BUJ10KeJxAk

[Music]
so hello there and welcome to another
tutorial my name is timy bakshi and this
time we're going to be going over
another math video and today we're going
to be talking about Transformations more
specifically rotations so today I'm
going to be covering a lot of different
types of Transformations four different
types to be specific uh but the main one
that I'm really going to be going in
depth about are going is going to be
rotations so just before I begin today
uh let's start off with transformations
in general now there are four main types
of
Transformations uh these are now I'll
get into each one in just a second and
what they mean but let me actually just
begin with uh Transformations there are
four main types uh there are
translations
stretches and compresses uh or stretch
and compression uh we also have uh
reflection and of course rotation now
the main reason that I'm talking uh
really about TR oh sorry rotations today
is going to be because well I find that
a lot of people have quite some
difficulty with rotations and I really
want to help as many people as I can uh
so I'm going to be covering the most
complicated topic out of them all which
would be rotations so let's actually get
right into this today uh as you can see
I have a few coordinate spaces uh drawn
out here uh and so today before I begin
the notations though I'm going to be
talking about lots of other types of
transformations in general so let's
actually begin with one of the simplest
types of Transformations translations
now translations are very simple so if I
just say so first of all we're going to
be talking about
[Music]
translations okay uh so let's actually
begin now quite simply let's say we have
a shape
okay we've got a shape over here on our
coordinate space and let's just say we
wanted to translate it well quite simply
all we do is move the shape we're not
going to manipulate the shape whatsoever
but we can move it left we can move it
right we can move it up and we can also
move it
down now again we're not manipulating
the shape or anything we are just moving
it around this coordinate space and that
would be a translation and this can go
left right uh up or down uh and remember
you can even move it diagonally uh
indirectly by going let's say you wanted
to move in this direction over here
you'd go left and then up uh or you
could go right and then down or really
however else you would like to move
essentially we're just moving the shape
which allows us to translate it
okay so that's what translations are
quite simply so I'm not going to be
covering that in too much detail in this
video but then again if I do get enough
response that you know people want to
learn much more about rotations that'll
be coming in another video but now let's
talk about stretch and compression so of
course our next topic is
stretch oh sorry uh so we're going to be
talking about stretch and compression in
this case both vertical and horizontal
stretch
and sorry and
come
compression Okay so we've got stretch
and compression
here and so let's just say you know what
the best way to explain this would be
let's say we have a
curve okay I'm going to draw nice big
curve over
here okay so we've got this nice black
curve and I'm going to say this is our o
curve meaning original curve let's take
a different color maybe blue I guess so
we've got our blue marker and let's just
say uh that we want to stretch this
curve vertically how would we do that
well seems a little weird at first but
think about
this
this would be our stretched curve
this is what would happen if we were to
stretch a curve vertically okay so this
is our S curve now you must be wondering
though how does this work this is
obviously compression well no imagine
this we've got a
curve and let's say we were to take this
curve grab it from the top and the
bottom vertically we were to grab it
vertically and stretch it out vertically
the top goes even higher and the bottom
goes down then of course from the sides
get thinner and thinner and thinner the
more I were to stretch stretch stretch
it vertically that means horizontally we
would be compressing it but vertically
we would be stretching it and so that's
what we're doing in this case we're
taking our curve and we're taking it and
we're stretching it from the from
vertically and we end up with this nice
horizontally compressed vertically
stretched blue
curve now the exact same concept applies
to compressing vertically so if I were
to take uh something of this sort so if
I were to just take like this red marker
and just this isn't exact unfortunately
uh because uh I mean of course I'm not a
computer I can't uh get perfect uh
however this is uh a quite close uh
imagination of how this curve would look
compressed so again the way we're doing
this is imagine we have our curve again
and we take it from both the top and the
bottom vertically and we compress it
vertically of course the sides will get
stretched out and so this is a
horizontal stretch but vertically we are
compressing this red curve we're
compressing this black curve this o
curve uh in order to get the red final
compressed
curve all right so now uh let's actually
try something else so before I get uh
into sorry uh before I get into rotation
now we're going to be talking about
rotations in a second but let's take
talk a little bit about
well
Reflections
so all right so Reflections what are
they uh well you know what this is best
explained with an example so let's take
an example uh let's say I were to draw a
little triangle here okay if I would
just uh finish this up so now we got
ourselves a nice little triangle uh and
so it's uh more towards the right than
it is up so it has more of an x value
than it does a y value and so let's say
we wanted to reflect this on the X AIS
of our coordinate space all we do is
negate the Y value sounds weird because
this is our x value right over here this
is our horizontal x value uh and so what
we have to do is negate the Y value in
order to get our reflected uh shape or
Point uh and so in this case let's just
say uh I were to roughly calculate here
uh we would end up with something around
here uh and then here and then like this
this is what would happen if we were to
reflect and technically if I had a piece
of paper and if I were to do it on a
piece of paper and if I were to fold the
paper in half along this x AIS then the
these two should be perfect perfectly
symmetrical this should be able to
overlap right on this perfectly if I've
reflected it nicely uh however if it's
even a bit off should but technically a
reflection would be completely
symmetrical on that axis uh and then
again the same concept applies to this y
vertical axis we've got here so what we
would do is let's say we wanted to
reflect this on our y vertical axis
quite simply all we need to do uh is of
course take this axis uh and uh of
course with the X axxis in order to
reflect on that we negate the Y value so
in order to reflect on the y axis we
negate the x value and so of course all
we need to do uh negate the x value and
say just um approximate this here uh it
would end up around somewhere uh here
is so again not perfect uh but this is
around where it would end up if I were
reflect on the Y AIS and again if this
was paper not AA solid whiteboard uh
then I could just you know fold this in
half and I should be able to see this
overlap perfectly onto this uh our
original shape or our original Point uh
and it should be perfectly symmetrical
uh of course though I'm not a machine so
I wouldn't be able to do that but now
I'd like to get into the main reason of
why I'm creating this video today
rotations and so without any further Ado
let's get right into rotations so let's
begin now again another thing I'm going
to be doing today uh is actually going
to be able to show you uh not just how
you can you know use those classic old
methods of rotating your notebook around
and doing all that sort of stuff instead
I'm going to be introducing you to a
generic algorithm that'll be that'll
allow you to calculate the coordinates
uh of how you could rotate a shape
counterclockwise or clockwise not just
about the origin but another point which
I will get to in just a second so now
before we begin uh I have to uh show you
something so of course we know what the
origin of our coordinate spaces the
origin would be right over here okay uh
and the origin is our zero zero point
okay now our this this is our origin
okay we agree this is the very center of
our coordinate space and let's just say
uh I were to draw a rectangle
or Square really any shape or even a
point on this coordinate space now let's
just say and you know actually in order
to help me do this I'm going to use my
uh trusty string so this is going to
help me rotate and so again this is one
of those classic methods right using one
of these strings or a compass or
rotating a notebook or the wax paper but
today I'm going to be showing you how
you can actually calculate those
coordinates but uh what I'm going to be
showing you on the Whiteboard here today
uh is just generally that this is a
rotation if I were to take my marker uh
say okay let's say this point we want to
rotate now let's say we want to go
counterclockwise about the origin quite
simply that's all we need to do and now
this is how we can go
counterclockwise uh over in this
direction uh that's how we can rotate
counterclockwise about our origin uh
however if I wanted to rotate clockwise
almost the same drill we just take that
little origin again uh I'm just going to
put that on the origin and then that
would be how we rotate clockwise about
the origin now that's quite simple but
it's not fun right and so of course I'm
going to be showing you how you can just
take those coordinates and calculate how
to rotate at 90° clockwise or
counterclockwise however something else
very very interesting that I'm going to
be getting into in just a second is
let's say we were to have another point
over here and and we were to label this
point B okay now this is a point in our
coordinate space what if we wanted to
rotate this same shape or point or
anything um about point B but not the
origin and so if we wanted to rotate
clockwise what would we do well I'm
probably going to have to covered this
just give me a second well I'm just
going to again use my string and this is
one of those old classic methods I'm
just going to do this and so this is how
we would go clockwise uh if we wanted to
rotate about point B right and we're not
rotating about the origin and if I
wanted to go
counterclockwise then it's again almost
the exact same thing uh all I do is put
the marker there uh take my string over
to point B uh and
then go over here okay so we have to
keep going in that direction uh and so
of course today I'm going to showing you
how you can rotate both clockwise and
counterclockwise about the origin and a
point about another point for in this
case point B uh on your coordinate space
and you'll be able to rotate anything a
shape a point anything and so of course
I'm very excited to get into a more
detailed explanation of rotations
themselves anyway though thank you for
the watching this whiteboard part let's
get right back into an actual in-depth
explanation of rotation themselves let's
get to it shall
we all right so welcome back to this
explanation part now and again we're
talking about Transformations and again
in this video specifically I'm talking
about rotations but if this video gets a
lot of you know support and people like
it and people want to know more about it
of course I will create more tutorials
uh on many different topics in
Transformations uh you know the
translations the stretching and
compression compression uh our
Reflections that type of stuff I'll be
I'll be creating videos about all that
sort of stuff uh if this video gets uh
you know a lot of uh support uh and
people actually want uh to know more
about this topic but anyway let's begin
so now as you can see in front of you
I've drawn a grid now in order to show
you what a rotation would be on a
coordinate uh you on a coordinate uh
space or coordinate um uh one of those
coordinate spaces we need to draw that
out so let's actually draw out one of
these coordinate spaces all right so as
you can see we've got four quadrants uh
and we've got our positive y our
negative y our Negative X and our
positive X and each one has their own
quadrants uh and so let's actually label
uh the origin of this uh coordinate
space uh which is0 0 in the middle you
know just uh in between everything uh
very middle everything uh and let's draw
Let's uh Mark a point on this coordinate
space uh which is 62 all right so this
is a 62 position 6x2 y so we are
essentially moving six in as X six
towards the right and two up as positive
2 y okay uh now let's say we wanted to
rotate this 90° clockwise how would we
do that well I'm going to give you a
quick uh sneak peek into the future and
I'm just going to give you the answer
right now the answer is 2 -6 okay we
rotate 6 2 90° clockwise we're getting
to
26 but how how is that possible well let
me show you let's draw a line with zero
y uh sorry with zero y but 6X so we're
drawing a 60 Line This is a blue line
and as you can see uh if uh we have uh
now as you can see all of these uh all
of these quadrants that we have here
these four quadrants are at right angles
and so if we were to rotate this blue
line 90° clockwise quite easily we could
say that it would end up here but this
still doesn't really prove how we could
have uh 62 trans you know rotated 90°
clockwise to 2 -6 how does that work
well let's put the this line back and
let's draw a latch from 62 to this line
another blue
line and now let's rotate this line 90°
clockwise and as you can see it ends up
at 2
-6 in fact in order to prove that this
is 90° we can actually draw lines
between them and as you can see the
angle here is 90° so it has been proven
that we have actually rotated this 90°
but actually Let's uh continue this
right let's say let's say we were to
again draw this line and put a latch to
2 -6 let's rotate this another 90°
clockwise all right we get to -6
-2 now again we can draw this line and
we can see that we have a 90° rotation
and we can do this once more and we get
to
-26 again we can draw this line and we
see that's another 90° but this is 270°
of course we went from 62 that's 190°
then we've got 180° then we rotated
270° from 62 to -26
clockwise and so after 3 after sorry 270
in order to get to 360 so a 360°
rotation is a full circle we know that
so if we were to have 62 and we were to
have a 360 degree rotation uh from 62
clockwise or even counterclockwise we
would reach back back at
62 and after 270 we get immediately if
we had another 90
360 and so in theory if this blue line
hits 62 again that means that we have
actually rotated
360° and that we have actually rotated
90° every single time and so as you can
see it does and if we draw this line you
can see that indeed we have been
rotating 90° clockwise every single time
we went from 62 to 26 to 66 -2 to -2 6
and then 360° back again to 62 and this
was all done in a clockwise
way but okay so this is great but how
would this really help us this is no
different than you know as I was saying
those classic techniques we need an
algorithm or some sort of pattern so
let's actually take a look at what this
pattern is so as you can see pattern we
have we begin with postive 6
pos2 now in order to get to the next uh
the next uh rotation the next rotation
the next 90° clockwise rotation what we
need to do is first of all we have to
have a negator okay and we also have to
have and what we're going to do
essentially is we're going to swap as
you can see we went from 62 to 2-6 and
so we're going to to
swap the six and the two but what we're
also going to do is we're going to make
it so that while the six the x value is
moving into the Y value into the Y
position we're going to negate it to
make it
A6 and so the positive2 over there the Y
value that's going into X that remains
intact the sign remains intact however
the positive six that goes into the Y
value that's currently X that goes into
y gets negated to -6 and we end up with
our
coordinate now another thing I'd
actually like to draw your attention to
uh is just before we continue with our
pattern if you take a look over here we
have 62 and we have -6
-2 now these two uh coordinates uh
what's happening is when you have uh
these when we have 62 which is on our
which is one of the Quant if we were to
move it diagonally if we were to reflect
remember when I explained Reflections in
the beginning if we were to reflect it
exactly diagonally to the to the to its
diagonal um quadrant we would have it uh
exactly uh placed diagonally and as you
can see if we just negate the six and
the two we have 6 -2 and it's placed
perfectly uh at a 180° angle uh from uh
clockwise or counterclockwise uh anyway
from 62 and that proves again that we've
been rotating 90° every single time but
continuing with the pattern though so
now we've got positive 2 -6 let's see if
the same pattern that we applied to POS
6 pos2 applies to this as well so let's
say we swap and negate the x value while
it's going to Y as you can see we end up
with -6 -2 because the -6 value its sign
stays intact while going over to X how
however the posit 2 gets negated while
going over to Y and we end up with -6 -2
which puts us at 18 180° angle with that
other quadrant over there which is a 90°
turn which is 90° uh rotate however from
positive2 -6 now again we put the same
pattern into play uh we have the Y value
going to x uh sign stays intact the X
going to Y uh it's it gets negated and
we end up with -2
pos6 because of course uh -6 negated
would be pos6 so now continuing though
let's say we were to put this again now
as you can see with -26 we've done as I
said a whole 270° of
rotating and if this pattern continues
again with 26 -26 sorry and we're able
to get 62 that means we've done a full
360 degree rotation and every single
time we rotated 90 Dees so let's
see positive 6 will remain intact while
going over to X from y but the negative
-2 will get negated to positive2 while
going over to the yv value and we end up
with POS 6 pos2 and we have made a full
360° rotation with this pattern which
proves that it actually does
work now let's actually take a look at
the algorithm though sure we've gotten a
pattern but what's our final algorithm
of how we can actually rotate uh a point
90° uh by from the origin well it's as
simple as this we have our original
starting point let's say positive 6
positive2 are positive X and positive y
values we essentially have the X going
over to the Y spot and getting negated
whereas the Y goes over to the X Spot
while keeping its sign intact
and we end up with positive y Negative X
because the negative X was
negated now you must be wondering though
sure we can you know apply this to
points but what about shapes how can we
rotate shapes from the origin well let's
put this simply all you need to do is
let's say you have a triangle with
points a b and c well what you do is
exactly that You' label those points and
you'd find their positions in the grid
what you do is You' actually rotate each
point individually using this algorithm
once you've done that and once you've uh
labeled those rotated points you're
going to connect those points just like
you would uh for creating the shape and
of course you have your shape that has
been rotated 90°
clockwise all right so that's the
algorithm for this but you know this is
nice and all we're rotating from the
origin but let's say we want something
more complex more fun
well what if we wanted to rotate from a
point that's not the origin sounds weird
doesn't it but let me explain so we have
this blank grid here again this blank
coordinate space and we have our origin
label as 0 0 and we have a red dot that
we want to rotate we're going to label
this as 1315 because that's its position
on the grid that is relative to the
origin
but now let's say we have a black dot
okay and we label as black dot 912 which
is its position relative to the
origin and we wanted to rotate this red
dot not from the origin but about this
black dot how would we do it well first
of all let's begin by drawing a circle
and let's talk about the algorithm let's
Dive Right In so to begin we have 13 15
now again we are not rotating about or
relative to the origin that's supposed
to be about the black dot and so in
order to do that we need to find the red
Dot's position with respect to that
black dot in
912 and so what we do is we subtract
that 912 value from 1315 and this
results in 43 POS 4 pos3 and this is our
final value which is actually you know
the 135
position according or sort of uh in
respect to the 912 Dot and so what this
means is of course the 4X 3 Y and so
this means that we're going four to the
right and three up that is our position
that is the red Dot's position uh like
relative to that black
dot and so what we can do now is use our
classic algorithm that we're so used to
we can swap these values and negate the
X going to the Y and we end up with pos3
-4 but wait a minute we have our
algorithm output that we've been using
for so long
now but this is not the value of 1315
rotated 90°
clockwise remember how we factored in or
we found 131 15's location relative to
that black dot well yeah we encode ened
that so I guess you could say encoded
it's not technically encoding but uh
quote unquote if you will uh we have
encoded uh 1315 with that 912 location
and we got
43 and so now we need to take this
location that this location that we've
got this coordinate pair and decode it
and we're going to put this on the side
okay so we don't mess up our current
coordinate pair uh and in order to
decode it well quite simply we just add
back the 9 and 12 this results in 1208
and in order to prove to you that 1208
is in fact that 90° uh that we've uh
rotated from
1315 on about the 912 well let's put
that on the grd uh sorry the grid and
let's label it as you can see that is
our 128 point and that is how we were
able to rotate 1315 90° clockwise about
the 9 12 or black dot Point instead of
the
origin that's pretty interesting but
let's say we wanted to continue we've
done 90° but let's do
180° which is 128 but another 90°
clockwise how do we do that well again
we're using the same uh positive3 -4
value that we got from our algorithm and
we're going to run the same classic
algorithm again the Y value going to X
will stay the the same not stay the same
the sign will stay intact whereas the x
value going to the Y will be negated and
finally we get -4 -3 but again we need
to decode it and so what we do is we
bring it to the side we add 912 and we
end up with
509 and using this value we can put that
on the grid label it and as you can see
that is indeed 90° clockwise from 128
and
180° from the clockwise uh from
1315 and then again that is about the
912 and not from the origin let's
actually continue this pattern once more
and we end up again with
-34 if we take this to the side again
decode it with a 9 and 12 and we end up
with 616 we put that on the graph and we
label it and as you can see
we have that 90° turn again so now what
we've done what we've done is we have a
90° clockwise turn from 59 about the 912
which is
616 but it's 180° rotation from 128 and
it's a 270° rotation from
1315 and so again we apply the same
algorithm as we did last time if we can
take 616
and apply our same algorithm in order to
rotate at 90° clockwise that would mean
from 1315 it should be
360° and since 360° is a perfect circle
that means we should end up with
135 and so we run our algorithm again
the positive4 sign stays intact and it
goes from y to X and negative3 sign gets
negated to three and goes over to Y and
as you can see we end up with 4
pos3 and that is the same value we got
when we took pos3 pos5 minus postive 9
pos2 we got 43 and now we got 43 again
and so there's no doubt that if we take
it to the side add 912 again we get
135 as our final result and as you can
see we have rotated 13 15 360° clockwise
about the point 912 not the
origin now this is pretty nice isn't it
how we were able to take something that
was so frustrating and simplify it down
into one
algorithm but we're not quite done yet
there's still one more
step what if we wanted to take all of
this and we wanted to condense it into
one final algorithm how would we do
it well let's say we have two different
algorithm one to rotate a point
clockwise and one to rotate a point
counterclockwise let's begin describing
how this would work well again this
isn't going to be about the origin it
could be about any point it could be
about the origin it could be about uh
another Point really anything and so we
have this red dot uh that we want to uh
that we want to rotate okay and we're
going to say its location is XP YP that
is the red dots location and this is
what we want to
rotate and this is one of the inputs
that both of our algorithms will
take but then we also label the the
black dot and this black dot is what we
want to rotate about now this black dot
could be the origin it could be where
the black dot currently is it could be
any place or any location on this
coordinate
space and we're going to label this
black dot as XO and
yo then what we're going to do is we're
going to take this uh this black dot and
what we're going to do is again just
like we did just a few seconds ago we're
going to take XP and YP and we're going
to subtract XO and yo from XP and YP
just like we did as I said a few seconds
ago and this is going to happen in both
of the algorithms and we're going to end
up with a result that I like to call X1
and y1 and this is where the algorithms
start to defer a bit uh and so what
going to happen
is uh let's just put our negators here
and so just like last time again what
I'm saying is we're using the same
algorithm uh we are essentially uh for
the clockwise one at least we're taking
X1 and we're putting it into the Y's
position while negating it so the
positive X1 becomes a negative X1
whereas the positive y1 switches over
with the X's position while keeping its
sign
intact however on the counterclockwise
side now I mean you've already seen the
clockwise side that's pretty you're fam
you're pretty familiar with that but
let's talk about
counterclockwise on the counterclockwise
side we were essentially able to take
the positive y1 value and we're
essentially doing the exact opposite
thing that we would do for clockwise we
are taking the we're taking the positive
X1 and we're swapping it with the Y
position all right we're doing that but
we're not negating we're keeping its
sign intact whereas we're taking the
positive y1 and we are of of course
swapping it out with the X position and
negating that instead so we end up with
negative y1 and positive
X1 and so finally as you can see we're
doing the exact opposite thing for the
counterclockwise algorithm than we are
for the clockwise algorithm and this
allows us to end up with our result now
again we have actually
found the location of the 90°
rotated uh point this Red Dot 90°
rotated we found its uh location
clockwise or counterclockwise but this
isn't our final location but why well
again I mean you should probably
remember this it's because we were we
did not yet decode it because of course
we found the red dots
location in you know with respect to
that black dot and so now what we need
to do is make it back in respect to that
uh Middle Point that origin
and so what we need to do is we need to
add XO and yo back to our result and we
should finally get an XA value a ya a
value uh for our clockwise algorithm and
an XB value and a YB value from our
counterclockwise algorithm and so of
course from our clockwise algorithm as I
said we get that XA and ya a uh which
which is our final 90° clockwise rotated
algorithm uh or Point uh and of course
from the counterclockwise algorithm we
get XB and YB which is the final point
of where uh we uh which is the final
Point uh of our 90° counterclockwise
rotation and that was all there was to
this video of course I really hope you
enjoyed and learned a lot if you did
please make sure to leave a like on this
video it really does help out a lot and
if you think this could help anybody
else uh please do consider sharing the
video again it really does help out uh
and of course if you have any more
questions suggestions uh or uh concerns
that type of thing you can leave it down
in the comments below you can of course
email it to me at tagim Manny gmail.com
or just straight tweet it to me at tajim
Manny of course my contact info will be
in the
description but if you really really do
like my content and you want to see a
lot more of it please do consider
subscribing to my YouTube channel as it
really does help us a lot again though
finally I really hope you enjoyed thank
you very much for watching the video
goodbye okay