1ilp fundamentals of maths for lighting revision for pdf

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ILP ELD Fundamentals of Maths for Lighting Issues 2.1 Autumn 2011 ILP

ILP Exterior Lighting Diploma

Fundamentals of Maths for Lighting

Mathematics to support the underlying Engineering Principles involved in Lighting

Design. Lighting designs are usually carried out using computer design packages, often

involving sophisticated modelling and rendering software. However, even with these design packages, it is still possible to achieve a technically acceptable lighting design

in terms of calculated values, which would not be acceptable from an installation, maintenance or aesthetic view point. Having an understanding of the engineering

principles and mathematics which support the design process will assist lighting designers in establishing the full acceptability of their design.

In addition, Lighting Engineers wishing to register with the Engineering Council (ECuk ) must be able to demonstrate a grasp of the underlying engineering principles

involved in lighting design, which in turn must be supported by an understanding of the mathematics involved.

For this reason, the attached notes set out the key mathematical principles and processes required to support the understanding of the engineering principles.

These notes are intended to refresh your knowledge of the mathematics involved. Please read the attached notes and worked examples. For further practice we

recommend BBC bitesize and Mathsisfun website, although there are now numerous interactive learning website available.

If this revision highlights areas where you feel you would like extra support, please email [email protected] who will liaise with course tutors and your employer to

assist in providing additional resources in terms of additional training, support and information to students requiring them.

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A Algebra

1. Definitions and Rules

Algebra is simply a way of expressing numbers, or relationships between numbers, in terms of letters, or a combination of letters and numbers.

The letters are often called variables (since they can usually vary). With a formula which contains two or more variables, providing all but one variable

have known values, it is possible to calculate the value of the remaining unknown variable.

i. Algebra uses a form of shorthand which omits multiplication and division signs and certain other signs e.g.

2a means 2 x a /3 means 3 ab means a x b b2 means b x b

Also where the result of a multiplication (or division is, say, 1 x b, then this

could be written as 1b, but the convention is to drop the number 1 and write it simply as b.

Finally, where positive and negative numbers are used, the convention is that the positive sign (+) is not shown, but the negative sign (-) is.

e.g. + 3y 4z is written as 3y 4z

ii. An Expression has no equals or inequality sign and can have

different value depending on the value we give to the letter in the expression e.g.

in the expression ( + 1),

if = 2,

then the expression ( + 1) becomes (2 + 1), which is 3.

iii. A Formula is written using an equals sign (=) or one of the

inequality signs, or a combination of the two: -

= means equal to (each side of the equation is balanced like balanced scales or a balanced see-saw).

means not equal to

> means greater than < means less than

=> means greater than or equal to

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(3 + 3) = 2 + 9, which holds true for only certain values of

(in this case, is 6), or, 2a + 7 = 3 b which can hold true for a range of difference values

(e.g. where b = 1, b = 2 b = 3 etc.)

A Formula can be re-written in different ways in order, say, to simplify it or for working out purposes e.g.

( + 1)2 = ( + 1)( + 1) So

( + 1)2 = 2 + 2 + 1

This can sometimes be called an Identity, since both sides of the equation are identical for all values.

2. Negative Numbers or values

Where negative numbers or variables are used, the following rules apply: -

Adding a negative number is the same as subtracting, e.g.

7 + (-3) is the same as 7 3 = 4.

Subtracting a negative number is the same as adding, e.g. (-5) - (-2) is the same as (-5) + 2 = -3.

Multiplying or Dividing the following also applies: -

Positive x positive = positive e.g. a x b = ab

Positive x negative = negative e.g. a x (-b) = -ab Negative x positive = negative e.g. (-a) x b = -ab

Negative x negative = positive e.g. (a) x (-b) = ab

3. Brackets

Where brackets (or parentheses) are used, the rule is simple - work out

everything inside the bracket first, where possible, then divide or multiply or add or subtract it, by everything outside the bracket.

Remember - BODMAS Brackets first then Outside then Divide then Multiply then Add then Subtract e.g.

4 (3), where = 12

Gives 4 (123)

Which is 4 x (4)

Which is 4 x 12 x (4) Which is 48 x (4) Which gives 192

Where brackets are used more than once, e.g. two sets of expressions

multiplied together, then everything inside one set of brackets is multiplied by everything inside the other set of brackets e.g.

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(a + 1)(b + 2) Gives ab + 2a + 1b + 2

Gives ab + 2a + b + 2

4. Transposing Formulae

When working with formulae, we are attempting to define relationships between variables. Depending on what we are trying to achieve, we may want

to work on the formula to write it in a different way. For example we may know the values of all but one of the variables, and the way the formula is written, does not allow us to easily calculate the remaining unknown variable.

What we are trying to do is to move the variable letter we are interested in to

the left hand side of the equation (or where we have only one variable, and the rest are numbers, have the variable (letter) on the left hand side of the equation and the number(s) on the right hand side. As we have seen in the

Definitions and Rules Section, an equation, or formula using an equals sign (=), is considered to be balanced.

In order to keep this balance, whatever we do on one side of the equation, we

must do on the other. So if we subtract a from one side of the equation, we must subtract a from the other. Similarly, if we multiply one side by a number or variable, we must multiply the other side by the same number or

variable. This is referred to as using Inverses, which can be considered as undoing part or all of the equation.

Adding or subtracting are the opposite or (inverse) of each other. Multiplying and dividing are the opposite of each other.

Examples of these follow below: -

i. 3b + 2 = 11 giving 3b + 2 2 = 11 2

giving 3b = 9 giving 3b/3 or 9/3

giving b = 3 You can see that we are undoing the equation, we are keeping the equation

in balance by doing the same thing on both sides of the equation (e.g. subtracting 2 from each side), and we have the variable on the left hand side

and the numbers of the right hand side. ii now consider 3b + a = 2c d

lets say we want to re-arrange or transpose the formula so that we

have the variable d on the left hand side, as that is the one we are interested in to calculate.

So 3b + a + d = 2c d + d Giving 3b + a + d = 2c

Giving 3b 3b + a a + d = 2c 3b a Giving d = 2c 3b a

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iii with division and multiplication, the same principles apply

So a = (-2c + 3)d

This time, lets say we are interested in c,

So, a/d = (-2c + 3) d/d Giving a/d = (-2c + 3)

Giving a/d = -2c + 3 Giving a/d + 2c = -2c + 2c + 3 Giving a/d + 2c = 3

Giving a/d + 2c a/d = 3 a/d Giving 2c = 3 a/d

Giving c = 3/2 (a/d)/2 Giving c = (3 (a/d))/2

A quick way of considering the inverse, is to remember that if you move a number or variable from one side of the equation to another, the inverse

applies e.g.

If we move a negative number it becomes positive If we move a positive variable it becomes negative If we move a number which is multiplying, it becomes a divider

e.g. 2a + 7 = 3C 2d

lets say we want to find d,

then 2a + 7 = 3c 2d so 2d + 2a + 7 = 3c so 2d = 3c 2a 7

so d = (3c 2a 7)/2

B. Interpolation Suppose we have conducted a scientific experiment in which we looked at the height

of a plant every day for a month. We have entered the heights into a spreadsheet for draw a graph. Unfortunately, there was one day when the measurement was not

taken. We could leave that data point empty or we could find a reasonable approximation to suggest what the height would have been on that day.

This process is called interpolation; estimating data that is within the range of data that we have collected. The opposite of this is where we look for information outside

the range of the data collected. For example, if we are interested to estimate the height of the plant after the last measurements we took, this is the process of extrapolation.

The simplest way of finding an approximate value between two existing values is to

assume that the function that connects them is linear - in other words that we can draw a straight line between them.

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Lets look at the following data:

1 1

3 7

If we want to find the value of , when = 2, as 2 is halfway between 1 and 3 you

might expect that the corresponding value will be 4, which is half way between 1

and 7. What we have effectively done is assumed the given data points can be joined by a straight line, and that the data point lies on that straight line.

However, rather than draw a graph each time, we can calculate the value

mathematically. If a straight line is joining points of our known data, then the

gradient is the same for all points on the line i.e. the difference between values

divided by the difference between values is the same for all points on the line.

So, for the difference in known values is 7-1 = 6

for , the difference is 3-1 = 2.

Now lets take our value of = 2 as the difference between the values, divided by

the difference between the values is the same for all points on the line

So 3+1 = - i.e. = 4.

Data produced for lighting equipment and solutions is often presented in a table for

specific values only. To find values in between those given, we can use linear interpolation e.g:

The Utilisation Factors (U.F.) for specified values of Room Index (R.I.) are as follows:

R.I. U.F.

3.0 0.58

4.0 0.61

If the calculated value of R.I. is 3.7, then the corresponding required U.F.

(U.F. req) can be found using the same calculation as detailed in the foregoing text:

(0.61 0.58)/(4.0 3.0) = (U.F. req 0.58)/(3.7 3.0) Giving, 0.03/1 = (U.F. req 0.58)/0.7

Giving, 0.03 x 0.7 = U.F. req 0.58 Giving, 0.021 + 0.58 = U.F. req

Giving, 0.601 = U.F. req You can also carry out the same calculation as follows:

(4.0 3.0/(0.61 0.58) = (3.7 3.0)/U.F.req 0.58

Giving (U.F. req 0.58) = 0.7 x 0.03

So U.F. req = 0.021 + 0.58 = 6.601

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C. Geometry and Trigonometry

Geometry is concerned with properties, relationships and measurement of points, lines, curves and surfaces. Trigonometry is concerned with angles and sides of

triangles. Angles

An angle is a measure of a turn between two lines. Angles are measured in degrees

().

Acute angle (less than 90) Right Angle (90)

Obtuse angle (between 90 and 180)

Straight line (180)

Reflex angle (between 180 and 360)

Full rotation (360)

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Right Angles, straight angles and full turns

A right angle is an angle of 90

Question 1

In this diagram a right angle is split into two other angles and 70.

What is the value of ?

How to work out the answer: We know that + 70 = 90

So = 20

A straight line is an angle of 180

Question 2

In this diagram a straight line is split into two other angles and 50. What is

the value of ?

50

The Answer.

70

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We know that + 50 = 180

so = 130

A full turn is an angle of 360

Question 3

In this diagram a full turn is split up into 3 angles , 100 and 230.

230

100

The Answer

We know that = 100 + 230 = 360

So + 330 = 360

So = 30

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Angles made by parallel lines

The two angles marked in the diagram below are called corresponding

angles and are equal to each other.

The two angles c and f marked in the diagram below are called alternate

angles or Z angles and are equal to each other.

The two angles marked in the following diagram are called vertically opposite angles and are equal to each other.

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The different types of Triangle

For any triangle the 3 angles add up to 180

So in the diagram below we know that + 50 + 70 = 180

+ 120 = 180

= 60

An equilateral triangle is one with all 3 sides equal in length and all 3

angles equal to 60.

70

50

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An Isosceles triangle is one with two sides equal in length and two equal

angles.

In any right angled triangle, the square of the longest side (the side opposite the right angle called the hypotenuse) is the sum of the squares of the other

two sides. This can be written in the formula. a2 + b2 = c2

c is the longest side

This is Pythagoras Theorem

Triangles determining angles or lengths of sides The relationship between angles and sides of triangles is determined using

formulae which describe tangents, sines and cosines. A

Hypotenuse

Adjacent side

B C Opposite side

c

a

b

a

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In the triangle on the previous page the angle between

AB and BC = 90 - a right angle. If we now look at angle a, the following relationships arise:

Sin a = Opposite / Hypotenuse = BC/AC

Cos a = Adjacent / Hypotenuse = AB/AC

Tan a = Opposite / Adjacent = BC/AB

These relationships can be remembered using the mnemonic: SOHCAHTOA

(Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse

Tan = Opposite/Adjacent)

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D. Radians

Radians take advantage of the relationship between and measures in circles.

There are 2 radians in a circle (i.e. a little over 6). We write this as 2cccc.

There are 2 radians in a full turn. We notate radians with a small superscript c.

To convert from degrees to radians.

Divide by 360 and multiply by 2 times ....

c = 2

360

To convert from radians to degrees.

Divide by 2 times and multiply by 360.

= 360yc 2

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E. Solid Geometry

A steradian is used to measure solid angles.

A steradian is related to the surface area of a sphere in the same way a radian is related to the circumference of a circle:

A radian cuts out a length of a circles circumference equal to the radius

A steradian cuts out an area of a sphere equal to the (radius)2

The name steradian is made up from the Greek stereos for solid and radian. The SI Unit abbreviation is sr.

Sphere vs Steradian

The surface area of a sphere is 4r2 The surface area of a steradian is just r2

So a sphere measures 4 steradians, or about 12.57 steradians.

Likewise a steradian is 1/12.57, or about 8% of a sphere.

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And because you are measuring an angle, it doesnt matter what size the

sphere is, it will always measure 4 steradians.

Example: a sphere with a radius of 1 (called the "unit sphere"):

has a surface area of 4, and a steradian would "cut out" an area of 1.

And now for some light revision ..

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The question paper should be treated as informal revision and not a test. It has been produced so that lecturers may have

an understanding of your requirements, and could possibly adapt areas of the course content to suit. For this reason we

would like you to return the completed answers to:

[email protected] no later than Friday 2nd September

2011.

H

3

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Question 1

What is the length H?

Question 2

What is the angle ?

Question 3

Show two ways to calculate angle ?

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Question 4

The above diagram represents a ceiling mounted downlight.

If the total beam angle is 30and the patch is 1m

diameter, what is height H?

Question 5

If we use a different downlight with a 60beam (same

mounting height)

a) What is patch diameter?

b) What is patch area?

H

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