If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

How one-point linear perspective works

Linear perspective, a method of creating a three-dimensional world on a two-dimensional surface, was discovered by Filippo Brunelleschi in the early Renaissance. This technique, which uses a vanishing point, horizon line, and orthogonals, was used by artists like Leonardo da Vinci to create realistic and expressive art. Created by Beth Harris and Steven Zucker.

Want to join the conversation?

  • female robot grace style avatar for user J Helston
    At Dr. Harris mentions that some may argue the ancient Greeks and Romans utilized some elements of linear perspective. My questions are:

    1) Can someone please provide links to these ancient Greek or Roman instances of linear perspective?

    2) More importantly, are there reasons why these techniques were "lost" or "under-utilized" after the ancient Greek and Roman cultures until the Renaissance?
    (23 votes)
    Default Khan Academy avatar avatar for user
  • leafers ultimate style avatar for user Scott
    We often talk about linear perspective (as is the case in the video) as a "discovery", but is this really correct? Can we call it a discovery in traditional terms, or is it more of an innovation, as other artistic techniques, such as shading, are used to make paintings more realistic?
    (15 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user WesleyBurchnall
      I think you have to examine the debate between mathematics being a discovery or an invention. The traditional view of 'copyright' as it pertains to mathematics. The general notion is that facts cannot be copyrighted.

      For example, the location of the white house or it's address cannot be copyrighted; nor can my phone number. Math is considered to be a 'fact' and that new mathematical discoveries are just 'uncovering' pre-existing facts. Like for example, a right-angle triangle having at least one degree with 45 degrees is considered a fact that was discovered and could have been found in 'nature' with careful enough observation.

      If you look at the book of 'steps' or 'rules' by which to create a linear perspective 'on painting' which is probably akin to a 3D to 2D translation section in a linear algebra textbook, I would describe that book as a publishing of a mathematic discovery of a fact. Infact, linear-perspective in mathematical terms would be an orthogonal projection.

      I don't believe there is a significant degree of interpretation or creativity of the linear-perspective process is applied, but rather, a degree of accuracy. The end-result of the linear-perspective painting if the rules/steps are followed I think is more dependent upon the skill of the artist in terms of artistic techniques.
      (24 votes)
  • leaf green style avatar for user 福龍丸
    Is it possible to create an illusion of a four-dimensional reality using our 3-d space (the fourth dimension not being time, but another spatial dimension)?
    (7 votes)
    Default Khan Academy avatar avatar for user
  • male robot hal style avatar for user Abe Vallerian
    Does mathematics have something to do with this Linear Perspective?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user Yabusame
    When selecting a Horizon Point (HP) outside of the picture, how do you know where to place this on your horizon? Surely, where you place the HP will affect the angle of the rays (~) and thus the height of the intersections therafter.
    (5 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Tatianna Jaynes
      From what I understand from the video the Horizon Point (HP) is actually what will determine the Horizon Line (HL). Now again, the way I understand it, the HL is meant to be eye level with the viewer. Take a world for example, using the Equator as the HL. Now if you put the HP Northwest instead of due West, you haven't changed it you have merely rotated it. . . If I really wanted to screw with the intersections, I'd move the vanishing point.
      (1 vote)
  • blobby green style avatar for user 124257
    I am very sorry to bother, But what are Orthogonals? (not Orthographic)
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Carlota Grs
    Does the figures eyes always have to be on the same level as the horizon line? What if in the composition, for example, there is one standing, one sitting, and one laying down? How do I know where to put them acording to the horizon line?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user jkimball
      You would make sure that the eyes of each figure would line up with rays drawn through the external point drawn on the horizon line, so that their perspective would be maintained no matter how big or how close they were.

      The reason they put the eyes on the horizon line is for aesthetic reasons.
      (3 votes)
  • aqualine ultimate style avatar for user Natasha Lahoff
    Did Alberti base his knowledge of art off the Greeks and the Romans, other painters in his time, or his own works when he wrote "On Painting"?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Kathleen Maisto
    Hi....I was sitting in my kitchen looking at a 4 shelf metal stand and trying to draw it...had a hard time in drawing the angles...can I use one-point linear perspective to draw this ?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • leafers tree style avatar for user Ben
    At , could the point on the horizon be put anywhere?
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

[MUSIC PLAYING] SPEAKER 1: So this is a video about the elements of linear perspective with a little bit of history thrown in. SPEAKER 2: I love linear perspective. SPEAKER 1: It's hard not to love linear perspective. It's like this magic formula. SPEAKER 2: Well, look what even Paolo Uccello was able to do just a few decades after linear perspective was first discovered. SPEAKER 1: So linear perspective is a way of recreating the three-dimensional world on a two-dimensional surface. And it's really accurate. SPEAKER 2: Well, look at this Paolo Uccello. Look at this Study of a Chalice. This wasn't done on a computer. This was done with pen and ink on paper. SPEAKER 1: No Photoshop. SPEAKER 2: No Photoshop. SPEAKER 1: So let's give a little bit of historical background, and then we'll talk about how it's done. SPEAKER 2: OK. So let's start first with what the problem was. SPEAKER 1: OK. So here we have a painting from the early 1300s by an artist named Duccio, who's painting at Siena. And you can see that Duccio's interested in creating an earthly space for his figure of the Angel Gabriel and Mary, but that the space doesn't really make sense. SPEAKER 2: OK. So what you're saying is that we have kind of a real room here. We can see the beams in the ceiling. We can see the architecture. We can see the doors. And so he's really interested in putting these figures in a real place. The problem is-- and by the way, don't get me wrong. I love Duccio. But the problem is is that Duccio is not constructing that architectural space in a way that looks logical to our eye. SPEAKER 1: And I think it probably wasn't a problem for Duccio. But it was a problem for artists about 100 years later who had a different goal. And their goal was a kind of really accurate realism on that flat surface. SPEAKER 2: OK. But before we leave the Duccio, let's spend just a moment being kind of unfair and finding what's wrong. SPEAKER 1: OK. SPEAKER 2: OK. So for one thing, the beams of the ceiling right up here don't agree spatially with the seat that the Virgin Mary is on or with this little stand for the Bible that we see here, or, for that matter, with the lines that are constructed by the top of the capitals of these balusters. So none of this is really making sense. SPEAKER 1: Right. It's not a rational space. And there's this increasing interest in the 1400s in rationalism. SPEAKER 2: That's the period that we really call the Renaissance. SPEAKER 1: Right. The early Renaissance. And so in Florence in 1420, Brunelleschi-- and let's put up a picture of Brunelleschi. SPEAKER 2: OK. So he's right here, Filippo Brunelleschi. SPEAKER 1: And he discovers-- or some would say rediscovers, because some think that maybe the ancient Greeks and Romans had this before-- but he discovers linear perspective. SPEAKER 2: So he was a genius. SPEAKER 1: He was a Renaissance man. SPEAKER 2: He was an architect. He was an engineer. He was a sculptor. And according to tradition, he had gone down to Rome, and he was studying ancient Roman buildings, ruins, and he wanted to be able to sketch them accurately. And he developed this system, linear perspective, as a way of doing that. SPEAKER 1: And in 1420 in Florence, he demonstrated this system. And 15 years later, another brilliant Renaissance man, Alberti, codified what Brunelleschi had discovered. He explained the system of linear perspective for artists. SPEAKER 2: So he publishes a book called On Painting in 1435, and we have a later version of that book right here. And inside that book, he really gives the formula for linear perspective, and that's what we have here. So let's just spend a moment talking about how this system works. SPEAKER 1: OK. So let's go down here, and let's actually do a diagram of linear perspective. SPEAKER 2: OK. Now I cannot do Paolo Uccello's chalice, but I can draw a basic linear perspectival structure. SPEAKER 1: OK, go for it. SPEAKER 2: OK. So first of all, we need to understand that one-point linear perspective, sometimes called scientific perspective, is made up of three basic elements. There's a vanishing point, there is a horizon line, and there are orthogonals. So let's start off with just creating a simple interior. I'm going to draw just a rectangle here. SPEAKER 1: So this is your painting. This is your flat surface. SPEAKER 2: That's exactly right. And I'm going to decide that the vanishing point needs to be pretty much in the middle. SPEAKER 1: OK. SPEAKER 2: So I'm putting the vanishing point right about here. SPEAKER 1: OK. SPEAKER 2: OK? Now let's see. SPEAKER 1: Why don't you label that VP so we remember it's vanishing point. SPEAKER 2: OK. So that's the vanishing point. Now what I want to do is I want to create a series of rays that move down to the bottom line. And these, one could think of as kind of floorboards in a room, right? And artists had been able to do this long before linear perspective. Artists had never had a problem with this. SPEAKER 1: Right. Well, that's because they were constructing it intuitively. And intuitively, when you look around at the world, you see walls in a room that look as though if they continued they would meet. Or the floorboards look as though they would meet. So it's kind of intuitive. SPEAKER 2: So I'm actually going to add not only a floor to this room, but I'm going to put in a couple of windows. We'll just make it very simple here. So I'll put in a couple more verticals right here. And then I'm simply going to have all of this meet in the middle at that vanishing point. Now I'm going to use an eraser here just to clean this up just a little bit so we can get rid of some of the extraneous lines just to make things a little more clear. And voila. You can sort of see a window-- SPEAKER 1: OK. I've got a window. SPEAKER 2: --beginning to form. But now here's the problem. The problem was if you didn't want to have floorboards and instead you wanted to have a tile floor, you had a problem. Because you know intuitively the horizontal lines have to get closer together as they go back in space. The problem is it's hard to exactly figure out what those proportions are as they get denser and denser as they go back in space so that the floor doesn't look like it's popping up. SPEAKER 1: Which happened often, actually, in paintings from the Trecento. So the idea is that the tiles get smaller and smaller because things generally get smaller and smaller as they move away from us in space. SPEAKER 2: Or appear that way, at least. SPEAKER 1: Right. SPEAKER 2: So what Alberti wrote down in On Painting was that you need to have a second point in space outside of the picture plane that was at the level of your eye. So I'm just going to put it here. It's at the same level as the vanishing point, right? And so we would call this, of course, what? This is H. This is the horizon line. And I missed it, but there it is. SPEAKER 1: OK. SPEAKER 2: OK. And then what I would do-- and I would, of course, do this more accurately with a ruler-- is I would draw another series of rays from that second point-- SPEAKER 1: From the exterior point. SPEAKER 2: That's right. And have it connect to each of those floorboards, right? And so as you can see, what's happening is that that angle becomes more extreme as I move across. Right? And I'm doing it freehand, so it's a little bit hard to see, but you get the point. Now something really interesting just happened, which is I can now create a horizontal line that is at that first intersection-- do you see that right there?-- going straight across. SPEAKER 1: I see it. SPEAKER 2: Then I can draw a second one at that second intersection right there, and so forth. And they get more and more compressed as I go back in space. And the illusion should be, then, a kind of compression in space. So I think this will become more clear if I just do a little bit of erasing now. SPEAKER 1: OK. While you're erasing, I want to talk about that word illusion. SPEAKER 2: OK. SPEAKER 1: Because I think it's key to everything here. SPEAKER 2: Absolutely. SPEAKER 1: What artists are looking to do is to create an illusion of reality on this two-dimensional surface. Alberti said a painting should be like a window. So in a way, you don't see the two-dimensional surface. A two-dimensional surface becomes something you look through to a world that is a continuation of our own world. So the idea of the illusion being incredibly convincing was so important to the artists of the Renaissance, artists like Masaccio or later Piero della Francesca or Andrea Mantegna. SPEAKER 2: And so now I'm just going to fill in a few of these tiles alternating so that you really can get a sense of that floor in space. Whoops. So is that working? SPEAKER 1: So even in this rough way here on this tablet, this is working, basically. SPEAKER 2: It actually couldn't be rougher, could it? But I think it still makes the point. If I were then finally to get rid of these lines and, in fact, get rid of the vanishing point entirely and instead now draw in a back wall, we have something that comes fairly close to looking like an interior space. SPEAKER 1: Now what about putting figures in? SPEAKER 2: Ah. So now you're really asking for trouble here. SPEAKER 1: I'm sorry. Can you do that? SPEAKER 2: I don't know. Let's see. So if I were to draw a figure, what I would like to do is make sure that the eye level of the figure is approximately at the horizon line. So I would put that figure in just about here. SPEAKER 1: And what if you put a figure more in the foreground or more in the background? SPEAKER 2: So if I put a figure that was more in the foreground, I would still want their eye level to be at that imaginary horizon line. But of course, now they would be larger. SPEAKER 1: Right. So I think this is the part that's counter-intuitive. The heads are on the same level, and it's the feet that are on different levels. SPEAKER 2: That's exactly right. And Alberti also said that that eye level, that horizon line would ideally also be the viewer's eye level so that the perspective would really work perfectly. SPEAKER 1: OK. So we have orthogonals, the diagonal lines that meet at the vanishing point. We know the vanishing point is a point on the horizon line, and we understand how these correspond to the viewer and to creating an illusion of space. SPEAKER 2: Let's take a look at what somebody who can really draw does with this. SPEAKER 1: OK. SPEAKER 2: Let's take a look at Leonardo da Vinci's The Last Supper. SPEAKER 1: OK. So not you. SPEAKER 2: Not me at all. SPEAKER 1: Someone who can really draw. OK. So here is Leonardo's Last Supper. Immediately, the interesting thing is that after Brunelleschi discovers linear perspective, artists like Masaccio begin to use it. But they realize that in addition to creating an illusion of space it has a way of bringing the viewer's attention to the vanishing point. So artists begin to use it not just to create that illusion, but they begin to use it expressively. And that's what we really see here with Leonardo. SPEAKER 2: So not only is Leonardo creating this beautiful perspectival space, but he's also focusing our attention on Jesus Christ at the center who is the vanishing point. SPEAKER 1: Right. It brings our eye, our attention to the divine. SPEAKER 2: So here we see Leonardo's Last Supper, and we can certainly just intuitively make out the orthogonals and the vanishing point. But let's go down and really look at the diagram. SPEAKER 1: OK. Here we are. SPEAKER 2: So it's interesting. Their eye level all across is basically at the horizon line. And of course, we see the vanishing point, the point where all of the orthogonals intersect, which is right here. And so we have all of these lines that are moving across the surface of this wall, and they are all bringing our eye right to Jesus Christ in the center. SPEAKER 1: And those lines are orthogonal lines. And there you have it. SPEAKER 2: That's how it works. SPEAKER 1: Linear perspective. [MUSIC PLAYING]