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okay so here we can see a little toy
frog and now I'm gonna slowly move back
this autofocus is stupidly slow to
adjust so I can't move too quickly and
it's gone well sort of so what we just
saw was an illusion that I bought for
like 10 bucks on Amazon which has to do
with the type of curve that pretty much
anyone watching this video has seen
before now how it actually works isn't
that complicated but we're gonna work
our way up to it instead to begin we're
gonna play a little scavenger hunt so
I'm going to hide some treasure on this
map and not tell you where but I will
allow you to guess and wherever you
guess I will tell you how far away the
treasure is from that spot the question
is how many guesses would you need to
make before you knew exactly where the
treasure is
well the answer is three but the real
question is how now since this is a
video I'll have to do the guessing for
you and I'll make this your first pick
to which I say you are a hundred
kilometers off so what do we do now
well let's just say that this line is a
hundred kilometers long for measurement
purposes that means the treasure could
be here since that's a hundred
kilometers away from your guess or it
could be here since that's also 100
kilometers away or it could be here or
here in fact the treasure could only be
on this circle of radius 100 because
this is every single point on the map
that's 100 kilometers away from our
initial guess now you can make a second
guess anywhere you like let's say here
and yes it doesn't have to be on the
circle now for this I would say you're
50 kilometres away then we can again
sweep out a circle and now we know that
the treasure is at one of these two
locations because those are the only
points on the map at the correct
distances from both of our guesses all
it takes is one more guess now and by
the way yes you could just guess one of
the two points but again it mostly
doesn't matter like we could guess here
and I'd say you're now this far away we
then make a circle that tells us the
treasures located at this common
intersection point now if you replace
those guesses with satellites we have
the basics of how GPS works and why it's
possible to figure out where someone is
on earth with such precision
well GPS we need three satellites in
space to mostly determine our location
however since we live in a
three-dimensional world
not a 2d map we do need a four satellite
to determine altitude based on how long
it takes a signal to travel between the
satellite and receiver along with
knowing how fast the speed of light is
onboard computers can calculate how far
away the receiver is with that they can
then determine a sphere where that
object could be located and where all
the spheres intersect is the location of
the receiver now there's much more that
goes into this because as you can
imagine these systems need to be
extremely accurate with signals
traveling at the speed of light a few
nanoseconds makes a difference so
there's atomic clocks being used air
calculations being made and even
relativistic effects being taken into
account since time takes slower for
objects moving at high speeds but for
this video I'm only trying to go into so
much detail and the same analysis can
even be used to detect the epicenter of
an earthquake which we can see here on
brilliant sight now since the math works
in the same way we know that the minimum
number of seismographs needed for this
would be three we can also see in the
explanation that the seismographs or our
guesses from before actually cannot just
be anywhere since it is possible that a
third circle will still lead to two
intersections so we've got to be a
little careful with location but as you
can see this is still a really powerful
technique that's being utilized every
day okay now let's take it up a notch
we're gonna go back to the map and do
the same scavenger hunt but this time
we're gonna make our three guesses all
at once just imply the visuals little
I'm going to make these our guesses
which I'll label points one two and
three now this time I'm only going to
tell you the difference in distances as
then I'm not going to tell you how far
the treasure is from this point which
we'll call d1 that's unknown I will not
tell you the distance from the second
point or d2 but I will tell you the
difference in those distances is 60
kilometers so if maybe this point is a
hundred kilometres from point two and 40
kilometers from point one that's a
potential spot for the treasure since
the difference in distances is 60 and
that automatically means this is a
potential spot as well just due to
symmetry but how do we find all
potential spots well when I just did was
randomly select two numbers 140 that
subtract a 60 and then found all points
100 kilometers from point 1 and also all
points 40 kilometers from point two
those intersections with MB are points
of interest
and now we can again pick just about any
two numbers let's subtract 260 like how
about ninety and thirty this time we
then make a circle with the radius of
ninety and another with the radius of
thirty those intersections will again
satisfy a difference of sixty kilometers
between points one and two we could then
just increase the radius of each circle
by maybe five and get another set of
potential points since the difference in
distances is still sixty in fact we can
just keep slowly increasing each circles
radius at the same rate to keep that
distance at sixty if we trace out the
intersection points we get a curve that
contains all possible places the
treasure could be located the name of
this curve is a hyperbola and each one
of our guesses is a focus now I know
most of you're saying wait that's only
half a hyperbola but if we had just
taken the circles we made and switch
them we would get the other half because
for an actual hyperbola we care about
the positive difference or absolute
value being a constant so for those who
only know a hyperbola by its equation
instead think about it for what it
really is it's every single point or the
distance to a certain location off the
curve minus the distance to another spot
off the curve is always the same
regardless of what point you select then
going back to our problem we can do the
same analysis for the other two points
like I'd say d3 minus d2 is some value
we'd make another half hyperbola and the
treasure would lie at the intersection
point this is exactly how hyperbolic
navigation works we're three
transmitters send out pulses at the same
time and a receiver measures the
differences in those times it can then
use those differences to determine the
hyperbolas we saw earlier which will
dust tell us the receivers exact
location the most famous use of this is
the low ram system which stands for
long-range navigation it was developed
during World War two and was in use for
many years but it's now out data as it
was replaced by satellite navigation and
techniques we discussed earlier now
moving on the next curve is an ellipse
which has almost the exact same
definition as a hyperbola the only
difference is if you pick a point on the
ellipse and determine the distance to
each focus the sum of those values is
always the same regardless of which
point on the ellipse you pick the most
interesting property of an ellipse in my
opinion is the fact that any beam of
light that passes over a focus will
reflect off the curve and go through the
other focus like at the inside of this
were a mirror or something it doesn't
matter where the beam comes from or its
direction this will happen in theory
every time and this even applies to
sound or solid objects as you man seen a
number files elliptical pool-table video
most of you have probably heard of or
been to one of those rooms where if you
stand at the right spot you can hear
someone whispering from very far away
which was even shown in an episode of
How I Met Your Mother
one example of this is Grand Central
Station in New York if this person here
whispers into the wall someone in the
opposite corner which is barely cut off
in this picture would be able to hear it
with no issues and this is due to the
curvature of the ceiling above them this
phenomenon occurs often due to
elliptical enclosures where if you stand
at one focus your sound which radiates
in many directions will ideally reflect
back to the other focus and the
reflected sound will reach the other
focus at about the same time because as
we saw the total distance a sound wave
has to travel or D 1 plus D 2 is always
the same regardless of which path it
takes one of the most impressive
architectural designs where this
phenomenon occurs is a Mormon Tabernacle
located in Utah where apparently tour
guides will drop a pin on the stand
where the preacher speaks at and the
resulting sound can be heard throughout
the entire building in fact if this
stuff interests you there's an entire
career beginning to known as acoustical
engineering one thing these engineers do
is design concert halls that provide an
optimal acoustic experience for the
audience things like metal panels along
the ceiling of this concert hall or the
transparent baffles in this auditorium
are all put into place for acoustic
purposes the Walt Disney Concert Hall
you see here is actually one of the most
sophisticated concert halls in the world
due to its architecture and layout these
engineers work on much more such as
ultrasound sonar technology for
submarines audio processing and more
but I just thought this was a pretty
interesting field then probably the most
famous application of an ellipse is seen
with an orbiting systems in fact all
bounded orbiting bodies like the Earth
Mars or even Halley's Comet follow
elliptical paths and what they're
orbiting around or the Sun in this case
is located at the focus of each
individual curve
something more interesting though is
that all conic sections yes every single
one shows up in orbiting systems
circular orbits are often taught at
first to students since the math like
you see here is easier to grasp however
these are extremely ideal and not seen
in a reality elliptical paths are again
what we observe for bound orbits however
if a celestial body like a comet gets
really close to Earth it will enter an
unbound orbit as in Earth's gravity will
affect it but due to its high speed the
comet will escape back into interstellar
space the path that it typically travels
what you can see here is a hyperbola but
if the speed of the comet just matches
the escape speed of Earth meaning it's
barely fast enough to escape Earth's
pull the shape of the path will be a
parabola which brings us to our last
curve now I'm sure parabolas are nothing
new to most of you guys but you may not
know that these have an extremely useful
property just like all other conic
sections parabolas have a focus and any
beam of light that goes through that
focus will reflect straight out parallel
to the axis of symmetry and this works
backwards as well where any beam of
light coming in head-on will reflect
through the focus so if you imagine
those beams are wireless signals like
radio waves and we just turn this
sideways you can now kind of see why
these antennas are shaped the way that
they are this thing here is located at
the focus and when signals coming in
from space or whatever interact with the
surface they all ideally reflect into
that one point that captures the signal
then again these also work in reverse
but the use of parabolas does not stop
there solar cookers make use of this
reflection telescopes use this curvature
to focus very distant light and even the
Olympic flame is traditionally late
using a parabolic reflector to
concentrate sunlight now again if we
take one parabolic reflector and scatter
light from the focus all beams will
reflect in the same direction but what
if we now put another reflector of the
same dimensions right here basically
where its own vertex and the focus of
the other parabola meet well since they
have the same dimensions that means the
bottom parabolas focus will be up here
overlapping the vertex of the other
curve this means the light will reflect
again and all intersect at this point
here all that light started journey down
here we can think of that as light
coming off an object like maybe a toy
frog if we cut a hole in the top the
reflected light will end up focusing
right at that location and if you don't
know intersecting light rays create an
image so now we've come full circle and
can finally see why this works with a
parabolic mirror at the bottom another
with a hole cut out of it on top and the
Frog place right at the vertex of the
bottom curve which is also the focus of
the other we get the illusion we saw
earlier now if you're a beginner and
want to learn more about these algebra
fundamentals or you're more advanced
want to explore how these are applied in
orbital mechanics and complex
oscillating systems then you can
continue to do so at brilliant org whoo
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