WEBVTT
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ok so last time we talked about certain kinds
of maps from r two to r two and we called
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these the rigid digit motions of the plane
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and we studied a few examples so the kinds
we looked where maps which we called translations
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then there where reflection the looked at
rotations so these were all examples of maps
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which had the following property that they
did not change either lengths or angles so
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they sort of just move the plane in some rigid
fashion and we also studied things like a
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invariant figures under these these type
of maps and so on so now what we want to do
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next is to talk about non rigid motions so
let's again start doing these by examples
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so now we look at things which do not preserve
lengths or angles or things like that in
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general so the first example will be the following
map so remember we are looking at the maps
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from the plane to the plane so let's define
f to be the function which takes the point
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x comma y on the plane to the point let
say three x comma two y so for let's start
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with two x comma two y which is again a point
on the plane and let's try and study what
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this function does ok the outside all it does
is maps multiplied the x co ordinate by two
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and multiply the y co ordinate by two now
what this does geometrically speaking is the
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following it takes this point x comma y so
i just joined at by line segment to the origin
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and it maps this point to the point along
the same line but twice as for so it maps
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it to the point two x comma two y so this
i should think of the functions it takes it
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point x y and sort of blose it up by a factor
of two are stretches that line segment by
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a factor of two and maps it to the the n point
of the resulting line segment so this is some
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times called are dilation or a stretch by
a factor of two dilation by a factor of two
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so dilation here is word which is use to mean
either sort of a stretch or a shrink by some
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factor ok so let's do the same sort of analysis
that we did for the rigid motions which is
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to try and understand what this map does to
other kinds of figures on the plane so for
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instants let's study the effect of this map
on lines ok so i will i will take a typical
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line on the plane so let say here's the line
on the plane and what we want to do is to
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apply this function f to each point of this
line ok so it take each point of the line
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apply the function f you get a new point and
you join all the the resulting the new points
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and see what sort of figure it becomes ok
so for instants for here it appears if you
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take this point here on the y axis the
function doubles this distance it maps in
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to something again on the y axis but twice
as for similarly this point on the x axis
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will map to something along the x axis but
which is twice the distance from here and
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the same sort of wholes for every point in
the middle you you just join it to the origin
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short of blow it by a factor of two so this
point maps to this point so on so if you short
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of try and join these what you will find
is that it's again a line ok it's a line but
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sort of twice as for from the origin as the
original line of course this is a just pictorial
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thing you know I am not really justify this
you could try and convince yourself of this
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by plotting of few points and so on but of
course the most conclusive proof is really
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by know writing down actually equations and
checking that this function does indeed map
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line to a line ok so how does one one go about
checking things like this so here's the
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well let's let's do the following let's call
this function x y what it maps to the point
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xy the image of the point x y under this map
let's call it x prime comma y prime s you
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want think of this function as short of sending
points here to points so it maps it to a point
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who's co ordinates are now x dash y dash this
point here is and so on ok so so it from avoid
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at function that's what the function does
so now let's do the following let's figure
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of what this this becomes on the on the new
axis so first let me take a line here so i
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will take a line l so what's the equation
typically equation of a line when the many
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different ways of writing it but so here's
one familiar form the equation of a line it's
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call it y equals m x plus c so this the equation
of the line so the line l satisfies this equation
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has equation y equals m x plus c in other
words a typical point x comma y which lies
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on this line will satisfy this equation ok
so that's what it means to say the line has
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this equation so let's take a point x comma
y on this line l so in other words x y satisfy
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this equations and apply the function f two
it ok i will apply f two this point xy there
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by obtaining an other point x prime y prime
on the plane and the question out really is
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the following as you let x comma vary along
this line as yo move them as you move this
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point along the line you want to know how
x prime y prime changes how does that move
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[vocalized-noise] in other wards if you know
the equation that x comma y satisfies can
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you some how reduce an equation that x prime
y prime satisfies that's really the question
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here so let's try and and reduce this so what
is x prime by definition it two x y prime
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by definition is two i ok now we know x and
y satisfy this relation so that automatically
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gives your x prime y prime because x is nothing
but x prime divided by two and y is nothing
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but y prime divided by two so from this you
conclude the following y prime divided
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by two which is this basically y must be the
same as m times x which is x prime divided
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by two plus c ok so observe that the point
x prime comma y prime satisfies that new equation
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here which of course you can simplify to y
prime equals m times x prime plus two c ok
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so this new equation is again the equation
of a line which we now call l dash the line
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line l dash and this new line observe is well
it's the equation looks like moralize like
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the the old line l except that c is replace
by two c ok so remembering that c is sort
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of like the y intersect it's a line who's
y intersect twice the y intersect of the original
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line ok so lines maps to lines that sort of
the the moral of this analysis lines map under
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this function f back to lines ok so that's
one one aspect of what this map to us the
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dilation map [vocalized- noise] of course
so it sort of also easy to see what other
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sorts of figures would transform to under
this map for instants if you take a circle
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say of radius one of or say any arbitrary
radius r and apply this function f to it at
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the circle has radius r then what the function
is going to do to each point of the circle
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it's going to map it to the point which is
at distance two r from the origin so all you
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going to get is a circle who's radius is twice
the original radius and so on so this is now
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circle who's radius is two r so let's going
to be the image of this this circle under
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this map f ok so
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some what better circle so it's call it circle
of radius two r and of course well other kinds
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of figures sort of triangles and so on so
so this where triangle on the plane then let's
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do the following let's think of what this
function does so imagine joining the three
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vertices to the origin in this way and what
this function does of course is to dilated
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by factor of two which sort of pushes everything
out by factor of two so imagine now extending
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these lines all are them to the length and
the resulting three points are the images
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of the vertices of the original triangle now
observe whenever you want to figure out
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the image of some sort of polygonal region
so at region which is bounded by lines so
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for instants here it's a triangle if i want
to know how what happens to this triangle
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under this map f it's actually enough to just
figure of out to what happens to the three
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vertices ok so if i have the three vertices
a b and c the map two let's call them a prime
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b prime and c prime now a maps to a prime
b maps to b prime and we now know the one
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sort of important fact that this map sense
lines to lines so the map so the line joining
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a b must therefore go again to some line and
it must therefore be the line joining a prime
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and b prime because line is uniquely determine
as one as you know two points on the line
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ok so this line had better map to this line
there are no choices similarly b maps to b
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prime c maps to c prime the line b c must
maps to some line that's the other fact we
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know therefore it first map to the line line
joining the b prime and c prime similarly
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ok so at least some what convincing argument
i hope which tells you that this triangle
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here will just map to this some what bigger
and notice this bigger triangle here has all
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side lengths being twice the original side
lengths ok so you can see this by using similar
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triangles for instants it that's familiar
the angles remain the same so these angles
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are all preserve the same angles as in the
original triangle but all side lengths are
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no twice the original twice and to proof this
for instants you need to you some elementary
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similar triangles because the side four a
is half the length of the side o a dash ok
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so you will sort of have to plane with some
triangles here so i leave that exercise for
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you but observe that this triangle a dash
b dash c dash has well it's a the side like
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this is similar to the original a b c and
the all side lengths are double the angles
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of course remain the same also because sort
of one of the properties on similar triangles
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ok so lengths double angles remain the same
and notice that what happens to the areas
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so what is this transformation do to areas
so that's the sort of the final point to note
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what what happens to areas so observe that
if i have circle of radius r it's areas pi
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r square and the circle of radius s two r
has area which is four times pi r square because
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i need to do five times two r whole square
so this resulting bigger circle here has four
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times the area similarly in the case of the
triangle all side lengths are double so if
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you wish the altitude of the triangle will
also the double the original altitude and
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so the area of which is sort of half base
time multitude is also multiplied by a factor
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of four ok so areas get multiplied by a factor
of four ok so lengths get multiplied by a
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factor of two areas get multiplied by factor
of four angles remaining on changed so the
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those are the various features of this
particular map it sense everything to twice
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twice itself so it's a dilations some means
called uniform dilation by a factor of two
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now it's taken an other so this is some times
called inhomogeneous dilation so this quick
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this slightly this is example two it from
call it inhomogeneous
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inhomogeneous dilation so what is this to
well f of x comma y is for instants three
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x comma two y so here's an example of a map
which sense x y to three x comma two y so
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to really understand why i called it an inhomogeneous
dilation known uniform by dilation so imagine
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what is map does to say a square of side length
one so imagine i have a square here one zero
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zero one one one that's the origin so the
first fact here is again that lines map to
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lines ok so this this dilation this inhomogeneous
dilation has the same property that it maps
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lines to lines so lines map to lines so that
i leave as an exercise for you to check pretty
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much by using the same calculation which is
if you write down equation of a line so
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you take a point x y on that line and then
you figure of what equation x prime and y
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primes satisfy x prime is three x and y prime
is two y so so may be will infact do this
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in a minute because we also want to see
what happens triangles so the key thing
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here is if you take a square of a side length
one and you map you figure out of what happens
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to it under this map so here's what you will
notice that the point one comma zero maps
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to the point three comma zero and the point
zero zero one maps to zero two so what you
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going to get now in know on the x square but
rather are rectangle right ok so and the point
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one comma one if you sort of figure out of
what it does it goes to three comma two right
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so the square of side one as now become a
rectangle of sides three and two [vocalized-noise]
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this was originally one map so this a reason
why and calling it inhomogeneous dilation
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meaning it is sort of like a dilation but
it is dilation by different amounts along
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different directions so the x axis suffers
a dilation by a factor of three there is a
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y axis is only stretch by a factor of two
ok so this stretching factors at different
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along the different actions and of course
so the shapes of the the original figures
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are very much going to disorsted here for
instants becomes a rectangle similarly
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if you try to figure of what happens to a
circle of this radius one like we did in the
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earlier example so again long the x axis it
gets stretch by the factor of three and the
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y axis by a factor of two and what you therefore
get is in fact in ellipse with major axis
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three and minor axis two ok so this map actually
sense a circle two and elements ok so again
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the result of the fact that it is stretching
differently along different actions and the
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other important thing is that angles so now
what can we say about lengths in this case
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as we said lengths do get stretch but by different
amounts along different directions so we can't
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say better much much more then there that
so various facts lengths get dilated by different
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amounts along different direction get dilated
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differently along different directions angles
so let's consider angle so here's an important
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feature of this angle are not preserved any
more
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ok so we looked at the earlier case of a dilation
uniform dilation by a factor of two that mapped
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any any figure to a similar figure ok where
all the angles send at up the preserve if
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you try doing this to the inhomogeneous dilation
you find that angles are not preserve so here's
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an example let's take the the line just
the x axis that's one of the lines and let
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say let's take the line y equals x as the
line l two so that makes of forty five degree
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angle so here's a forty five degree angle
between the lines l one and l two and now
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if you figure of what happens to these to
lines under this inhomogeneous dilation so
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l one just maps back to l one so i am just
going to tell you the answer check that this
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is in fact correct l one the x axis maps back
to the x axis where as the the line l two
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whose slope is one ok y equals s now maps
to something of slope two by three ok so it
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it will sort of move a little closer to the
x axis this is what happens to the line l
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two so check that l two becomes a line who's
slope is two by three the original slope here
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is y equals x so that has slope one
ok so this angle here theta is strictly smaller
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than the forty five degree so what it's done
is because of this sort of stretching differently
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along the two different directions it has
ended up making the angle different so here
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an example where angles are not preserve so
pretty much are first example because of course
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rigid motions that we talked about last time
preserve everything preserve lengths angles
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areas pretty much every where you can think
they in the preserve length dilations they
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don't preserve lengths of course they don't
preserve areas but they do preserve angles
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now inhomogeneous dilations are the first
examples of things which also don't preserve
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angles ok so the desktop thing sufficiently
that even angles change and because we talk
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about lengths or angle so what are what about
areas ok so let's just look at the two examples
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of figures which get transformed under this
map so one we said you take the square of
21:15.070 --> 21:22.000
side one it becomes a rectangle of sides three
and two so the area here is a one where it's
21:22.000 --> 21:28.720
there is a six similarly a circle here of
radius one has area pi or square it's pi where
21:28.720 --> 21:34.611
is ellipse has area well it may not have a
formula which is very well known but the area
21:34.611 --> 21:40.270
of ellipse if you know the minor and major
axis so in this case area in the ellipse times
21:40.270 --> 21:45.930
pi times out to be pi times instead of r square
it's a into b the a and b are the minor and
21:45.930 --> 21:52.980
major axis so in this case area of ellipse
is time pi time three times two so let six
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pi so observe again that the ellipse has area
which is six time the area of the original
21:58.770 --> 22:04.960
preserve that ok so both this this square
becomes a rectangle of area which is six times
22:04.960 --> 22:09.920
the circle becomes an ellipse again who's
a area is six times so the areas in fact get
22:09.920 --> 22:14.890
multiplied by the factor of six so areas at
least looking at are two examples so i am
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not really proving anything here it seems
at least from this example that the area is
22:19.790 --> 22:30.730
multiplied by a factor of six so those are
the futures again of the inhomogeneous dilation
22:30.730 --> 22:41.550
now let's do one more example sort of in three
the function f of x y just define t be the
22:41.550 --> 22:54.120
x minus y comma x plus y so here's an formula
for what the function does again let's study
22:54.120 --> 23:00.790
this pretty much using the the sames at a
point for a start what does this function
23:00.790 --> 23:09.090
do to line on the plane and again i claim
that lines map to lines ok so again exercise
23:09.090 --> 23:18.040
check that this is in fact correct so it is
very similar to everything we have seen until
23:18.040 --> 23:24.280
now in the sense that line certainly map to
lines ok so that of course makes it easier
23:24.280 --> 23:29.990
to figure out what it does to polygonal region
regions which are bounded by lines so let's
23:29.990 --> 23:45.610
do that next so again and take square of side
one it's take so here are the four vertices
23:45.610 --> 23:51.150
of the square of side one and let's figure
out what happens to each of these four points
23:51.150 --> 24:06.650
under this transformation f so it's apply
the transformation f to this
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so for a start let's apply so observe the
origin zero comma zero if i apply f two it
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i will just get zero minus zero zero plus
zero so the origin goes to the origin so the
24:19.130 --> 24:27.110
origin just remain where it is the point one
comma zero maps to one minus zero and one
24:27.110 --> 24:35.630
plus zero so that's just one comma one so
the point one one is one divided this is now
24:35.630 --> 24:42.460
the point one comma one under this maps to
zero comma two so that now point on the y
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axis so for is i have gotten so so for yes
what i have gotten one on the side like this
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other like that and if you look at what happens
to the third point zero comma one that maps
24:57.560 --> 25:06.760
to minus one comma one so that's this guy
ok and again as we said if you figure of what
25:06.760 --> 25:11.809
happens to the four vertices you are more
or less done because the the line segment
25:11.809 --> 25:17.450
joining them has no choice but to map to the
line segment joining the the the resulting
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points ok so what we have really is this
maps to the square here and observe that what
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is this this is again a square but the square
has side which is well what's the length of
25:32.580 --> 25:38.080
the side it's square root two times the length
of this side o this is a square of side one
25:38.080 --> 25:45.740
where as this is a square of side root two
ok and well what else is there it the area
25:45.740 --> 25:51.600
therefore is so what happens to areas the
area of this square becomes twice the area
25:51.600 --> 26:00.200
equals twice the original then
26:00.200 --> 26:06.180
ok and what else to seems to be happening
here so the length of this line segment this
26:06.180 --> 26:12.760
was a line segment length one it map to line
segment who's length is square root two times
26:12.760 --> 26:20.220
so len length is dilated by a factor of square
root ok so at least for for this square all
26:20.220 --> 26:27.540
of these ok so at least for this square all
of these sides are length one and the resulting
26:27.540 --> 26:32.930
sides there have length square root two ok
so it seems at least that length of line segment
26:32.930 --> 26:38.020
is multiplied by a factor of square root two
therefore the area is multiplied by a factor
26:38.020 --> 26:42.510
of two but one nice thing here seems to be
the angle are unchanged so the angles this
26:42.510 --> 26:48.280
was a square and well that seems to remain
is square in the other case of square so at
26:48.280 --> 26:53.490
least in this example the angle are unchanged
and but in the same time observe this has
26:53.490 --> 26:57.980
you know this is sort of the kind of thing
that happen if you when hard dilations when
26:57.980 --> 27:02.240
you dilated something by of factor of square
root two you would pretty much of have all
27:02.240 --> 27:07.030
these things happening lengths go by square
root two angles don't change area becomes
27:07.030 --> 27:12.600
twice and so on but this map is clearly not
just a dilation the dilation would had multiplied
27:12.600 --> 27:18.471
this would have map this square just to a
square in along the x and y axis of side length
27:18.471 --> 27:25.300
root two instead this is mapping it two some
sort of a rotated square ok so you you still
27:25.300 --> 27:29.240
have a square but it's rotated by an angle
forty five degree so observe this is a forty
27:29.240 --> 27:39.070
five degree ok so what it seems to be really
is the following and we look at this next
27:39.070 --> 27:46.310
time this map f really does two things at
once ok the first thing it does is it takes
27:46.310 --> 27:53.299
the original square here and rotates it
by an an angle of degrees and then it dilates
27:53.299 --> 27:58.340
lengths by a factor of square root two ok
so at least here's what it seems like f is
27:58.340 --> 28:06.850
doing it is doing two thing first it is a
rotation so f seems to other following description
28:06.850 --> 28:12.740
it seems like a rotation by forty five degrees
at least for the unit square followed by at
28:12.740 --> 28:26.080
dilation followed by dilation by factor of
root two so it's a some sort of a composite
28:26.080 --> 28:31.059
map it it something which have two things
that once and so we see that the natural way
28:31.059 --> 28:37.380
to thing about this in terms of compositions
right so we will in fact revisit what compositions
28:37.380 --> 28:43.700
mean and rewrite f in in terms of a composition
of two maps ok so this is something we will
28:43.700 --> 28:44.049
do next time