## Live Perspective: A New Approach to Depth in Drawing

Perspective. The word freezes the blood in the veins of every aspiring artist (and even many of those who seem to be pretty good at what they do). This “method of drawing 3D forms in 2D space” is full of confusing mathematical rules that seem to have nothing to do with carefree, passionate drawing. Even if you manage to grasp these rules, you might still wonder how they apply to the real world. When you look around, do you see one-point perspective or two-point perspective? If the horizon is always at eye level, what happens when you look down? What actually are vanishing points? And can you forget about perspective as long as you don't draw architecture?
In this article I won't explain all the rules of modifying an object in linear perspective. There are a lot of tutorials about it, so you can look them up. Instead, I'll explain to you where these rules come from and why someone needed to invent them. The rules, after all, are only a way of describing a fascinating phenomenon, one present in nature since the day our brains started to process the signals from our eyes. After reading this article, your world will never be the same!
Forget about math and geometry. Go back in time and remember those days when you were traveling and observing the buildings and objects moving with you. Those closest to you were moving the fastest, and these in the background were scarcely changing position. And the furthest of them, the moon, wasn't moving at all—it was, and still is, always there, no matter where you go.
But, of course, it was very silly of you to think the objects actually moved when you did. It was just an illusion, like how your monitor or a table looks skewed when you look at it from the side. Of course, it's a rectangle, so it's only an illusion. We're so used to these illusions that we don't see them any more, and if a child asks why the buildings move, or why a table is so skewed, for a while we may even not understand what they're talking about.
“Illusion” is a word we use to explain things that our brain makes us believe in, although they're not real. A table looks skewed. A building looks as if it's moving. The problem is that everything about looking is an illusion! Colors, position, length, width, height, rotation, even texture don't exist in the way we see them. The image in our head is just an interpretation of reality—an interpretation irrevocably relative to us.
How big is this object? Can you really tell?
Let's add something else to the scene. Now it's a small square, right?
Or... maybe it's huge.
Size doesn't exist without a relation. Nothing is big or small by itself—you need to compare it to something to define the size. Usually, we use a "default" size of something as a reference (a big apple is bigger than most of the apples you've seen).
But where is our square? Is it far or is it close?
It looks far now...
But it may be close, too.
Is it high?
Or maybe low?
An object isn't anywhere until you define a reference point. You need to create a relation between `x` and `y` to say where `x` is. Unintuitive? Keep on reading. I'll explain it all later.
Does this square move? Probably not, right?
Wait... Did I see something?
But... what's really moving here? The pink square or the ghost in the background? We will never know! And even in the first picture the white background may be sliding all the time, but you won't notice the movement until something changes in the picture.
You can tell whether something is moving by comparing it to another thing that isn't moving. The change of distance between them is the way you measure the speed. People used to believe that the Sun revolved around the Earth, and now we believe it's the opposite. The truth is neither of those is true—or they both are.
All these examples have one thing in common: a relation must occur for them to exist. Perspective is just a name for a relation between the observer and other objects. See? No math at all.
You may think, "But objects are somewhere just like that, they don't wait for us to tell them they are there!" It may look unintuitive, but there are a lot of expressions created by humans in relation to us:
• If I need to move a lot to reach it, it's far.
• If my arms get tired quickly when I carry it, it's heavy.
• If I barely feel it in my hand, it's light.
• If it burns when touched by me, it's hot.
They can be easily translated to a simple pseudo code:
 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 `If (distance(my.position,x)) > 100*my.steps ` `then x=far; else x=close;`   `If (weight(x)) > my.strength ` `then x=heavy; else x=light;`   `If (temperature(x)) > my.temperature ` `then x=hot; ` `else if (temperature(x)) == my.temperature ` `then x=normal;` `else x=cold;`   `If (size(x)) > my.remembered_size_of_x ` `then x=big; ` `else if (size(x)) == my.remembered_size_of_x ` `then x=normal;` `else x=small;`
Depending on what "me" you use, the actual result will be different. For most humans they will be similar, but you can be a strongman and call a fridge "light"—and you won't be wrong when saying it! What we call "true" is just a set of properties that the majority of humans would agree with. A fridge is heavy, because most people would have problems lifting it—not because it's heavy by itself.
What's interesting is that the expressions "far", "close", "big", "small", "heavy", "light", etc., change their meaning all the time depending on the variables. A remote control is far away from you when you need to get up to switch the channel (let's say, 3 meters), but at the same time a restaurant on the next street (300 meters) is close to you.
It may look like philosophy to you, something conceptual, one of many ways to describe reality. The fact is that all these things—size, position, distance, motion—are nothing but concepts. Imagine you're a kind of god and suddenly you can observe a world without it all! In fact, you can't imagine it—when trying, you're most likely "flying out of your body", but still being and observing everything from one point. We are our own references, and it's impossible—at least for a mentally healthy, sober person—to imagine the universe without any reference point. What's more, "feel", "touch", "observe", and other expressions like these imply an analyzing tool and analyzed object. We can't, in any way, sense objects without using us as a reference—as long as we are humans, we can't know what anything really is. Math is the closest we can get to its image, but the more accurate and complete it gets, the less people are able to understand it.
More specifically, perspective is a relation between a certain sense of a particular person and an object. Every sense may have a different perspective. That's where illusions come from—if an image received by one sense doesn't match the others (or our knowledge about it), we say it's not true. You can check it by closing your eyes in a small room with white walls. Spread your arms and you'll be surprised how tiny it became!
We use vision as the most important sense, so we tend to imagine that reality is just like what we see. The world of darkness, when our eyes stay closed, is a different world that we like to call incomplete. The fact is that what we see is incomplete too—our eyes and brain process only a small fraction of all visual signals available. We live in a reality that exists only for us, and is similar—but not necessarily identical—among humans. We don't know what the world looks like. It's being rendered right in front of you with every move of your head. That's why objects around you change form when you move—it's not an illusion, they really do. Shapes and forms exist only in your head, as an interpretation of certain information processed by your brain. There's no "true" form, one not created by your brain. All of them—straight and skewed—are the same. Either you call them all illusions, or they're all true.
What does it all have to do with art? And where is perspective, the one sketched with clean lines and vanishing points, in all of this?
I hope I didn't bore you with this lengthy explanation, but I think it's crucial to truly understand what I'll say next. As an artist you create an optical illusion—you use lines and patches of pigment to make people believe they're looking at something they know from reality. This illusion needs to take into account every single vision mechanism we know to be complete. You can't draw a bowl of apples, because, as we know now, we have no idea what it really is. You draw a seen bowl of apples—seen by someone's eyes.
This is where it all begins. When you draw from a reference—be it a photo or reality—you simply copy the image you see in your head. That's why it's relatively easy to achieve amazing results from this—you only need good manual skills and hand-eye coordination, both simple to learn.
Most people see this process as "copying reality". Again, it's impossible to create a copy of a bowl of apples with your brush (A). You can only create a visual copy (3) of the image created in your mind (2) when looking at the bowl of apples (1).
We're getting closer to the actual meaning of perspective now. The position of the observer, the distance between their eyes and the object, the health condition of their eyes, and the mental health of the observer all create the seen image. There are two important conclusions:
• The image of an object is an interpretation of one's brain.
• The same brain will create a countless number of different images of the same object when the position of the eyes changes.
Now, to the point. When looking at a picture, you don't see the pictured object (A)—you see a brain image you'd create if you were looking at the object from one strict position, angle, in certain light and brain conditions (B).
If you're confused, look at the illustration below. When looking at a picture, you imagine yourself as the observer. In your mind you reconstruct the conditions and position, and then you're able to imagine the object as a whole.
A set of variables of the observer (position, the angle and range of view, etc.) in relation to the environment is the meaning of perspective we use as artists.
It's still quite confusing, isn't it? Let's learn some more about depth.
How is it possible you can see 3D in a 2D picture? The same way you can see depth with only one eye! In fact, binocular vision is the most useful at a very short range—you can use it to thread a needle or do some other precise tasks. For other cases, like distinguishing "close" from "far", we use our observations from the past. We know how big an apple is when we hold it in our hand, so when it's much smaller than this, it must be far. For a complete image, we also use eye accommodation, comparison, and light and shadow.
The observer has only one eye, as long as we don't have the technology to draw 3D pictures in an easily accessible way. But it doesn't really matter! When you see a 3D model on your screen, it's 2D. The illusion of depth is made when you start to rotate it. The same trick is used when you've got one eye—you move your head to change the perspective and suddenly depth is created. Why? Because 2D pictures have only one perspective. If you can easily switch between at least two of them along some common dimension, it becomes 3D for your mind. It's because in a 2D scene, an object can only move up-down and left-right, or across. When it moves in some other direction—towards or away from you—another dimension is added. This third dimension is depth.
But why do some drawings look 3D, when they've all got only one perspective? It's because some perspectives suggest the existence of other perspectives. You look at them and it's very easy for your brain to imagine what would happen if the observer moved. Others don't give any hint about additional perspectives, so it's impossible for us to imagine them properly. If you've ever wondered why it's so easy to draw one side of the character, and so hard to make it more dynamic, here's the answer:
There are perspectives that convey only two dimensions. Let's call them 2D perspectives. Since a sheet of paper is 2D too (at least from our perspective...), conveying only two dimensions on it is very intuitive. However, you can't get round the third dimension and expect it still to be readable! Drawing in a 2D perspective inevitably leads to a flat picture—something that maybe has a third dimension, but we can't know anything about it, so we assume it doesn't have any.
2D perspective, as I call it, is known in technical drawing as orthographic projection. By drawing at least two sides of the object we're able to determine how it's going to look in 3D. However, none of the projections is a default perspective—because there's no such thing as a default perspective. Again, as humans, we have no sense that would let us "process" a whole object. For us, every object is made up of countless perspectives—and we can only see one at a time.
So, here's the problem—you can't draw something without any perspective. It would be like trying to draw an object as seen by nobody! Therefore, every time you draw something, you convey some kind of perspective—no matter if you know what you're doing or not. Unfortunately, when you try to learn something about perspective, you stumble upon a technical approach with a bunch of weird, stiff rules. Here's how you draw horizon, here's a vanishing point, one, two, three of them, right angles, walls, repeatable shapes, order... You look at it, you learn it, but you can't see any relation to what you draw for fun. Eventually, you decide it doesn't, indeed, have anything to do with your hobby, and you can ignore it.
I've been there, too. But let's say it once again: an image is created when it's seen. When something is seen, a perspective is created automatically. Therefore, perspective is sewn into everything you draw. You can either learn it or not—but there's no way to avoid it.
Cheer up! Luckily, it's not so hard to learn. After all, you've been doing it intuitively for years! You just need to organize your knowledge—then you won't need to guess any more. Perspective will work for you!
Finally, the part we've all been waiting for! We've already clarified that perspective is a crucial part of every drawing, not only technical ones. But where does it come from? How is a single perspective created? How and when does 2D perspective transform into 3D? And why do 3D objects on a 2D picture look distorted?
Open your mind—this is something you might never have thought about. It will be counter-intuitive, because you've been using Euclidean geometry all your life, and, as we're going to learn soon, vision doesn't work that way. It's not easy to jump from one way of thinking to another after all these years, but it's certainly worth it!
Let's start with the explanation of dimensions. You may know 2D is flat and 3D is...well, 3D, but how does it work? What's the difference between flat and three-dimensional objects?
Let's start with a probably shocking fact—objects aren't 2D, or 3D, or 5D—they're only immersed in dimensions and are perceived by us as a complete image made by parts from every dimension. That's why a cube can be a square, a square can be a line, and a line can be a point. We call an object "3D" if it exists in a third dimension as something more than a point.
It doesn't matter what we call the dimensions. What matters is that for us there are three of them. Let's start with two dimensions.
This is a 2D sheet, right? We know it well. It has width and height, and that's all we need to draw anything flat.
Actually, no. Two individual dimensions don't give us anything as long as they're separate. A line has a full length in the dimension only when it's parallel to it. In other cases it's shorter, and when it's perpendicular, it becomes a point! Not to mention how lines lying in a perpendicular row become one.
To create a real 2D space we need to add the second dimension to every point of first dimension...
...and the first dimension to every point of second dimension.
It may look like a mess, but now we've created a space shared by two dimensions. Now, no matter where the line goes, it will be registered by both dimensions. We're able to determine the length of the line even if it isn't parallel to any dimension!
For example, when a line isn't parallel to any dimension, the final image is created by merging a fraction of information from every dimension it has crossed.
In 2D space we can go left, right, up, down, and everywhere in between. However, there's no "forward" and "away", no "close" and "distant" here. Distance will be our third dimension—when you move one 2D sheet under or over another, depth is created.
To create 2D space, for every point of one dimension we added another dimension. It's the same with 3D space—for every point of third dimension we need to add a slice of 2D space.
However, both a sheet of paper and your screen are 2D. We can't picture the third dimension here! The illustration below is just a concept, not a reflection of reality.
If we want to draw a line just as it looks in one dimension, no problem. The same with two dimensions. But that's where it ends—we can only draw two dimensions at the same time on a 2D sheet. When we want to add the third one, it will squeeze to 2D space—the lines will be distorted, just like when we wanted to picture a 2D line in a single dimension.
It's also important to notice that there isn't any particular side of the object that is "third dimension". Now, when you can easily slide between three dimensions, with simple rotation a front side can become a back side, and the top may become the bottom. All the dimensions register the object, but the object isn't part of them.
An interesting fact is that we could add more dimensions—one 3D space for every point of fourth dimension and so on. It's very simple in math, but we humans perceive only three dimensions, and any more are nearly impossible to imagine. That's good for us—three dimensions are hard enough to grasp in art!
Our eyes aren't the most perfect of all the animals; they're actually pretty bad. Although with both eyes we have about 120 degrees field of vision, only in area 1 we can see sharp details and colors. In area 2, colors and blurry shapes are all that's left, and area 3 is mainly used to see motion only. However, our brain fills in gaps and we believe the image in our head is as good as a photo—with colorful, sharp details in every point. It also persuades us there's no blurry, double nose right in the center of our vision.
Our FOV cone is made of an infinite number of 2D (horizontal and vertical) planes placed along a line (distance—depth) between the eye and infinity. For our convenience, we're going to call the 2D planes frames. The second cone is what we usually imagine it to be, but it's closer in look to a camera's FOV than ours.
That's right—there are no "corners" of vision. We look around, not along vertical and horizontal lines.
Why a rectangle, then? Probably because it's a regular shape, easy to create as a painting canvas or a data array. It has nothing to do with our vision; it's just practical to use elsewhere.
Here's a symbolic interpretation of a FOV in the simplest configuration (only one eye used, we don't need more).
1. Glasses: I used them to show you where your eye is.
2. Nose: it's always there, but your brain tells you it isn't.
4. It's the area of your height.
5. Ground: place objects here for them to stand steadily.
6. Underground: if there's a hole in the ground, or the ground is actually water, you can make a good use of this area.
7. The edge of your upper eyelid.
8. The edge of your lower eyelid.
9. Certain distance between the eye and the ground.
It's important to remember where the level of the ground is. If you're using a human as the observer, imagine a person talking to them eye to eye, with the face covering a big part of the frame. Where would they stand? That's where the ground should be.
You don't need to use the whole FOV for your picture. You can crop it however you want, rotating the horizon for a feeling of lost balance and placing the center away from the middle. Feel free to experiment with it!
The most characteristic feature of perspective, objects getting smaller with distance, can be easily explained with the FOV cone.
While the cone gets wider with distance, the size of every frame stays the same for our brain. When you look at something very close to you, you don't see that your field of view suddenly got smaller—you only notice that the object got bigger in comparison to it. Objects don't change when getting closer or farther, they only land on different frames. The bigger the frame, the smaller the object seems in comparison. That's why you can cover the whole world with a hand—at one point it's indeed able to cover the rest of the cone.
Scale has to do with the perceived speed of objects. The farther the object, the longer the perceived way between both sides. Just compare a length of three cars in a row and a dozen big buildings—and they're both squeezed into a line of the same length.
It also explains why the back side of a cube seems to move at a different speed than the front—they're both on different frames!
Because of what we've just described, eventual changes are the most conspicuous in the narrowest part of the cone. Right in front of your eyes an apple can cover the whole world, but with distance it becomes less and less significant. That's why we can usually ignore the motion of eyeballs and assume that the FOV cone starts in the front of our head—and you can freely rotate your eyeballs when keeping your head still without changing the perspective.
Now we know why the size of an object changes with distance. But how can we determine the "default" size? At what point does the object look as big as it really is? If you've been reading carefully, you should know the answer—there's no such thing as a "real size". When you measure something with a ruler, you compare it to a model size of 1cm—a model that changes with distance too, so it's not constant for your eyes. There's no way to measure an object changing in perspective.
However, there's a trick our eyes use to overcome this inconvenience. The first clue to estimate the size is to notice how big a part of the frame it takes.
We've already noticed that even big objects get smaller with distance. How can we tell a big, distant object from a close, small one, then? We need some kind of depth indicator, which is what our eyes use when the distance is too big for the binocularity to be useful.
This is the most basic one. You know that a building is big enough to store you inside, so when it looks too small for it, it must be far.
Since the frames' size changes regularly, we can use proportions to estimate the size. It means that everything inside one frame will get smaller according to some kind of a factor that you can use in your equation to come back to the primary outcome. That's why we often use a human silhouette somewhere in the scene to stress the size of it. You can also use other well-known objects, like trees or mountains (when they're small in comparison to the main object, it must be huge), or grass (when it's huge, the main objects must be tiny).
When using shallow DOF you're able to separate close objects from distant ones. An easy trick is to draw some negligible objects just in front of the observer and blur them, to show the distance between the observer and the scene. Even if you don't want to use blur, the areas out of focus should be less detailed.
One object can cover another only when it's closer to us than it. It speaks a lot about distance and it's the simplest, most intuitive method for creating depth.
You can read more about it in my other article, but here's the point: The further something is, the more the color of the sky is being scattered between you and that object. It doesn't work when the air is very clear, but in most cases a bluer, lighter object = a distant object.
By mixing all these tricks you're able to achieve the same kind of depth monocular people observe. There's also a cool experiment to see how good your brain is at recreating depth from a 2D picture. Find a big, good-quality photo (it may be on the screen), close one eye and make a "telescope" out of your hand. Look through it at the photo, to see only the image and nothing else. There's a good chance you'll see it in 3D!
If you looked carefully at our cone, you should notice an odd thing. The 2D planes aren't really flat—they're like shallow bowls. It means they're spherical like the Earth, and just as we can't create a perfect, undistorted 2D map, we can't create a 2D frame without distortion.
The illustration below shows clearly that the line, although perpendicular to the sight line, lands on separate frames. As we know, the farther the frame, the smaller the object—so a part of the line will become smaller, making the line shorter and turned away from us!
To obtain a possibly undistorted image, the object needs to be placed right in the center of the FOV cone, with all its sides perpendicular to the sight line. It's impossible in the case of 3D objects—that's why they'll always look distorted.
By the way, a camera lens catches this distortion too, but it's usually undesired and cropped by the sensor. Wide-angle lenses accept part of this distortion, while fish-eye lenses take it all. In fact, our eyes work as fish-eye lenses—that's our brain that tells us we're seeing straight lines! Don't believe me? I'll explain it better soon.
Let's see how it works. When we want to see another side of the cube, we need to rotate it. However, at the same time the perpendicularity of the first one is being lost—both sides are sliced across multiple frames on a different distance (depth). Therefore, part of them looks shorter and more distant—they look rotated.
That's the first mystery solved. But is there any way to foresee the distortion without drawing a 2D view with all these curves first?
First you need to remember we've got two horizons—horizontal and vertical. We're so familiar with the horizontal horizon that we don't even notice the other one. But of course, that doesn't stop it from existing!
Both horizons cross right in the center, at the point you're looking at. You can move along the horizon, up and down, which is the same as sliding left and right. For now let's assume that left and right refers to the horizontal horizon, and up and down refers to the vertical one.
You can also move across, for example sliding up one horizon and left on the other.
The center area looks the closest to us. It's also the least distorted area. That's why it's used as a full frame and a base for linear perspective. However, this approach doesn't explain why the lines bend!
Remember that the image in your head is spherical; your brain only persuades you it is completely straight. When focusing on a small area in the center (A) the bending isn't that noticeable, but on a bigger scale it's crucial for a proper 3D look. Take a good look at the illustration below.
Imagine a row of cubes standing along the horizon, parallel to your eyes. The one at point `A` will look the closest to you, while the others will be observed as receding.
Why? It's the same distortion we were talking about earlier. Let's talk about the front sides of these cubes. Both points `A` lie on the same frame, so they're perceived from the same distance. However, between points `B` and `C` there's a difference in depth. For points `E` and `D` this difference is huge!
If you're still wondering how it's possible that we get a convex image on a concave frame of the cone, here's the answer:
The ultimate conclusion of all of this is the illustration below. The best and simplest lesson of perspective you can get is:
The higher the object over* the horizon, the more of its bottom** and the less of its front is visible
You can now create analogous situations to this, with "*below" and "**top", or with "*to the left of" and "**right", etc. Just make the pairs of opposite sides and it will work! An addition to this lesson is:
The farther the line from the center, the shorter
That would be all. What? Too simple? Where are the vanishing points and all...? If you really want to know, here's the answer:
Linear perspective is a simplification of everything we've been talking about. Let's see how it's possible.
In this perspective all the straight lines are parallel or perpendicular to each other. They don't converge at any point. This is the perspective we can observe when looking at the center area of our FOV, when the object stands in front of us.
In this perspective all the lines that aren't parallel or perpendicular to each other converge at one point on the horizon. This is an effect similar to that observed in the center area, except in reality a slight distortion will occur. The objects need to stand perpendicularly to the sight line for this.
In this perspective there are two points on the horizon where all the lines that are not parallel to each other converge. We can observe this effect when expanding the center area. Here the objects are allowed to be rotated.
In this perspective there are no parallel or perpendicular lines. They all converge to one of the two points on the horizon or to the third point on the vertical horizon. This effect can be observed when looking peripherally, especially up/down (e.g. observing a high building). Rotation is welcome.

### TDasany

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