How to Draw a Cube from Different Angles. Linear Perspective Basics (Part 1)

Guess what? Cubes are awesome!

Seriously! If you know how to draw a cube, you will also be able to sketch any 3-dimensional form. Of course, there are nuances in every object you choose. Still,  the core principles, the nuts and bolts of draughtsmanship, stay the same.

They all are in this box.

Let’s get them out.

So, this is not one of those ‘1-2-3 tutorials’. There are thousands of them on the Internet. My goal is to provide you with a skillset enabling you to draw anything. This approach will be helpful if you want to deal with concept art, industrial design, fantasy/cartoon art. They all require sketching from your imagination. It means that you need to be able to draw any object from any angle. You will get this knowledge from the lessons. And strengthen it with homework aasigments, provided after each lesson

Now you will learn the basics of linear perspective through an example of a cube. In later articles, we will get to much more complicated subjects. If you want to gain as much benefit from this article as possible, please, kill all destructing factors. Take a piece of paper and a pen, experiment with the concepts you are learning. Believe me, mere reading won’t help much.

What Is Perspective?

Perspective is a system of representation of three-dimensional world on a flat surface.

Not very complicated, right?

So, here is the most logical and useful concept to keep in mind about perspective.

The cube is seen through the piece of flat glass. The camera is pointed straight at this glass.

The glass is called Picture Plane (PP). The line drawn from the camera through PP is called Line Of Sight (LOS). I want to stress that LOS is always perpendicular to PP.

“So, why the hell are you telling me all that?” – you may ask.

And here is why.

We need to know how our lines are positioned in space, relative to something. Camera position is our guiding star. The main idea is that drawing in perspective is a representation of an image from a particular viewing position. There is no image without a viewer.

Drawing a Square

What is a cube? Basically, it consists of 6 square planes fitted together. To draw a cube, we need to know how to position a square in any and all cases, from any possible viewpoint.

Now we need to add a new nerdy word to our vocabulary – Normal Line or simply Normal.

If you put the bottom of a pencil on the surface, it will match the direction of a normal line. Normal is a line perpendicular to the surface.

Simple as that.

Every plane has an infinite number of these normal lines. For the sake of simplicity, we will draw only one.

Normals help us determine how we see the surface.

We take a rectangle and put it in front of the camera. If the normal is pointing straight at the picture plane (90 degrees), then we see the surface without any distortion. Rectangle – as it is.

And here is the crucial part.

If we tilt our rectangle in any direction, the normal line will be no longer perpendicular to PP. The surface of the rectangle shrinks.

How do we determine the direction of a compression? The normal line tells us.

The principle is called Shrink Along Normal.

Every plane is compressed only along its normal line.

We can see that plane has changed in proportions. It is narrower because it shrinks along the normal line.

This is always the case.

There is no other way to shrink a plane. Only along it’s normal. The more the plane is tilted away from the camera, the more it is compressed along its normal line. I will repeat this concept again and again because it’s very important. One tends to forget these simple things when it comes to drawing.

Converging of Parallel Lines

Another phenomenon. In the example above, you might have noticed that left and right margins of a rectangle are narrowing towards the top (from the camera view). It happens because they are parallel lines, which recede in space (relative to the picture plane). The lines which are parallel to PP, don’t converge.

Sounds difficult? Well, it absolutely should be like that. Learning new things can’t be easy.

I remember my first steps in drawing – it felt like a freakin’ hell. It will become easier over time, I promise you.

In the articles about the perspective, we will dive deep into the subject. This will help you to improve your draughtsmanship abilities in the shortest time. Not all readers have a burning desire to improve their drawing skills.

But if you do, then let’s go further.

Where do the parallel lines converge? I’m sure that you know – on a horizon line. It is written in every art book. Generation after generation. This concept is so popular and dogmatic that authors never reconsider its explanation.

But we will.

What Horizon Line Really Is?

We assume that the bottom of our camera is parallel to the ground plane.

We have a few horizontal planes in our scene. As these planes lift up, they gradually shrink.

Normal line of a horizontal plane is vertical. That is why every horizontal surface shrinks along the vertical line.

When the plane shrinks completely, it turns into a single line. Due to a vertical compression this line is horizontal.

This is our Horizon Line.

If your plane cuts the camera right in the center when you extend it, it means that the plane is shrunk to zero. It is the horizon.

Parallel lines situated on the horizontal planes (no matter which one) converge on the horizon line. Points of converging are called Vanishing Points (VP).

You can see that every set of parallel lines has its own VPs. Typical perspective construction can have Central (CVP), Left (LVP) and Right (RVP).

Ellipses

I didn’t tell you that a plane must have a rectangular shape only.

Why I didn’t?

Because a plane can be of any shape.

You just think of it as a flat piece of paper. Any shape can be cut from it. So, we can make a plane in the form of a circle. Then we put it in space and get…

Yes! An ellipse.

Here I pretend, that you’ve read my article about basic line work. If not, you may want to do it now. Because it has some, well, basics that we will need.

A plane in the shape of a circle also has its normal line, and this normal is also perpendicular to the plane’s surface.

Here is a trick about ellipses.

The normal line of ellipse’s plane always goes in the same direction as its minor axis.

The principle is just the same as in the case with rectangles.

But.

The circle has an equal diameter all the way. As the result of a compression (even the slightest one), the diameter will be represented by two most important lines: the longest and the shortest ones.

The biggest diameter is a major axis. This line doesn’t change its length, no matter how strong you tilt the surface.

The smallest diameter is a minor axis. It is perpendicular to the major axis. It has the same direction as the normal line. It changes the most when the circle is tilted away from the camera view.

The level of ellipse compression is called the degree of an ellipse.

If normal of an ellipse’s surface is pointing at 90 degrees to the picture plane. Then camera sees a completely open circle.

If the normal is pointing at 45 degrees at PP, the ellipse is called “45-degree ellipse”.

The fewer degrees, the more compression is there.

Ellipses Are Your Rescue Rangers

We will use ellipses even in cases when there are no visible circular planes in our drawing.

Why?

There are some great advantages.

  • Ellipse helps you determine the direction of the surface’s normal line.

Thus, you know in which direction to shrink your plane when it’s tilted away from the viewer.

  • Ellipse helps you determine the plane’s tilt relative to the viewer.

Stronger tilt = stronger shrink.

I didn’t tell you this till the last moment hoping that you have noticed it. I believe that the deepest knowledge is gained from your own experience. You have to fight for it. You have to be courageous and curious. There is no other way.

So, what I hoped that you guess?

  • Ellipse helps you to find the proportion of a perfect square.

A circle inside the square touches each of 4 borders at the midpoints. A circle in perspective (ellipse) does absolutely the same.

You see, every ellipse touches square’s borders at midpoints.

This is always the case.

The cube consists of perfect squares. Knowledge of how to construct a square is very important for our end goal – cube drawing.

This article may seem a bit too long compared to the end result.

“A cube drawing? Seriously? But it should be so simple… Why do I need to waste my time reading all this bullshit?”.

Keep your hair on.

I promise you that the stuff we are learning will soon help you to draw such cool things as tanks, dragons, robots, etc. Of course, if this is really interesting to you.

90 Degree Angle

Apart from square’s proportion, we must make sure that the square has four right angles (90 degrees). We need a proper construction of at least one angle. Three other will fall into place.

And again ellipse is your rescue ranger here.

It will help you to construct a proper 90° angle between two lines on a single plane. Let’s draw the right angle on the ground plane.

Determine proportions of a square from the given viewpoint using your ellipse.

Put a normal line (in this case it’s vertical because the plane is horizontal) in position, depending on how the angle should be turned.

If you want the angle to point straight at the viewer, the normal must be drawn right through the middle of the ellipse. Then draw two lines tangent to ellipse from the point on a vertical line. It will result in an equal shrink of the left and right sides (both 45°) of our box.

If you want to turn the angle clockwise, then you move the normal to the left.

To turn it contraclockwise, use the same principle.

How much of this normal line should extend beyond the ellipse?

It depends on the degree of an ellipse. The more it opens – the longer is the line. You will get it better with practice. Just be aware of touching the square at the midpoints.

There are several ways to complete the construction of a square in any possible position. It depends on the level of desirable perspective extremeness.

The closer the horizon line is to the ellipse (in relation to its size) – the more extreme is the perspective. The lines converge quickly which means that the object is close to the viewer or it is big. The image looks as it is shot with wide angle lens.

If the horizon line is far away from the ellipse, perspective will be flattened. Lines converge slightly. The object looks small or is far away from the viewer.

It’s a long lens effect.

Here you see that a vertical line sticks outside the ellipse on the same distance in both cases. The bottom angle is the same. The difference is only in the perspective extremeness. And again. The horizon line is perpendicular to the ellipse’s normal (minor axis).

Horizon is an essentially flat plane which is parallel to that ellipse. This plane is just tilted completely away from the viewer’s observation.

Drawing a Cube (Finally)

A cube has six sides, only 3 of them can be visible at a time. So, for the sake of simplicity, we will focus only on the visible sides (for now). You already know how to construct a horizontal square in every possible position.

This will be our top plane.

What is left? The remaining two side planes. Vertical edges are known. They are normal lines to the top plane.

What is not known is the length of a vertical line. The vertical line is parallel to PP, so it can’t be foreshortened as much. It gets bigger, when it advances in space (like any other object) due to a convergence of the lines. So, we assume, that it is a little bit longer than the major axis of our top ellipse because its major axis can’t be foreshortened as well.

There is a trick to check if the side planes are correct.

Can you guess what it is?

Yep, it’s ellipse.

Let’s put an ellipse with a minor axis pointing at RVP. Ellipse should touch midpoints of the plane’s edges. Try to ghost that ellipse between lines, visualize it. Then we just close the bottom edge of the left side with a line going to LVP. Finally, we close the right side with a line going to RVP.

The box is done.

Shrink Acceleration

Now we have our cube. But you may wonder: how to rotate it in space, how to tilt it? This is a good question. I like when you ask interesting questions! 😀

To resolve this problem, we must know about Shrink Acceleration. According to this principle, the shrink of the plane is not linear. In other words, it progresses faster and faster as it goes.

We put a cube on the ground plane in a position where the both side planes are at 45 degrees to PP.

Now we turn a cube clockwise. It is still standing on the ground.

You can see that the left side shrinks more, then the right side fattens in both absolute and relative values.

The more the plane (or a line) is shrunk, the faster it shrinks.

Even a slight turn of a completely shrunk line makes a dramatic change in its length. As it opens more and more, the speed of opening decreases.

shrink acceleration

Normal lines do the same thing. When they are parallel to PP, it’s hard to shrink them. As they turn away from the viewer, it becomes easier and easier to do it.

normal lines shrink acceleration

How to Draw a Cube from Any Angle in 5 Steps?

The time has come.

Take my apologies for such a long foreplay. All that stuff is crucial not only for our box drawing. You will appreciate this material when you get onto more complicated subjects in my later articles.

So, let’s do it!

  • Drop an ellipse. It can represent any side of a cube. Your only trouble here is the shrink and direction of a normal.
  • Drop normal line according to the desirable front edge position of a cube. Horizon line for this cube will be actually not horizontal (ha-ha). It just needs to be perpendicular to the normal of our plane. This is the only requirement.
  • Decide on perspective extremeness. Here horizon is out of frame, and the perspective is quite flat.
  • Measure the correct length of a “vertical” corner using an ellipse or just eyeball it. Draw a line to RVP to close the bottom border.
  • The last side will fall into place by itself. Just construct correct parallel lines to the ones that already exist.

Homework Assigment

Now it’s time to solidify your knowledge.

Do it even if you think that you already understand how to draw a cube from any angle.

Practice makes perfect, huh? But there should always be a good knowledge base.

I recommend to read this lesson again after the homework is done.

Here I’ve made a demo for you.
You can see that cubes can overlap each other.

Feel free to play with different sizes and angles.

Try diverse perspective extremeness. Big boxes can have dramatic convergence of lines.

See you next time.

 

 

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