my perspective corner

The shapes of circles are intuitive, the angles of them are less so.

The angle of an ellipse should match the angle of the surface that it is drawn on. However, the angle of a surface is not as simple as vertical or horizontal when the camera is at an angle. What makes things worse is that the lines surrounding a surface do a really bad job of telling you the angle of it. In perspective, we would look at the angle of a surface’s horizon to figure out its angle. Unfortunately, we don't have horizons in orthographic projection. Our best option is to look at the lines that are perpendicular to that surface instead.

To figure out what angle you should draw an ellipse on a specific surface, try to figure out what lines would be perpendicular to that surface, and draw the ellipse perpendicular to those. I’ve heard a lot of people phrase this like “when drawing a cylinder, the minor axis of the ellipses should align with the center of the cylinder”. I do not like this phrasing. I think that it focuses too much on cylinders specifically. I believe that phrasing it as finding the angle of a surface makes it easier to intuit how all 3d shapes should be drawn. Because if you want a drawing to feel like it has volume, then you’re going to use this rule constantly.

I’ve already talked a bit about the shapes of circles in perspective in ‘the basics part one’. However, just like in orthographic projection, figuring out what angle to draw them in is a bit more complicated. Many people seem to have adopted the orthographic method into perspective and will tell you that the minor axis of the ellipses should point towards the vanishing point that is perpendicular to it.

This is incorrect.

The method that I have found that seems most accurate is to have their major axis align with a hyperbola that gets steeper the further away from the horizon it gets.

I have unfortunately not been able to find a satisfying explanation as to why it works (a proper explanation would require you to calculate the intersections of cones at different angles, and I’m not a mathematician) so I will unfortunately not be able to provide one either. What I can do is prove that the other method is wrong.

When a horizontal surface is at eye level, it should perfectly align with the horizon. When using the first method, the ellipses will always be on a slight curve,  meaning that they can never align with a straight horizon.

Since we are seeing the cube slightly from above, we are also seeing the sphere slightly from above as well, meaning that the bottom of it is slightly behind it. 

One mistake people make when they don't understand this is one that I intentionally replicated with the image above. As you can see, the sphere isn’t centered on top of the cube properly, it is slightly towards the back.

Since it can be difficult to tell where the sides of spheres are in 3d space compared to other shapes, it is easy to end up drawing it slightly off from where it should be.

Another similar mistake that I see a lot of people make is drawing cylinders thinner than they should be. When you look at art made by beginners, you’ll sometimes find some people drawing the far leg of people a lot thinner than the close leg.

Many people will look at this and think it is because the artist simply wasn’t paying enough attention. But I find that it is usually because they are making the mistake above. When the base of the far leg is obscured by the close leg, they try to think three dimensionally to figure out where the lines should go. But then they assume that the silhouette of the leg needs to be drawn at the front and the side.

When in reality, where these lines actually need to be drawn have nothing to do with any of those directions.

One way to get a better feel for how round objects should be drawn is what I did in the image above. Imagine the thing you want to draw from a different angle first, where it’s easier to see the necessary widths and heights, and try to keep them when returning to the angle you actually want to draw from.

I could probably make an entire post covering different things you could learn from this technique alone, so instead I’ll just quickly cover a couple of them.

These might seem obvious, but you’d be surprised by how many professional artists I’ve seen who didn’t have a solid grasp on at least one of these.

So far I have mostly covered things in orthographic projection. Things are a lot more complicated in perspective. One difference is that, whereas in orthographic projection, everything is always seen from the same angle, in perspective, things are seen from different angles depending on where they are relative to the camera. Everything that is above the camera is seen from below, and everything that is below the camera is seen from above. I have made a modified version of the method above that works in perspective. It’s this one that I talked about over here.

Most of the methods I know for drawing spheres in perspective will start by having you draw a circle in perspective, which itself can be a lot if you want to be precise. Which is why I haven't really gone into depth about them. However, there is one misconception that I would like to clear out. Most people tend to believe that, just like in orthographic projection, a sphere in perspective is just a circle. When in reality, this is only true if it is perfectly centered on the center of vision. The moment that it moves to the side even slightly, it will become an ellipse. The best way to understand why is to take a look at the picture plane and station point. 

If you imagine the light traveling from our sphere towards our station point, the light will take the shape of a cone, meaning that a sphere's shape in perspective will be that of a cones intersection, just like circles

If the sphere is centered on the center of vision, it is a circle, if it is off to the side, it is an ellipse, and, just like a circle, if it is reaching behind the station point, it is a hyperbola.

This technique can be used in a variety of ways even outside of perspective. People have used similar techniques to transfer small sketches onto larger canvases, or when drawing from photos.

With the right setup, you can even use this to help draw from real life.

This device is called an albertian veil or alberi’s veil. It's apparently been used since the 15th century, but I’ve had a hard time finding a lot of information about them.

The next two ones are different executions of the same idea. That is, lines that are parallel with the picture plane don't experience any distortion. By storing information about a pattern as intersections on a horizontal line, we can transfer this information into perspective.

The first one is the easiest to set up. It takes advantage of the fact that if you take a series of parallel lines, an intersection of them will always have the same relative distances between each other, no matter the angle of the intersection.

The idea is that if you use the wall and a horizontal line as intersections, we can add these parallel lines afterward to make sure the spacing between certain points remain the same.

start by taking a pattern and draw vertical lines through the parts you want to transfer to a wall. Now, draw a horizontal line through them that we will bring into perspective.

Put the edge of the line next to the close edge of the wall you want the pattern on. Now, draw a line from the other edge of the horizontal line through the far edge of the wall, and extend it until it crosses the horizon. This is where we will put our vanishing point for the parallel lines

From this vanishing point, draw lines towards the rest of the intersections on the horizontal line. where these lines cross the wall, add vertical lines.

You can now use these to help draw the rest of the pattern.

As you've probably noticed, this only helps you find vertical lines. Horizontal or diagonal lines need to either be eyeballed or you need to find another method, like the next one.

The last method I’ll cover can be a lot trickier to set up. The theory is probably simpler than the one above, and you’ll get more information out of it, but it relies on you knowing the  angles of the lines you are working with with your vanishing points. Making this technique less reliable whenever you are not working with a full setup.

You start with the initial pattern, as usual, as well as a horizontal line above it, like the second method. However, instead of just drawing vertical lines, we draw a bunch of lines at different angles.

The idea is, if you redraw these lines from these intersections at the same angles, you should be able to recreate the pattern.

Now, all we need to do is add this line to our drawing, and draw those lines from their respective vanishing points.

Fun fact: Literally every book I've read that covered this method has referred to it as a method used by architects to draw floor plans.

textures:

some notes:

The red lines are just there to help you count.

The lines that are moving vertically across the lines on the bottom image are there to show you which directions are up and down relative to the camera.

The grid behind the horizontal lines on the top image consists of perfect squares drawn in perspective, This is so you can use them as a unit of measurement when drawing backgrounds.

There is a gray dot in the center of the top images. This is the center of vision. Try to keep the focal point close to it.

These templates are intentionally made with a wide field of view. This is so that you can crop the canvas to move the focal point without the field of view becoming too narrow.

Try to break the figure you’re drawing on the bottom image into spheres whenever you can if you struggle to figure out how wide things should be.

You can move things from side to side without worrying about how high up it should be on the canvas, but it will change how far apart things are. So be careful when doing it with body parts, or else the proportions might get weird.

The location of the floor on the bottom image doesn't actually matter. Moving it up and down is indistinguishable from rescaling the image, unless you move the floor to be above the camera.

You can flip the top image both vertically and horizontally without things breaking. But if you're going to flip it vertically, then you’ll need to flip the bottom image vertically too, of course.

Perfect perspective does not guarantee good poses. You will still need to make sure the silhouette is good and all that stuff if you don't want the pose to feel stiff and awkward. I find that body parts that are pointing directly towards the camera are really hard to work with.

By drawing a line from the light source through the corner that is casting the shadow, we will be able to see where the shadow should be by where the line lands on the ground. Sounds simple, but if you try to do this in perspective then you'll bump into a problem.

Where does the line cross the ground? To solve this we’ll need to add some more points. First we'll need to add a reference point for where the light source is relative to the ground, then we’ll need another reference point for the corner casting the shadow. If the box is on the ground then you can just use the corner beneath it as the reference point. By drawing a line through these two points we get a reference line for where the light beam is relative to the ground, allowing us to see where it lands.

Repeat this with every corner of the object and you can connect them to get a full shadow.

This works well with a close light source. But what do we do when the light source is far away, like the sun? That's simple, we just put the reference point for the light source on the horizon.

If the sun is behind us then we won't be able to draw it on our canvas. To get around this we’ll simply need to create a vanishing point for the other end, instead of creating one for where the light is coming from, we create one for where the light is heading towards.

When drawing these for a while you might notice two patterns. First is that all lines that are perpendicular to the surface that the shadow will be cast on will always have their shadows point directly towards the reference point. Second is that all lines that are parallel with the surface will always have their shadow be at the same angle, and therefore pointing towards the same vanishing point.  And since most of the things we tend to draw are boxes where all lines are either parallel to a surface or perpendicular to it, you can easily use this to just eyeball shadows without all the setup. The only thing you really need the actual location of the light source for is the length of the shadow.

This is also useful when you are drawing shadows that cover multiple surfaces. Normally, when you draw a shadow that moves across the floor and up a wall, the “proper” way of doing that would be to find multiple reference points for the light source, one for each surface. This can be tricky because you need to start by drawing a line that is perpendicular to the surface from the light source, and see where it meets the surface. But then you end up with the same problem we had at the start: How do you know where it meets the surface? The best way to solve this is to draw more reference lines, have it start from the initial reference point and have it “crawl” across the surfaces whilst making sure that it is always beneath the line from the light source.

However, most of the time, you can usually get away with following the two patterns I mentioned above. Just draw the shadow the way you usually would, and when they hit a wall, just continue the lines at the appropriate angle, and use the lines from the light source to determine how far they should go.

To find a vanishing point you simply need to position your eye at the station point, and look at an angle that is parallel to the lines you want to find a vanishing point for. If you want to find a vanishing point for lines that move straight up, you look straight up. If you want to find a vanishing point for lines that move straight forwards, you look straight forwards, and so on. Whatever point you are looking at on the canvas when you do this is where that vanishing point should be. However, this requires us to measure angles in 3d space above our canvas, which can be a bit difficult. So to make it easier we can rotate the station point around the horizon until it is flat with the canvas. allowing us to easily draw and measure angles from it.

This works well when the horizon crosses the station point. But when it doesn't (like in three point perspective) you'll bump into a problem. If you look at a side view you'll see why.

When the horizon line does not cross the center of vision then we can no longer use the distance between the station point and the canvas to find how far below the horizon our station point should be after being rotated. The distance and therefore our vanishing points will always be off. There are two ways to solve this. The first one is to use the pythagorean theorem. The second one is to do some more rotating. 

If we take the picture above and rotate it to the side first, basically rotating the station point twice, then we can easily measure the distance between the station point and the horizon on our canvas.

(note: It doesn't matter which direction you rotate the station point. It won't change the location of the vanishing points. Which direction I rotate them in depends on whether I want to prioritize saving space or reducing clutter.)

Once we've done this, we can also use this vertical station point to find the third vanishing point. The third vanishing point is always perpendicular to the ground, so all we need to do is add a line that is perpendicular to the line moving towards the horizon to our vertical station point.

And that's really most of it. As long as the initial station point you rotate is rotated around the center of vision, and the distance between them is the same, then any vanishing point you find using them will always feel like they are in the same 3d space, allowing you to rotate things however you want.

However, when most people use this type of setup they will usually want to find diagonal vanishing points for vertical surfaces. And to do that we’ll need to do some more rotating.

First you'll need to find the horizons for these vertical surfaces. That's the easy part, you just need to draw straight lines through all vanishing points.

Now we’ll need to rotate the station point around them. The best way I can describe how to do that is to first rotate the canvas so that one of these new horizons is horizontal, and then do the same things you did with the first horizon.

The first step in doing this is to add what I've heard some people refer to as the gravity line. It is the line that you draw from the horizon that the station point will be on after being rotated. I think the best way to understand it is to imagine the gravity line as the shadow of the station point as it is being rotated.

This gives it two properties that you'll need to keep when drawing it for other horizons in order to make sure everything is consistent: It will always start on top of the center of vision, and it will always be perpendicular to the horizon it is being rotated around.

Luckily these are pretty easy to draw due to an interesting phenomenon that occurs when all vanishing points are set up properly. That being that the center of vision will always be in the orthocenter of the three vanishing points.

This means that to set up the gravity lines, all you need to do is draw a line from a vanishing point through the center of vision, and it will always cross the horizon opposite to it at a 90 degree angle.

Next you'll need to find where in the gravity line the station point is. There are multiple ways of doing this. One comes from remembering that all vanishing points we currently have are 90 degrees apart from each other. So you'll just need to find where on the gravity line two lines drawn from each vanishing point on it’s horizon meet at 90 degrees. You can either mess around with a set square for a while, or you could draw some circles. Turns out that if you draw two lines from opposite ends of a circle towards any point on its edge then they will always meet at 90 degrees.

If you use the halfway point between the two vanishing points as a center, and the distance between it and the vanishing points as a radius, and draw a circle, then that circle will always intersect the gravity line where the station points should be.

However, the easier and more common technique comes from remembering that, as the station point is rotated around different axes, so are the lines connecting it to vanishing points. All lines that connect different station points to the same vanishing point will always be the same length. So instead of finding the halfway point between two vanishing points, you can just find a vanishing point, take the distance between it and any station point it is connected to, and draw a circle with that radius around the vanishing point.

When done correctly with all three horizons, you should be able to rotate every station point around their horizon, and have all of them meet at the same point directly above the center of vision.

Once you've completed a full setup you can find diagonal vanishing points for all surfaces.

Realistically, you probably won't draw this too often. At least not a full one. Perfect perspective is not something people tend to look for in art nowadays, making this level of precision unnecessary. The method I described at the top of this post for how to eyeball the locations of vanishing points will probably be enough for most drawings. Unless you are trying to show off some advanced perspective then close enough is good enough. I barely draw this in full. However, I do often draw it in parts. Whether it's just the two first station points to set up three point perspective, or if it's just rotating the station point around one horizon for a wall so I can find some diagonals. You technically don't even need to find all three diagonal vanishing points to draw a perfect cube, you just need two of them, and the rest will kinda just fall into place. I want people to understand the theory because that is more useful than just memorizing what lines to draw where. I don't want you to view this as a bunch of steps to get one final setup. I want you to look at each step as its own technique that you can use when you need it, because it is by taking this apart that you'll get most use out of it.