weight does not put the ship in danger of sinking.

The extra weight pushes the ship deeper into the water, which displaces more water and increases the buoyant force.

Balast tanks are filled in order to add weight to the ship once cargo has been discharged, and improve its stability.

When a ship is sailing with only Ballast Water, and no cargo or passengers, it is called a “Light Ship”; as in it is light in terms of weight.

In some extreme conditions, ballast water may be introduced to dedicated cargo spaces in order to add extra weight during heavy weather or to pass under low bridges.

 
 

Effect of Ballast Water

painting of the titanic
Image Source: http://www.plantoholiday.com

Most of the time, extra weight actually helps a ship remain stable.

The weight gives the ship a lower center of gravity – the point around which the weight of an object is concentrated.

This means the pull of gravity is greater near the base than the top, so the ship is less likely to topple over.

To stabilize a ship, shipbuilders place the vessel’s heaviest machinery and equipment, like the engines and fuel, at the lowest level, below the waterline.

As the ship’s load lightens, from using up fuel, for example, the ship takes in ocean water for ballast, or added stability.

But the water needs to be in the right place on the ship.

Too much water in the wrong place can make a ship sink.

Water entering a ship’s lowest level can be disastrous; especially if the water fills up only one end of the ship. That’s what happened in the sinking of the Titanic in 1912.

the normandie sunk tipped over
Image Source: http://upload.wikimedia.org

The Normandie, was an ocean liner built in the early 1900s.

While the ship was docked in New York Harbor in 1942, a fire broke out and Firefighters sprayed so much water on the ship’s top deck that the ship became top-heavy.

The Normandie rolled over and sank, right at the dock.

 
 

The Safe Loading Level

ship sailing with red hull exposed
Image Source: http://ebrd-stories.com

Ships have a red bottom painted onto them, and when a ship is sailing without cargo, the water level will be at the top of this red line boundary along the length of the ship.

A ship can be sailing out of the water partially exposing the red bottom without any risk of sinking, but might be less stable with more side to side roll occuring in big seas.

It is important to realise that not all water weighs the same, and this effects buoyancy.

Freshwater is less dense than sea water, and so a ship will sink down lower in the water, when it sails into a freshwater port.

So when loading in a fresh water port, especially in the tropics (warm water is less dense than cold water), the ship will be lower in the water, (but will not sink).

Then when it sails out into tropical sea water it will rise up higher out of the water, because sea water is more dense (heavier than fresh).

Ships also rise up out of the water as they consume fuel, and their fuel weight consequently decreases.

The safe loadng level in a Port involves making use of markings on the ship’s hull which are called “The Plimsoll Line”.

 
 

The Plimsoll Line

Plimsoll line with markings
Image Source: http://www.marineinsight.com

The Plimsoll Line was originally a long horizontal line painted on the side of merchant ships.

When a ship was loaded, the water level was not supposed to go above the line.

However, the water could reach different parts of the line as its temperature and saltiness varied with season and location.

The basic symbol, of a circle with a horizontal line passing through its centre, is now recognised worldwide.

The Oxford Companion to Ships and the Sea defines the Plimsoll Line as:

A mark painted on the sides of British merchant ships which indicates the draught levels to which a ship may be loaded with cargo for varying conditions of season and location. The Plimsoll Mark shows six loading levels, those which may be used in tropical fresh water; fresh water; tropical sea water; summer, sea water; winter, sea water; and winter, North Atlantic, for vessels under 100 metres (330 ft) in length.

Mathematics of Ships 3

The maximum load capacity of a ship is determined by loading the ship up so that the Plimsoll Line styas just above the water.

Sinking is the most dramatic form of overloading. Before the ship sinks, it will sit so low in the water that even the smallest wave will wash over its deck, flooding the ship and possibly causing it to sink. To avoid this, ships are loaded so that their decks are well above any wave the ship is likely to encounter.

This safe height is identified by the Plimsoll line. The Plimsoll line usually is marked by a color change on the hull that differentiates the portion that is meant to be underwater from the portion that is meant to remain above the water. An empty ship floats high in the water. As cargo is loaded, the ship sits lower. Eventually only the portion that is supposed to remain above the water can be seen. Most prudent owners will not load cargo beyond this maximum.

The original “Plimsoll Mark” was a circle with a horizontal line through it to show the maximum draft of a ship. Additional marks have been added over the years, allowing for different water densities and expected sea conditions.

Letters may also appear to the sides of the mark indicating the classification society that has surveyed the vessel’s load line. The initials used include AB for the American Bureau of Shipping, LR for Lloyd’s Register, GL for Germanischer Lloyd, BV for Bureau Veritas, IR for the Indian Register of Shipping, RI for the Registro Italiano Navale, NK for Nippon Kaiji Kyokai, and NV for Det Norske Veritas. These letters should be approximately 115 millimetres in height and 75 millimetres in width.[7] The Load Line Length is referred to during and following load line calculations.

The letters on the load line marks have the following meanings:
TF – Tropical Fresh Water
F – Fresh Water
T – Tropical Seawater
S – Summer Temperate Seawater
W – Winter Temperate Seawater
WNA – Winter North Atlantic

The following five minute YouTube video which explains the Plimsoll line markings including summer winter tropical etc.

It also covers the the 1/48th of Draft Rule for Winter and Summer Markings.

 
 

Fresh water is considered to have a density of 1000 kg/m³ and sea water 1025 kg/m³.

Fresh water marks make allowance for the fact that the ship will float deeper in fresh water than salt water. A ship loaded to her Fresh Water mark in fresh water will float at her Summer Mark once she has passed into sea water. Similarly if loaded to her Tropical Fresh water mark she will float at her Tropical Mark once she passes into sea water.

The Summer load line is the primary load line and it is from this mark that all other marks are derived. The position of the summer load line is calculated from the Load Line Rules and depends on many factors such as length of ship, type of ship, type and number of superstructures, amount of sheer, bow height and so on. The horizontal line through the circle of the Plimsoll mark is at the same level as the summer load line.

The Winter load line is one forty-eighth of the summer load draft below the summer load line.

The Tropical load line is one forty-eighth of the summer load draft above the summer load line.

The Winter North Atlantic load line is used by vessels not exceeding 100 metres in length when in certain areas of the North Atlantic Ocean during the winter period. When assigned it is 50 millimetres below the winter mark.[8]

The calculation of the Fresh Water load line is quite complicated.

The Fresh Water load line is an amount equal to \tfrac{\triangle}{4T} centimetres above the summer load line where \triangle is the displacement in metric tonnes at the summer load draft and T is the metric tonnes per centimetre immersion at that draft.
In any case where \triangle cannot be ascertained the fresh water load line is at the same level as the tropical load line.
The position of the Tropical Fresh load line relative to the tropical load line is found in the same way as the fresh water load line is to the summer load line.

Information Source: http://en.wikipedia.org/wiki/Waterline

 
 

That’s it for our look at some of the Geometry and Mathematics associated with ships.

If you want to look further into some of the Hydrodynamics and Mathematics of ships at sea, then here are some weblinks:

Centre of Buoancy and Centre of Gravity

 

Ship Hull Measurements Diagram

 

Mathematics of Hull Shape

 
 

MAV Conference PowerPoint

To download the PowerPoint Presentation from the Mathematics Association of Victoria Conference, click the link below and save the file to your computer.

Mathematics of Ships at Sea

Click the link below to download this 4MB PPT File:

http://passyworldofmathematics.com/MAVconfPPTs/ShipsMathPPTv1.pptx

 
 

AMC Teacher Resources

AMC website Image

The Australian Maritime College has an excellent set of free resources for Mathematics Teachers.

Among these are some very good mathematical exercises related to Ships.

The AMC is planning to add more online interactive lessons, but as of late 2013 the current the interactive lessons are as follows:

A Study of Similar Vessels (an application of Curve Fitting) – 5mins

Wind Farm Feasibility Study (an application of Probability) – 12mins

Vessel Speeds in Waves (an application of Differentiation) – 5mins

Ship Hydrostatics (an application of Integration) – 12mins

Wave Refraction (an application of Trigonometry) – 12mins

Ocean Waves (an application of Superposition) – 5mins

Scaling Laws (an application of Algebra) – 8mins

 

Go to the following link to find their Resources page:

http://www.amc.edu.au/why-study-maths

To get the workbooks and other resources, it is necessary to register your school and complete an order form online.

These resources are free to Australian schools, and if you would like to see what some of the workbook exercises look like, then check out the following link:

Click here for AMC Online Maths Workbook PDF

 

The AMC can be contacted about this program at the following email address: whystudymaths@amc.edu.au

 
 

Related Items

Ocean Mathematics – Overview
Mathematics of Ocean Waves and Surfing
Surfboard Geometry and Design
Tsunami Mathematics
Wave Power Mathematics
Shark Mathematics

 
 

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