What your teachers (probably) never told you about the parabola, hyperbola, and ellipse

Summary

This video, sponsored by Brilliant, explores various mathematical concepts and their real-world applications. It starts with a treasure hunt analogy to explain GPS and satellite navigation, moves on to how hyperbolas help in navigation and detecting earthquake epicenters, and highlights the unique properties of ellipses in acoustics and orbital mechanics. The video also covers parabolas, demonstrating their role in antennas, solar cookers, and telescopes. Finally, it delves into the creation of illusions using parabolic mirrors and promotes Brilliant's educational courses for deeper learning in math and science.

Takeaways

• 🎥 The video is sponsored by Brilliant, an educational platform offering a variety of math and science courses.
• 🐸 An optical illusion involving a toy frog is introduced, which is later explained using the concept of parabolic mirrors.
• 🧭 The video explains the basic principles of GPS navigation using the analogy of a scavenger hunt and circles drawn on a map.
• 📍 It takes a minimum of three guesses to pinpoint an exact location on a map, similar to how three GPS satellites are needed to determine a location.
• 🛰️ A fourth GPS satellite is necessary to determine altitude, using the time it takes for a signal to travel and the speed of light.
• 🔍 The precision of GPS systems requires accounting for factors such as atomic clocks, air calculations, and relativistic effects.
• 🌐 The concept of hyperbolic navigation is introduced, which uses the difference in distances to determine a location along a hyperbola curve.
• 🚀 The video mentions the Long Range Navigation (LORAN) system, a hyperbolic navigation system used historically before GPS.
• 🌑 The properties of ellipses and their use in architecture and acoustics are discussed, such as in Grand Central Station and the Mormon Tabernacle.
• 📡 Parabolas are shown to have applications in various technologies, including antennas, solar cookers, and reflecting telescopes.
• 🔄 The video concludes by connecting the parabolic properties back to the initial optical illusion of the toy frog, demonstrating how light rays create an image.

Q & A

What is the purpose of the scavenger hunt in the video?

The scavenger hunt serves as an analogy to explain the concept of triangulation used in GPS technology to determine location.

How many guesses are needed to pinpoint the exact location of the 'treasure' in the scavenger hunt analogy?

Three guesses are needed to determine the exact location of the treasure using the method of triangulation.

What is the minimum number of satellites required to determine a location using GPS?

A minimum of three satellites is required to determine a location in two dimensions, but four are needed to also determine altitude in three-dimensional space.

How does the GPS system calculate the distance between the satellite and the receiver?

The GPS system calculates the distance by measuring the time it takes for a signal to travel between the satellite and the receiver, using the known speed of light.

What is the significance of the hyperbola in the context of the scavenger hunt?

A hyperbola represents all points where the absolute difference in distances to two fixed points (foci) is constant, which is used to find potential locations of the treasure based on the differences in distances from two guesses.

What is the principle behind the use of hyperbolic navigation systems like LORAN?

Hyperbolic navigation systems work by having transmitters send out pulses simultaneously, and a receiver measures the differences in arrival times to determine its location using the hyperbolas formed by these differences.

Why is the ellipse relevant to the discussion of acoustics in the video?

An ellipse is relevant because it has the property that any beam of light or sound that passes over one focus will reflect off the curve and go through the other focus, which is used in architectural designs for acoustics.

How does the shape of an orbiting body's path relate to conic sections?

Orbiting bodies follow paths that are conic sections, with bound orbits being elliptical and unbound orbits being either parabolic or hyperbolic depending on the speed of the object.

What is the practical application of parabolas in the context of wireless signals?

Parabolas are used in antenna design because any signal that goes through the focus of a parabolic reflector will reflect straight out parallel to the axis of symmetry, allowing for the capture and concentration of signals.

How does the parabolic mirror create the illusion shown in the video?

The illusion is created by placing a toy frog at the focus of a parabolic mirror, with another parabolic mirror above it. The light from the frog reflects off the mirrors and converges at a point, creating the appearance of the frog being in a different location.

What is the educational platform mentioned in the video, and what does it offer?

The educational platform mentioned is Brilliant.org, which offers a wide variety of math and science courses, practice problems, and daily challenges to deepen understanding and apply concepts in real-world situations.

Outlines

00:00

🎨 The Illusion of GPS and Scavenger Hunt

This paragraph introduces an optical illusion toy, a frog, to illustrate the concept of GPS technology. The presenter uses a scavenger hunt analogy to explain how GPS works, involving guesses on a map to pinpoint a treasure's location. It starts with one guess, which defines a circle of possible locations 100 km away. A second guess narrows down the options to two points on the map. The final guess intersects these points, revealing the treasure's exact location. This process is likened to using three satellites to determine a location on Earth with precision, highlighting the need for four satellites to account for three-dimensional space and altitude. The explanation simplifies the complex technology of GPS, atomic clocks, and relativistic effects that ensure accuracy in positioning.

05:02

📡 Hyperbolic Navigation and the Geometry of Sound

The second paragraph delves into hyperbolic navigation, where three transmitters send out signals and a receiver measures the time differences to establish hyperbolas that intersect at the receiver's location. This method was used in the past in systems like LORAN (Long-Range Navigation), which has since been replaced by satellite navigation. The paragraph also explores the properties of ellipses, particularly their reflective nature, which allows light or sound to travel from one focus to the other via the curve. Examples include Grand Central Station's whispering gallery and the Mormon Tabernacle's acoustics, demonstrating how sound waves travel specific distances to reach a focus. The discussion highlights the field of acoustical engineering and its applications in various settings, including concert halls and submarines.

10:03

🚀 Orbital Mechanics and the Power of Parabolas

In the final paragraph, the focus shifts to the application of conic sections in orbital mechanics. It explains how celestial bodies follow elliptical orbits around the Sun, which is at one focus of the ellipse. The paragraph also touches on hyperbolic orbits for unbound celestial bodies like comets and parabolic paths for objects escaping Earth's gravity at the exact escape velocity. The unique property of parabolas, where light reflects in parallel lines from a focus, is highlighted, with applications in satellite dishes, solar cookers, and telescopes. The video concludes with an explanation of how intersecting parabolic reflectors can create an image, linking back to the initial illusion of the frog toy. The sponsor, Brilliant.org, is acknowledged for their support, and their platform for learning math and science is promoted, with a special mention of their offline courses for iOS and Android.

Keywords

💡Autofocus

Autofocus refers to the camera's ability to automatically adjust the focus to keep the subject sharp without manual adjustment. In the video, the script mentions the autofocus being 'stupidly slow to adjust,' illustrating a moment where the technology did not meet the presenter's expectations for quick movement, affecting the demonstration of an illusion.

💡Illusion

An illusion is a visual phenomenon or sensory distortion that makes an object appear different from reality. The video script describes an illusion involving a toy frog, which is a central theme of the video, demonstrating how our perception can be tricked by certain visual setups.

💡Scavenger Hunt

A scavenger hunt is a game where players search for objects or solve clues to find a hidden 'treasure.' In the context of the video, the scavenger hunt is used as a metaphor to explain the concept of triangulation in GPS technology, where guesses represent the location of the treasure, and the distances from known points help pinpoint the exact location.

💡Triangulation

Triangulation is a method of determining the location of a point by measuring the angles to it from two known points. The script uses the concept of a scavenger hunt to explain how GPS uses triangulation with signals from three satellites to determine a precise location on a two-dimensional plane.

💡GPS

GPS stands for Global Positioning System, a satellite-based navigation system that provides location and time information in all weather conditions. The video script explains the basic principles of GPS, emphasizing the need for signals from at least three satellites to determine a location on Earth.

💡Seismographs

Seismographs are instruments that measure and record seismic waves caused by earthquakes. The script mentions seismographs in the context of using triangulation to determine the epicenter of an earthquake, highlighting the application of mathematical principles in real-world scenarios.

💡Hyperbola

A hyperbola is a type of conic section defined as the set of all points for which the difference of the distances to two fixed points (foci) is constant. The video script describes how hyperbolas are used in hyperbolic navigation systems, such as the LORAN system, to determine a receiver's location based on the difference in signal arrival times.

💡Ellipse

An ellipse is a conic section representing the set of all points where the sum of the distances to two fixed points (foci) is constant. The script explains the properties of an ellipse, particularly its reflective properties, which are used in various applications like whispering galleries and architectural acoustics.

💡Parabolic Antenna

A parabolic antenna is a type of microwave antenna that uses a parabolic reflector to direct the radio signals. The script describes how parabolic antennas work based on the principle that signals entering the focus of a parabola will reflect in a parallel direction, which is crucial for wireless communication systems.

💡Acoustical Engineering

Acoustical engineering is the branch of engineering that deals with the control of sound in various environments. The video script mentions this field in the context of designing spaces with optimal acoustics, such as concert halls and whispering galleries, where the shape of the space influences the propagation of sound.

💡Orbital Mechanics

Orbital mechanics is the study of the motions of celestial bodies under the influence of gravity. The script touches on the fact that all bound orbits, like those of planets and comets, follow elliptical paths with the central body at one focus, illustrating the application of conic sections in astronomy.

💡Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. The script explores different types of conic sections—circles, ellipses, hyperbolas, and parabolas—and their various applications in technology, architecture, and astronomy, showing the mathematical principles behind everyday phenomena.

Highlights

The video demonstrates an optical illusion involving a toy frog and a parabolic mirror, purchased for $10 on Amazon.

Explains the concept of GPS by comparing it to a scavenger hunt on a map, using circles to pinpoint a location.

Details the minimum number of satellites required for GPS: three for location and a fourth for altitude.

Introduces the concept of hyperbolic navigation using the difference in distances from multiple points to determine a location.

Describes the use of hyperbolas in the LORAN system, a long-range navigation system developed during World War II.

Explains the properties of an ellipse, including its use in acoustics and the phenomenon of sound reflection.

Mentions Grand Central Station and the Mormon Tabernacle as examples of architectural designs utilizing elliptical properties for acoustics.

Discusses the application of elliptical orbits in astronomy, with the Sun located at one of the foci.

Introduces the concept of parabolic orbits for celestial bodies like comets moving at escape velocity.

Describes the use of parabolas in wireless signal reflection, such as in satellite dishes and radio telescopes.

Explains how parabolic reflectors are used in solar cookers and telescopes to focus light or capture signals.

Demonstrates the illusion of the toy frog using two parabolic mirrors to create a focused image.

Promotes Brilliant.org for its wide variety of math and science courses that provide both theoretical knowledge and practical problem-solving.

Highlights Brilliant.org's courses on real-world applications of math and science, such as determining the direction of a bike by its tracks.

Announces the availability of offline courses on Brilliant.org for iOS and Android, allowing learning on the go.

Offers a 20% discount on the annual premium subscription to Brilliant.org for viewers of the video.

Transcripts

this video was sponsored by brilliant

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

I'd like to thank for sponsoring this

video

brilliant hosts a wide variety of math

and science courses that will not only

take you through the necessary technical

information but also give you constant

practice problems along the way to

ensure you understand everything

fundamentally one thing I love doing on

this channel is showing unique

applications of math and science that we

don't learn in school and a lot of ideas

have actually come from brilliant

physics of the everyday was one course I

really enjoyed going through which

showed things like how to determine the

direction of bike is traveling just by

its tracks or how to recreate a crime

scene using blood stain pattern analysis

learning about these very real-world

situations for a more mathematical

viewpoint is something that always has

piqued my interest they even have daily

challenges that turn learning into a

habit these questions can range from

what happens when you cut a mobius strip

in half - how did prehistoric humans

keep track of time and much more to give

you a range of topics look forward to

learning plus they just came out with

offline courses for iOS and soon Android

so you can download some of your

favorite courses right to your phone

you can then solve math or science

problems and learn something new whether

you're traveling commuting to work or

school or even if you're somewhere with

terrible internet connection so if you

want to get started right now and

support the channel you can click the

link below or go to brilliant org slash

major prep to get 20% off your annual

premium subscription and with that I'm

gonna end that video there if you guys

enjoyed be sure to like him subscribe

don't forget to follow me on Twitter and

join the mid for a Facebook group for

updates on everything hit that bell if

you're not being notified I'll see you

all in the next video

Oh