Download ILP Exterior Lighting Diploma Fundamentals of Maths for Lighting

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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.
ILP ELD Fundamentals of Maths for Lighting
Issue 2.2 Autumn 2016
© ILP
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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
𝑥/3 means 𝑥 ÷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 (𝑥 + 1),
if 𝑥 = 2,
then the expression (𝑥 + 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
<= means less than or equal to
Formulae (the plural of Formula) are referred to as Equations where the
equal sign is used e.g.
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(3𝑥 + 3) = 2𝑥 + 9, which holds true for only certain values of 𝑥
(in this case, 𝑥 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.
(𝑥 + 1)2 = (𝑥 + 1)(𝑥 + 1)
So
(𝑥 + 1)2 = 𝑥2 + 2𝑥 + 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.
4𝑥 (𝑥÷3), where 𝑥 = 12
Gives 4𝑥 (12÷3)
Which is 4𝑥 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.
giving
giving
giving
giving
3b + 2 = 11
3b + 2 – 2 = 11 – 2
3b = 9
3b/3 or 9/3
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
let’s 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
Giving
Giving
Giving
ILP ELD Fundamentals of Maths for Lighting
3b + a + d = 2c – d + d
3b + a + d = 2c
3b – 3b + a – a + d = 2c – 3b – a
d = 2c – 3b – a
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iii with division and multiplication, the same principles apply
So
a = (-2c + 3)d
This time, let’s say we are interested in c,
So,
Giving
Giving
Giving
Giving
Giving
Giving
Giving
Giving
a/d = (-2c + 3) d/d
a/d = (-2c + 3)
a/d = -2c + 3
a/d + 2c = -2c + 2c + 3
a/d + 2c = 3
a/d + 2c – a/d = 3 – a/d
2c = 3 – a/d
c = 3/2 – (a/d)/2
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
let’s say we want to find d,
then
B.
2a + 7 = 3c – 2d so 2d + 2a + 7 = 3c
so 2d = 3c – 2a – 7
so d = (3c – 2a – 7)/2
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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Let’s look at the following data:
𝑥
𝑦
1
3
1
7
If we want to find the value of 𝑦, when 𝑥 = 2, as 2 is halfway between 1 and 3 you
might expect that the corresponding 𝑦 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 𝑦 values divided by
the difference between 𝑥 values is the same for all points on the line.
So, for 𝑦 the difference in known values is 7-1 = 6
for 𝑥, the difference is 3-1 = 2.
Now let’s take our value of 𝑥 = 2 – as the difference between the 𝑦 values, divided by
the difference between the 𝑥 values is the same for all points on the line –
So 3+1 =
- i.e. 𝑦 = 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.
3.0
4.0
U.F.
0.58
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,
Giving,
Giving,
Giving,
0.03/1 = (U.F. req – 0.58)/0.7
0.03 x 0.7 = U.F. req – 0.58
0.021 + 0.58 = U.F. req
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
ILP ELD Fundamentals of Maths for Lighting
Issue 2.2 Autumn 2016
© ILP
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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
Full rotation (360°)
ILP ELD Fundamentals of Maths for Lighting
Issue 2.2 Autumn 2016
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360°)
8
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 𝑥° and 70°.
What is the value of 𝑥°?
70°
𝑥°
How to work out the answer: We know that 𝑥 + 70° = 90°
So 𝑥 = 20
A straight line is an angle of 180°
Question 2
In this diagram a straight line is split into two other angles 𝑦° and 50°. What is
the value of 𝑦°?
50°
𝑦°
The Answer….
ILP ELD Fundamentals of Maths for Lighting
Issue 2.2 Autumn 2016
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We know that 𝑦 + 50° = 180°
so 𝑦 = 130°
A full turn is an angle of 360°
Question 3
In this diagram a full turn is split up into 3 angles 𝑧°, 100° and 230°.
𝑧°
230°
100°
The Answer
We know that 𝑧 = 100° + 230° = 360°
So 𝑧 + 330° = 360°
So 𝑧 = 30°
ILP ELD Fundamentals of Maths for Lighting
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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.
ILP ELD Fundamentals of Maths for Lighting
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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 𝑥 + 50° + 70° = 180°
70°
50°
𝑥°
𝑥 + 120° = 180°
𝑥 = 60°
An equilateral triangle is one with all 3 sides equal in length and all 3
angles equal to 60°.
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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
c
b
a
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
aa
Hypotenuse
Adjacent side
B
C
Opposite side
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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 𝜋 and measures in circles.
There are 2𝜋 radians in a circle (i.e. a little over 6). We write this as 2𝜋c.
There are 2𝜋 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 𝜋.
𝑦c = 2𝜋𝑥°
360
To convert from radians to degrees.
Divide by 2 times 𝜋 and multiply by 360.
𝑥° = 360yc
2𝜋
ILP ELD Fundamentals of Maths for Lighting
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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 circle’s 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 4𝜋r2
The surface area of a steradian is just r2
So a sphere measures 4𝜋 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 doesn’t matter what size the
sphere is, it will always measure 4𝜋 steradians.
Example: a sphere with a radius of 1 (called the "unit sphere"):
has a surface area of 4𝜋, and a steradian would "cut out" an area of 1.
And now for some ‘light revision’ ……..
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The following questions are for revision only and not a test.
𝛳
H
3
∝
4
Question 1
What is the length H?
Question 2
What is the angle θ?
Question 3
Show two ways to calculate angle ∝?
ILP ELD Fundamentals of Maths for Lighting
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𝛳
H
Question 4
The above diagram represents a ceiling mounted downlight.
If the total beam angle θ is 30°and the patch is 1m
diameter, what is height H?
Question 5
If we use a different downlight with a 60°beam (same
mounting height)
a) What is patch diameter?
b) What is patch area?
ILP ELD Fundamentals of Maths for Lighting
Issue 2.2 Autumn 2016
© ILP