Modelling optical lenses with Dynamic Geometry Software
Andreas Ulovec defined functions
Piecewise
1 Introduction
Freyja
Hreinsdóttir
In optics wof
hen Iceland
it comes down to show the path of rays of light through glass, lenses or systems of University
lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke 1 Introduction
glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y problems
removing one piece and putting another piece in. often
To see woccur
hat happens if you me.g.
ake afunctions
lens In applied
piecewise
defined
functions
naturally
describing prices
thicker, you These
have to take out the current lens and put in used
the new ne. Students can tlimits
he
n observe the Using a
of postage.
functions
are also
often
to odemonstrate
and discontinuities.
situation before the like
change and after the hange but it is these
not exactly a gradual inchange lets them dynamic
software
GeoGebra
itcis
possible toñ study
functions
morethat detail
and to use sliders
observe h
ow t
he p
ath o
f l
ight a
ctually c
hanges. W
e w
ant t
o d
emonstrate h
ow y
ou c
an s
how t
he p
ath o
f to examine how their definition can be altered slightly to create continuous functions, differentiable
light through dynamic geometry software (DGS).
functions
etc. a lens with the help of
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in 2 Defining
functions
in
GeoGebra
simulation, the pedagogic
value is not on
quite tintervals
he same), but complementing
it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of In GeoGebra
are otwo
to define
or by
light hits the gthere
lass surface f an oways
ptical lens, a part of functions
it gets reflec on intervals; using
ted back the
in a command
certain angle, Function
and another part penetrates The same happens when the using
the command
If.the glass and continues there, in another angle. light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings ñ light hits a plane surface
2.1 Reflection
When a ray of light hits a plane glass surface, a part of it is reflected. T he law of reflection
says that the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
Fig.1 To define the function above we write If[x < 2, 2x, x^2] in the input field.
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
optical lenses with [2x,-∞,2]
Dynamic and
Geometry Softwareto get the same graph although
It isModelling also possible
to write
Function
Function[x^2,2,∞]
this will result in two functions being defined. The If command has the advantage that the derivative of
Andreas Ulovec
the function can be calculated directly.
1 Introduction
If there
are three different intervals a nested If command can be used e.g.
If[xIn <optics 2, x²,
If[xit c<omes 3, xdown + 2,to 10
- x]]
in othe
figure
below:
when show the presults
ath of rays f light through glass, lenses or systems of lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become m
ore c
omplex, a
nd f
rom t
he e
quations a
lone i
t w
ould b
e d
ifficult t
o s
ee w
hat Fig. 2 A function f(x) defined on three intervals. Its derivative g(x) ishappens. shownWinith red. The derivative is
DGS it is possible to simulate the properties o
f a
l
ens w
ithout a
ctually h
aving t
o u
se a
l
ens, laser light, undefined at x = 2 and at x = 3.
etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings 3 Sliders
ñ light hits a plane surface
2.1 RGeoGebra
eflection it is very easy to investigate the effect of changing the value of one (or more)
Using
parameters
theglass definition
To define suchTahe parameter
select the
When a ray occurring
of light hits ain
plane surface, a of
part aof function.
it is reflected. law of reflection
says slider
that tool
the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
and click on the Graphics view. When that is done a small window opens:
Fig. 3 This window is used for setting the interval of the parameter, the increment etc.
After a slider b has been defined it can be used in the definition of a function, e.g. 𝑓 𝑥 = 𝑥 ! + 𝑏,
and its value can be changed using the mouse.
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling sliders
optical lenses ith Dynamic Geometry Software
4 Using
to w
define
continuous
functions
Andreas Ulovec
We consider the following problem: determine values of the parameters a, b and c such that the
function
1 Introduction
𝑥 − 𝑎 if 𝑥 < −1
In optics when it comes down to show the path of rays of light through glass, lenses or systems of ℎ 𝑥 ñ= the 𝑥 !experiments + 𝑏 if are −q1 𝑥 ≤ 3
lenses, many physics teachers groan uite ≤complex
, and you need a lot of 𝑐
−
𝑥 otherwise equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke is continuous.
glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be Wedone solve
problem
by defining
sliders
using iaf ynested
command to define h(x).
b ythis
removing one piece and putting three
another piece in. a,
To b,
see cwand
hat happens ou make aIf lens Thethicker, values
of
the
sliders
can
now
be
changed
to
make
the
function
continuous.
you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings ñ light hits a plane surface
Fig. 4 We change the definition of the function by moving the sliders.
2.1 change
Reflection
If we
the values of a, b and c the graphs move up and down so it is very easy to find values
such
that
the
graphs
connected.
When a ray of light hits are
a plane glass surface, a part of it is reflected. T he law of reflection
says that the Define
angle of the
incidence (between the in
ray GeoGebra
of light and the ) is equal to the angle oof
f reflection:
Task:
function
above
and findnormal
the appropriate
values
the sliders. Do you
get more than one solution? Solve the problem algebraically as well.
A more complicated situation arises if the parameters not only have to do with the location of the
graph but also its shape.
Task: Define the function
𝑥 + 5 if 𝑥 < −1
𝑔 𝑥 = 𝑎𝑥 ! + 𝑏 if − 1 ≤ 𝑥 ≤ 3
10 − 𝑥 otherwise in GeoGebra and find values of a and b such that the function is continuous. Solve the problem also
algebraically.
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling optical lenses with Dynamic Geometry Software
5 Differentiable
functions
Andreas Ulovec
In problems like the ones given above it is very noticeable that at the connecting points we get a
corner
or a break in the function. This is because even though the function is continuous at the points it
1 Introduction
is not differentiable.
In optics when it comes down to show the path of rays of light through glass, lenses or systems of ! !!! ! !(!)
Thelenses, derivative
of a teachers function
𝑓(𝑥) at xñ=the a eisxperiments definedare asqthe
of a lot of as ℎ
many physics groan uite climit
omplexof the quotient
, and you need !
equipment. I
t i
s d
ifficult e
nough t
o s
how a
r
ay o
f l
ight i
n a
ir ñ
you n
eed s
moke, d
ust o
r a
ny o
ther w
ay goes to 0 and the function is differentiable at x = a if the limit exist. This limit is the slope of a tangent
making light visible. To show the path of light in materials, you need special equipment ñ smoke lineof to
the graph of 𝑓(𝑥) at the point (𝑎, 𝑓(𝑎)). For the limit to exist it needs to be the same
whether
glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be we done approach
the value a of x from the right or from the left. We can think of the limit as the slopes of
b y removing one piece and putting another piece in. To see what happens if you make a lens secant
lines
going
points
+Students ℎ, 𝑓(𝑎can + tℎ))
the
tangent
thicker, you have to tthrough
ake out the the
current lens (𝑎,
and 𝑓(𝑎))
put in the and
new (𝑎
one. he approaching n observe the line as
h goes
to
0.
In
the
cases
we
have
a
break
in
the
function
these
limit
exist
from
the
left
and
from the
situation before the change and after the change ñ but it is not exactly a gradual change that lets them right
but
they
are
not
equal.
observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings ñ light hits a plane surface
2.1 Reflection
When a ray of light hits a plane glass surface, a part of it is reflected. T he law of reflection
says that the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
Fig. 5 Secant lines approaching the tangent line at a certain point
Task: Demonstrate the above by defining the tangent to f(x) at x = 2 (using the tangent tool
),
defining a slider h and a line through the points (2 + ℎ, 𝑓(2 + ℎ )) and (2, 𝑓(2)) and then using the
mouse to change the value of h and watch the secant approach the tangent.
Below we describe how we can join two different quadratic functions to get a piecewise defined
function that is both continuous and differentiable at the meeting point.
Open a GeoGebra worksheet and define four sliders b, c, d and e . Define two quadratic functions
𝑓 𝑥 = −𝑥 ! + 𝑏𝑥 + 𝑐 and 𝑔 𝑥 = 𝑥 ! + 𝑑𝑥 + 𝑒 and find values of the sliders such that 1, 𝑓 1 =
1, 𝑔 1 . This ensures that the graphs of the two functions intersect at this point. Now use the tangent
tool to get the tangents to both functions at the meeting point.
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling optical lenses with Dynamic Geometry Software
Andreas Ulovec
1 Introduction
In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the Fig.6
pedagogic
value intersect
is not quite tat
he the
same), it. Inot
t can the
as wsame.
ell be useful The graphs
𝑥 b=ut 1 butcomplementing
the tangents are
for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of If we
now
piecewise
function
light hits tdefine
he glass saurface of an optical lens, a pℎ(𝑥)
art of isuch
t gets rthat
eflec
ted back in a certain angle, and another part penetrates the glass and continues there, in a𝑓
nother a
ngle. The same happens when the 𝑥 if 𝑥 ≤ 1
mathematics is required to calculate the angle in light reaches the other surface of the lens ℎ 𝑥 ñ=again 𝑔 𝑥there otherwise
which the light is ref
lected and refracted. For ideal lenses, is an easy equation calculating these effects ñ
but t
his i
s j
ust a
m
odel, a
nd i
t d
oes w
ork w
ell o
nly w
ith t
hin lenses and with light falling in we get the function below:
near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings ñ light hits a plane surface
2.1 Reflection
When a ray of light hits a plane glass surface, a part of it is reflected. T he law of reflection
says that the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
Fig. 7 The function ℎ(𝑥) (black graph) is not differentiable at 𝑥 = 1.
To redefine the function ℎ(𝑥) so that it is differentiable we need to change the values of the sliders
such that the tangents are the same.
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling optical lenses above
with Dand
ynamic eometry Software
Task:
create the
worksheet
find G
values
of the
sliders such that the function h(x) is
differentiable at every point.
Andreas Ulovec
1 Introduction
In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings ñ light hits a plane surface
2.1 Reflection
Fig. 8 Here we have one solution to the problem. The function ℎ(𝑥) (pink graph) is differentiable
When a ray of light hits a plane glass surface, a part of it everywhere.
is reflected. T he law of reflection
says that the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling optical lsecond
enses with Dand
ynamic Geometry Software
7 Connecting
third
degree
polynomials
Andreas Ulovec
We can make constructions similar to the ones above using other types of functions e.g. second and
third degree polynomials as seen below.
1 Introduction
In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations Fig.be 9difficult to see what happens. With become more complex, and from the equations alone it would it is possible to simulate the properties of a lens without actually aving to use Aa so
lens, laser light, TheDGS functions
𝑓(𝑥)
and 𝑔(𝑥)
have a common
tangent
at hthe
point
we
can
define a differentiable
etc. B
ut e
ven f
or
the D
GS, w
e n
eed m
athematics t
o c
reate t
he s
imulation i
n t
he f
irst p
lace.
function using pieces of 𝑓(𝑥) and 𝑔(𝑥) on intervals separated by 𝑥 = 0.
Task:
make
your own examples
like this
using
third degree polynomials.
2 Easy beginnings ñ light hits a e.g.
planetwo surface
2.1 Reflection
6 Removing breaks
When a ray of light hits a plane glass surface, a part of it is reflected. T he law of reflection
says that angle incidence method
(between tto
he rredefine
ay of light aand function
the normal
) is equal to the ain
ngle of reflection:
Wethe can
useof a similar
on
a small
interval
order
to remove a break in
the graph of a function.
Fig. 10 A simple example of a function that has breaks at several points
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling optical lenses with Dynamic Geometry Software
Andreas Ulovec
1 Introduction
In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the Fig.the 11other A break
graph removed
bymredefining
function
on athe small
light reaches surface inof the
the lens ñ again athematics is rthe
equired to calculate angle interval
in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in the centre of the lens. ith tH
hicker lenses and light m break we want
ore tooffremove
-­‐centre, and
the calculations Wenear define
two points
GW
and
on opposite
sitesbeing of the
a point I on the
become m
ore c
omplex, a
nd f
rom t
he e
quations a
lone i
t w
ould b
e d
ifficult t
o s
ee w
hat h
appens. W
ith line x = 4. We then use the command FitPoly[{G,I,H},2] to get a second degree polynomial that goes
DGS it ithese
s possible to simulate properties of a lens without actually having to laser through
three
pointsthe and
the tangent
tool
to get
a tangent
touse thea lens, graph
oflight, this polynomial at the
etc. B
ut e
ven f
or
the D
GS, w
e n
eed m
athematics t
o c
reate t
he s
imulation i
n t
he f
irst p
lace.
point G. We then move the point I (it is fixed on the line 𝑥 = 4) until this tangent coincides with the
segment from (0, 0) to (4, 5). After removing help lines, changing colours etc we get the graph below.
2 Easy beginnings ñ light hits a plane surface
2.1 Reflection
When a ray of light hits a plane glass surface, a part of it is reflected. T he law of reflection
says that the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
Fig. 12 One of the breaks has been removed
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.
Modelling ptical lenses with Dynamic Geometry Software
Task:
create theoconstruction
above
Andreas Ulovec
References
1 Introduction
[1]
GeoGebra, downloadable from http://www.geogebra.org.
In optics when it comes down to show the path of rays of light through glass, lenses or systems of lenses, many physics teachers groan ñ the experiments are quite complex
, and you need a lot of equipment. It is difficult enough to show a ray of light in air ñ you need smoke, dust or any other way of making light visible. To show the path of light in materials, you need special equipment ñ smoke glass lenses etc. Now thatís
not always available, and adjustments to the system can usually only be done b y removing one piece and putting another piece in. To see what happens if you make a lens thicker, you have to take out the current lens and put in the new one. Students can the
n observe the situation before the change and after the change ñ but it is not exactly a gradual change that lets them observe how the path of light actually changes. We want to demonstrate how you can show the path of light through a lens with the help of
dynamic geometry software (DGS).
This material can be useful for science teachers, who can use it to model experiments with lenses, reflection and refraction ñ not instead of the actual experiment (if one sees experiments only in simulation, the pedagogic
value is not quite the same), but complementing it. It can as well be useful for mathematics teachers. Well, now where is the mathematics? There is a lot of it in there! If a ray of light hits the glass surface of an optical lens, a part of it gets reflec
ted back in a certain angle, and another part penetrates the glass and continues there, in another angle. The same happens when the light reaches the other surface of the lens ñ again mathematics is required to calculate the angle in which the light is ref
lected and refracted. For ideal lenses, there is an easy equation calculating these effects ñ but this is just a model, and it does work well only with thin lenses and with light falling in near the centre of the lens. With thicker lenses and light being m
ore off -­‐centre, the calculations become more complex, and from the equations alone it would be difficult to see what happens. With DGS it is possible to simulate the properties of a lens without actually having to use a lens, laser light, etc. But even for the DGS, we need mathematics to create the simulation in the first place.
2 Easy beginnings ñ light hits a plane surface
2.1 Reflection
When a ray of light hits a plane glass surface, a part of it is reflected. T he law of reflection
says that the angle of incidence (between the ray of light and the normal ) is equal to the angle of reflection:
Fig.1 Reflection of light at a plane surface.
This project has been funded with support from the European Commission in its Lifelong Learning Programme
This project has been fund
ed with suppo rt from the This
European Commissionreflectsin the
its Lviews
ifelong Lonly
earning (510028-LLP-1-2010-1-IT-COMENIUS-CMP).
publication
ofProgramme
the authors, and the
(510028
-­‐LLP
-­‐1-­‐2010
-­‐1-­‐IT
-­‐COMENIUS
-­‐CMP). T
his p
ublication r
eflects t
he v
iews o
nly o
f t
he a
uthors, a
nd he Commission cannot be held responsible for any use which may be made of the information tcontained
therein.
Commission cannot be held responsible for any use which may be made of t
he information contained therein.