Map Projections Explained

Correction: The azimuthal equidistant projection is actually defined by θ = a and r = 90° - b. I got b mixed up with phi.
This video was made for 3blue1brown's summer of math exposition contest. It was selected as one of the top 100, and that has really helped kickstart my channel, so a huge thank you goes out to 3blue1brown.
Note: If the title changes, it's because I'm testing out different video titles to see what works the best.
In this video, I explain map projections: what they are, how they work, and the math behind them. This video connects 2 of my favorite subjects: math and geography. If you want to do further research, this wikipedia page is very helpful:
en.wikipedia.org/wiki/List_of...
The images of each projection come from that page, and are made by Daniel R. Strebe.
Chapters:
0:00 Intro
0:41 The Problem
3:37 Equirectangular Projection
7:15 Literal Projection
9:49 Mercator Projection
18:51 Sinusoidal Projection
21:24 Lambert Equal Area Projection
23:33 Azimuthal Equidistant Projection
24:30 Compromises
29:30 Perspectives
31:21 Conclusion

[@PrimalPeashoter]

I kill for anything geography in-depth without unecessary bullshit and to me you pulled it off perfectly. Concise. I'm saving this in my internet gems folder

[@philippfrogel9355]

What else is in this gem folder, if I may ask?

[@ignvb]

wow, you may share that list dude!

[@a-cd2lw]

Wait, you have a folder of amazing videos?

[@error.418]

"You might think math and [insert literally anything] are [not connected], but they are more connected than you think" true.

[@bettercalldelta]

Lol I was going to comment about flat earthers claiming the earth is the Azimuthal projection but then you said it. Awesome video btw

[@Kuvina]

thank you haha :)

[@HelmutNevermore]

I like the azimuthal equidistant projection, its only flaw is that it's centered at the North pole, whereas it should be centered at the point that the world revolves around - which is, of course, me. Joking aside, such map centered at any given point of particular interest may distort shapes and areas and directions and distances between two random points on the map, but it tells you accurately how far things are from you and in which direction you should go to reach them. What else could you possibly want from a map?

[@angelradness5542]

How does this channel only have less than 2000 subscribers? This is fantastic Level of presentation for such a low subscriber channel. Great job!

[@swagatochatterjee7104]

The Kavrayskiy is way way simple to compute without scarificing loads of quality. That's huge!!

[@andrewchapman2039]

Actually needed a map for a project that didn't distort distances or shapes, and came back to this video for a recommendation! I'll have to add arrows so I don't forget which bits connect where, but Dynmaxion's the map for me!

[@borja4015]

Great job, I wish I had found a video like this years ago, when I first tried to understand about the Mercator projection. There were some Interesting factshere , and I think you got a nice balance between the rigour/math and a good pacing.

[@FreyrDev]

I always love to see videos on map projections, so this is great, I need more places to spread the word of our lord and saviour the Kavrayskiy VII Projection lol

[@Kuvina]

Yeah, I didn't mention it in the video because I think the total scores shouldn't be taken too seriously, but the Kavrayskiy 7 got the highest score out of all the projections at 14. In my opinion, the other compromise projections sacrifice too much in terms of orientation distortion, but the Kavrayskiy 7 has the perfect amount.

[@alleycatsphinx]

@@Kuvinaaww, spaceforce!!!

[@zyxw59]

Web maps are a use case for Mercator that it (obviously) wasn't designed for, but suites well - its conformal and orientation-preserving properties mean that you can have a map where you can zoom in at any point without reprojecting, all while maintaining directions and shapes.

[@TeslabladePlaysMC]

It's worth noting that some web maps, especially Google Maps, have started using a perspective projection since the 3D graphics required to do that is trivial, and basically all modern browsers support 3D graphics. It's also just a much better way to show Earth accurately, especially at the poles.

[@sinox5]

Keep doing that good videos! I personnally think this channel is one of the best channels in all KZbin, featuring all of the interesting subjects ever. Great job! I love your videos!

[@ocamlmail]

Probably the best video on this topic I've seen so far, thank you very much!

[@rv6amark]

Outstanding documentation of map projections, with a bit of history thrown in. This is the best explanation of a very complex subject that I have ever seen. Thank you for making it.

[@DialecticRed]

My favorite map projection is the Peirce Quincuncial projection. I think it is potentially the best map projection of the earth because even though distortion exists, it is relegated to relatively insignificant areas (all in oceans), and it has the great property that it can tile the plane making it a perfect fit for spherical geometry in 2d video games, working in a similar way to the naive approach of teleportation on either side to create a torus. It is also just very aesthetically pleasing imho

[@mengbomin]

I'll add my view to this one, adding that since the distortion is continuous (except at those four points of extreme distortion), it doesn't have artifacts from the edges of the icosahedron that the Dymaxion has. Further, it's the only projection I know of that can be tessellated.

[@DialecticRed]

@@mengbomin Yeah the tesselation is extremely powerful. It basically has the properties of being the spherical version of the very popular torus projection, where you just "teleport" from one end to the other. With the Quincuncial Projection, you more so get "reflected" across the side you go "off of." It's difficult to explain in writing but becomes apparent if you just follow where things ought to go.

[@akeem2983]

​@@mengbomin There is at least one more projection that's tesselatable - Lee World in Tetrahedron, which is also conformal and is shaped like a triangle, which could also be folded into a tetrahedron - hence map's name. I think that its distortions are even smaller, but at the expence of convenience, because it seems like it's hard to use it to map all of the Earth's landmasses without either major interruptions in them or repeats

[@ckq]

I've always been fascinated by map projections, but never found any resource with explanations about the math of it. Great video

[@fredg8328]

If you look at the Wikipedia page for map projections it brings you to the pages for each projection where you will find the maths most of the time. Unfortunately it is not always easy to follow and it misses some informations when you want to code a program to display them. This video is not really more helpful for that purpose. For example when you code it on a computer your map is a rectangular texture with a width and a height, but the video does take that into account in the equations. For some projections it is impossible because you would need an infinite map. The angles are switching between degrees and radians too...

[@YgorCortes]

Great video! My personal favourite projections are the Hammer and the Érdi-Krausz ones. I like how they have relatively proportional sizes and their oval shapes just look so good to me

[@dcterr1]

Excellent comprehensive discussion of various types of map projections and their advantages and disadvantages! Besides being a mathematician, I've always loved geography as well as cartography. I already knew most of the mathematical formulas for these map projections, but I never thought too much about the overall advantages and disadvantages of each one, and I think you nailed this part! And I also agree for the most part with the scores you gave each one. Great job!

[@8GigueBits]

I remember I had a toy map which was a "can" projection instead of a cylindrical projection: there was a central stripe in cylindrical projection, but the poles were projected separately onto circles that you could rotate across the top and bottom edges of the map. I wonder how all these maps would look if you could rotate the earth before projecting. For one, if you rotate the earth longitudinally by 180 degrees you automatically avoid eurocentrism. But a projection of a latitudinally rolling earth would also be fun to watch.

[@saharapengu]

Very nice video and I definitely share your sentiment towards the mercator projection.

[@alex.g7317]

27:57 the big three: Mercator, Dymaxion, Lambert… cool. Bare minimum of maps I need for everyday life.

[@kennyearthling7965]

amazing, second vid I saw, so great. subbed

[@thomasbartscher7764]

I love the Azimuthal Equidistant projection, and I think it may be extremely useful when making textures for spheres. You can basically create an arbitrary amount of these projections from different points on the sphere, and then interpolate between them depending on how close you are to the respective center point when determining the color of a point on the sphere. Seriously though, thank you for the great video, and thank you *especially* for making me finally understand Mercator!

[@MC-br1gk]

Just found you. What an awesome job for your 2nd. Looking forward to watching others.

[@unrealzocker]

Just stumbled upon this now. Thank you for the video!

[@Kuvina]

no, thank _you_

[@weguloler322]

Very interesting video! Nice job

[@tlmoller]

Really nice video! Like it a lot. I do miss the map type that is created by cutting slices up from the sphere along the medians. Like 10 slices of 36 deg. Looks somewhat strange when you add it to a flat surface and distances are hopeles.

[@inversebrah]

this is amazing, learned a lot, thank you

[@Kostyurik]

I love Peirce quincuncial map projection the most, it is super aesthetically pleasing

[@atzuras]

Fantastic video. I saw a lot of confussion time ago with TV screen ratios when projecting cinematic ratios.. same issue. shape distortion or cutting things out or adding black stripes in. Many people could not realize that you cannot fit a 3:1 ratio in a 4:3 or 16"9 screen without loosing something in the way.

[@trombonedude5312]

My favorite projection is the gnomonic projection because it portrays all great circle paths as straight lines. It's a perspective projection but it projects from the center of the earth to an infinite plane tangent to the earth. As such, it can't show the whole earth, only up to half, but it is incredibly useful for optimizing long distance navigation routes

[@Zero_Contradictions]

Thank you for making this! In my opinion, this video is one of the best submissions I've seen for the 3b1b SoME1. Have you heard of the Danseiji N Map Projections by Justin Kunimune? They are among some of the most balanced map projections I've ever seen.

[@Kuvina]

thank you so much! I had never heard of those, but they're definitely very unique and interesting.

[@mari-with-a-gun]

I think you should double check the first projection direction. It preserves NESW directions but not anything in between (I think)

[@wizengy]

Great video! There should also be a globe in every elementary and middle school classroom. How do you get from Beijing to New York City by the shortest path. Put a sting on the globe at though two points an pull it tight. Wow, it almost goes to the north pole. What is the state closest to Africa? Wow, the string shows that it is Maine.

[@Qsalis]

you put a lot of work into editing this. the conclusion was interesting to be sure your audio sounds a bit uncertain but you'll definitely improve over time in that matter, so no sweat. what software do you use?

[@Kuvina]

Thank you so much! I use a lot of different software for different parts of the production. blender for 3d images gimp for drawing libreoffice for adding text and stuff audacity for audio shotcut for video editing And they're all free!

[@171151]

Good video man I learned a lot

[@sephirothjenkins6783]

How do you only have 47 subscribers? I mean, that's obviously rhetorical, but damn that number should be bigger

[@Kuvina]

Thank you!

[@XDinky]

Finally somebody that can explain what that school class wall poster projection is calculated, great video!

[@baksoBoy]

How the hell does this video only have 6K views?? This video is really high quality and super interesting! I was honestly shocked when I looked at the view count

[@Dalroc]

We all know why. It's a great video. It's not the video that's the issue.

[@petterlarsson7257]

I NEeD more OF THIS

[@elliebakonoe4252]

Great video! ❤️

[@Kuvina]

thank you so much!!

[@guessundheit6494]

17:10 - Only true if you're trying to place the entire world on a map. It's better to use multiple maps around smaller areas based on great circles, never more than 4000km from the circle or a 4000km radius. Plane and shipping routes could be based on such maps. 28:37 - I live in an Asian country. Maps here put the Intl Date Line slightly off centre, which puts the edges in the Atlantic Ocean. These maps do a great job of showing proximity and connection between countries. Most already know all the European/North American trade routes, so there's no need to show them.

[@andryxmartinez9602]

Great video!!

[@cupostuff9929]

I love that your #Some2 vid got reccomended to me I love your channel

[@EduNauta95]

Super cool video. I ended up using the robinson projection for a video i made drawing the political world map by memory

[@spacesandwich5593]

Is that jingle at the beggining from that video where they turn the map of Europe into a midi? (It sounds just like the left half of iceland pitched down)

[@Kuvina]

Well many years ago I had made a little song where my favorite part was like C C# E E D# C. But then I was inspired by Europe in midi art to change the last note to a D!

[@LuftpolsterManu]

I didnt know maps could be so much fun. :D

[@deathracoffee]

Awesome video

[@wickederebus]

do you have a video about the statistics of drawing particular cards from a deck of a specific size? example: Yugiho has a starting hand of 5 card, with Maximum 3 copies of a card per deck, and a deck size of 40 to 60 cards. Magic: The Gathering has a starting hand of 7 (with a rule about shuffling back and drawing fewer cards), Maximum 4 copies per deck, and a deck minimum of 60 with no hard limit on deck size.

[@ferudunatakan]

Equirectangular makes most sense to me.

[@benhur2806]

So, I did some poking and prodding on the Lambert Equal-Area Projection, and spent quite a bit of time working out the "optimal" aspect ratio, which minimises the square of the logarithm of the ratio between the change in y vs the change in latitude and the change in x vs the change in longitude... The constant k for this turned out to be 1/(pi*e^(1-ln(2*pi)), or about 0.735759... Just figured I'd share.

[@benhur2806]

Addendum since I was a bit tired from all the maths and didn't see it while writing, so here you go, k actually turns out to simplify extremely nicely to simply 2/e, with the aspect ratio of the map coming out to 2pi/e, which in my opinion is kind of pretty. xP

[@ericplays2021]

What happens if you cut the sphere into 2 hemispheres, flatten the hemispheres, and put the flattened hemispheres next to each other? I think it has low distortion

[@groszak1]

Just like the central cylindrical is projected from the center, Gall stereographic is projected from the opposite side of each longitude being projected.

[@Kuvina]

That's right! I considered mentioning that explanation, but I figured it would be too confusing to have a moving projection origin point.

[@natsuzkan]

Probably the best option overall would be a bunch of perspective views of overlapping areas, so not really a "map" of the planet in the sense of mapping the globe to a single flat surface. More like imagining a bunch of Google Earth screenshots pasted next to each other. The edges of each individual image are distorted, but adjacent perspective views will show those same areas more accurately.

[@elistidham8494]

nice video!!

[@Trylo-bot]

Hey this is great!

[@Kuvina]

thank you :)

[@jademonass2954]

why cant we take the mercator projection (the best looking one, that doesnt distort the actual shapes) and manually edit it in photoshop so the sizes are relatively correct? i know nobody would use it for navegation and whatever else, but for most people the visual is all we need

[@alan2here]

Dymaxion is nice :) It'd look even better if the background colour were blue like the oceans, and then it could be rectangular in shape. Also when choosing how to unfold the icosahedron making it more square in shape would be nice.

[@Kuvina]

Yeah I think if you want to eliminate bias (North being up, the prime meridian going through London), then Dymaxion is your best bet. But it is unconventional, making it hard to implement. Unfortunately, an icosahedron has edges everywhere, so the only way to get a more square look would be cutting apart some land.

[@hazza2247]

such a good video

[@marvinochieng6295]

deserve more subs

[@AOOA926]

Did you know that there are flat earthers all around the globe

[@gabrielbarrantes6946]

You actually don't need the fundamental theorem of calculus to resolve differential equations... Sometimes just guessing is enough... But even that antiderivates were known for quite a while before the "oficial" invention of calculus... To be completely fair, the only thing that was missing at that moment in history was noticing that the Area could be get from an antiderivate... Derivates and integrals were known for a while probably since ancient Greek...

[@wickederebus]

24:10 Funnily enough, the Flat Earther at my Yugioh local tournament store thinks the Flat Earth is just the Mercator projection, but that there is secretly some extra landmass West of America, and that you cannot get to Japan/Asia by going West from California. I don't like trying to talk to him.

[@trentHV]

20:22 Australia is italic

[@pyrokitfx5834]

Aren't meridian lines actually parallel and "parallel" lines not, because on a sphere two straight lines that start parallel will cross, and those "parallel" lines are actually curved because you know they're on a sphere

[@Kuvina]

Correct! A technically straight line on a sphere is known as a great circle, and the meridians are great circles, but not the parallels.

[@caty863]

In this digital age where our devices have gotten so powerful to effortlessly display virtual globes, there is no need to use map projections anymore.

[@Miabeloved]

Nice

[@Neptoid]

What is up with the authagraph projection?

[@Kuvina]

I didn't include it in the video, but it's an equal area map that's rectangular in shape. Unfortunately, There is a lot of direction and shape distortion, and this distortion is inconsistent across the map, so for those reasons, I think there are better options if you want an equal area map

[@mug-o-tea9517]

good video

[@cupostuff9929]

I feel like Navigation should've been a category due to how major maps are in that field

[@krzysztofq7420]

So when sailors used Mercator, they drew a straight line on a map and sailed in costant direction, byt in reality they weren't travelling the shortest path, but their trace was a curve? And it was great for sailors because they didn't want to travel the shortest distance, but they wanted to sail in constant bearing easily controlled by looking at the compass?

[@Kuvina]

Exactly! It may not have been the shortest distance, but it was pretty close, and it was the only way they had to make sure they know exactly where they land.

[@JR13751]

Tetrahedron could be considered to unfold nicely. Any projection that maps whole globe to a tetrahedron can continuously tile a plane.

[@George83_Thomas]

More mapping more winning

[@LeoStaley]

I'm disappointed you didn't include continuity in your categories. Breaks and gaps in a map are problematic for many use cases for maps..

[@gz6616]

I remembered seeing one numberphile video talking about a weird, totally unpractical projection (forgot its name) which looks like the integral symbol but with tight spirals on both ends. I'd love to hear your opinion on a problem: If I use a disk of certain radius in the projectioned x-y plane to perform a search, say within 5 pixels around a point P, what projection gives the best result if P is at relatively high latitude, say 60-80 degree? My context is that my data are on a regular lat/lon grid, and the grid size is ~ 25 km. I'd like to do a search for a 800 km radius, for each pixel. Using image processing libraries I'll have to use some map projection so my data array is a 2d matrix.

[@Kuvina]

First of all I would like to make it clear that I don't fully understand the context of your problem, so don't take my advice too seriously, but my advice would be the following: It seems to me that you want a projection that gives the least amount of distortion of any given circle. To find that, you can look at Tissot's indicatrix of deformation, which shows you what various circles on earth's surface would look like mapped out on each projection. The projection with the least distortion of these circles is technically Dymaxion, but it would be hard to translate latitude and longitude into coordinates on the Dymaxion map. Also, the edges could mess with the algorithm. For that reason, I think the best approach would actually be to use the equirectangular projection, but change the shape of the search area. If you're given a point a, b with a = longitude and b = latitude, then you can easily find it on equirectangular. If you then want to search in a circle around that point with radius r, then that circle will actually look like an ellipse on the projection. Specifically, its height will equal 2r, and its length will be 2r/cos(b), where b is latitude. This image represents this perfectly: upload.wikimedia.org/wikipedia/commons/1/17/Plate_Carr%C3%A9e_with_Tissot%27s_Indicatrices_of_Distortion.svg

[@gz6616]

@@Kuvina Thanks for the very detailed reply! Yes, I think my goal is to preserve the shape of a circle. Thanks for pointing me to the Tissot's indicatrix.

[@aidenaune7008]

why do we even use latitude lines if they are so messed up? surely using longitudinal lined rotated 90 degrees up would work better. on top of this, one could then create a third set with a 90 degree longitudinal offset, creating a cubic expression of point lines, with an intersection point every 90 degrees. the map would be a bit of a mess near the center points, but only if you use a large number of lines. if one then takes this and slices along the lines directly between adjacent center points, they would get triple right triangles, which could be flattened to equilateral triangles with only minor distortion (90 to 60 at the most extreme, meaning only a shrinkage to 2/3rds, and only at the points, with rapid dropoff). these would be fairly easy to line up on a per use basis, and an easy to use generalist map could be made simply by forming two halves.

[@Kuvina]

Latitude and longitude make the most sense when you factor in the rotation and orbit of earth. Latitude corresponds to climate because the equator receives sunlight most directly overhead. And longitude corresponds to the time of day, hence why we have time zones.

[@aidenaune7008]

@@Kuvina but latitude can be expressed as a product of the two new line projections, and we would still have longitude.

[@tristanridley1601]

Can't agree at all that you should compromise between the three distortions. Almost all purposes for maps have way more to do with area than anything else. Balance between the other two distortions, while keeping area perfect.

[@anayagrawal6550]

30:35 In the visual, the denominator should be the area of the entire Earth not just the blue area. 😀

[@dyld921]

For the orthgraphic projection, why not show it as 2 circles, for 2 halves of the Earth? Then you get the entire Earth

[@Kuvina]

Good idea! That is done occasionally, and it does show the whole earth, but it makes it a discontinuous surface, and it has many of the same issues as before, like area , direction, and shape distortion. So I probably wouldn't rank it very high, but just like every projection, it has its advantages and disadvantages.

[@groszak1]

It's also possible to do a two-circled azimuthal equidistant, or two-circled stereographic (projecting from the back of the sphere)

[@philippfrogel9355]

I got a bit lost in the making of the differential equations (which I always do), but other than that, phenomenal video, such a good structure and abstraction, you said everything at the right time etc etc. Wow & subbed ❤

[@nikolaslehto8827]

Did you know that "kuvina" means "as images" in finnish?

[@johnsavard7583]

The usual convention is to use theta and phi the opposite way around, with phi for longitude and theta for latitude.

[@sakesaurus1706]

i wanted to see the leaderboard for scores

[@helloiamenergyman]

OMG THEY MENTIONED THE WATERMAN BUTTERFLY XHN GYJHBNFHNBVGJCFYBVTUJCUT

[@helloiamenergyman]

HOW DARE YOU GIVE IT A 2 IN AESTHETICS ITS THE BEST FVUCKINGF PROJECTION EVER IN TH|AT CATEGROY

[@amatiasq]

I liked it but it was too slow for me, I watched it at 1.5x and was totally understandable

[@Kuvina]

Thank you for the feedback! I will keep this in mind.

[@vaclavtrpisovsky]

@@Kuvina The speed is fine for me but sadly, a lot of enthusiasm you obviously have for the subject gets lost in the toned-down narration and vocal hoarseness. I would suggest giving presentations in front of audiences to exercise your voice and expression. You may start with an imaginary one, and then give talks to your nerdy friends. You seem to have lots to talk about and will definitely capture the interest of any savvy crowd. People will love your ability to convey knowledge, and you will gain confidence in your voice, develop intonation and feel happier by seeing acceptance of yourself and your interests. It may take a while before you get comfortable, especially if you are still shaping your gender expression. But don't let that stop you - once you feel like taking the stage, you will take it by storm.

[@maddoxmonteza]

that's why the globe is best

[@SuperMarioOddity]

Watch in 1.25x for best experience

[@Kuvina]

I agree. Looking back at my older videos, I can't believe how slowly I talk!

[@SuperMarioOddity]

@@Kuvina It feels like it's slowed down, rather than initially slow narration. It has the type of slurring that you get when you slow down footage

[@xrete]

lol subbing

[@TheGamingG810]

31:00 this looks 3d

[@vincentcleaver1925]

Simple rectangle and we hates it!!!

[@leewilliam3417]

Mmmm 😊

[@ArconicTower]

9:01

[@ArconicTower]

32:23

[@tjjones621]

@@ArconicTower 33:04

[@xXJ4FARGAMERXx]

Great video! The only nitpicks I have are: 1-have you considered... maybe using.... _Dark mode?_ looking at constant white isn't so good for the eyes 😅 2-I felt like the explanations relied a lot on knowledge of trigonometry (sin(x), cos(x) etc..), could you have explained what sin(x) and cos(x) actually mean? I know, The problem isn't actually you, I should actually learn more about trigonometry instead of just "it kind of has something to do with angles? Theta? Circles?" But regardless, even just a quick 40 sec detour to explain what sec(x) and cos(x) basically mean would've helped a lot! That's all, Great Video!

[@Kuvina]

Thank you for the constructive feedback! I will definitely consider these things for the future.

[@falpsdsqglthnsac]

imagine a right triangle with an angle called theta. sine (sin) is the ratio of the leg opposite theta to the hypotenuse. cosine (cos) is the ratio of the leg adjacent to theta to the hypotenuse. tangent (tan) is the ratio of the opposite to the adjacent. you remember this with the acronym "sohcahtoa". it's more helpful than it looks. their reciprocals are cosecant (csc), secant (sec), and cotangent (cot) respectively.

[@TheGamingG810]

imo mollweide is best

[@k.k.9378]

Love the content. Would be easier to listen if you moved your mouth more. Still love the content.

[@PrivateLZG]

Probably stupid assumption, but couldn't you take the Mercadian projection and take every infinitely small, horizontal line (maybe pixels) and scale it down on both axes? Like, for a Robinson-like projection that is done, but just horizontally. If you scale everything correctly, won't the shape of everything stay the same while making everything sized correctly?

[@bumbleandsimba]

889th like

[@Maukustus]

why dymaxion look like a bird lol

[@chengong388]

The perfect projection is an iPad with maps, perspective projection where every angle is accessible.

[@cmyk8964]

You could benefit a lot from voice lessons.

[@Mag3.1415]

YOUR MIC WAS SO BAD