The Definitive Guide To Drawing Interlocking Three-Dimensional Forms From Imagination

Open as Flashcard

Examples of interlocking forms

Figure 1: Examples of interlocking forms.

Several courses have exercises where you draw forms as they interlock in space, most notably Peter Han’s Dynamic Sketching course and Antonio’s ArtWod. However, I have seen no resources that explain how to do that other than using your intuition.

For the general case, you do need to develop an instinct or intuition for where the Crossing Lines go. A good way to do that is to use Timed Drawing, Flashcards, or Memory Drawing.

But while it is true that you need to do it mostly from intuition, there are construction techniques that can aid your intuition. Different situations require different approaches. I made this definitive guide to solve that and present the various methods and when to use them.

 

What Are Interlocking Forms?

If you take two forms, say, two cubes, and place them near enough to each other, parts of their insides will overlap, and they will share contour lines, lines on a face of both forms. See, for example, Figure 1.

If you have two planes in space that are not parallel to each other, they intersect at a line, and so if the forms intrude into each others’ spaces, there will be lines that define where these faces cross. See Figure 2.

Example of lines formed by faces crossing each other

Figure 2: Example of lines formed by faces crossing each other.

We will call these ‘Crossing Lines’.

The task is to find these Crossing Lines where the faces of the two forms intersect.

 

Why Do This Exercise From Imagination?

Many things around us consist of interlocking forms. It is helpful to be able to visualize them. The exercise develops your instinct and intuition for how forms typically interlock in space.

It is also an exercise you can do without reference, and it is fun to be able to do it from imagination.

 

How To Do These Exercises

We will go from simple cases to more elaborate cases. If you already have some knowledge of this subject, please review the simple cases, as some concepts are introduced there. In later sections, you are expected to know these concepts.

The Crossing Lines can be constructed precisely in the first two simple cases.

For the cases discussed after that, it’s not easy to construct these Crossing Lines accurately, and you have to develop an instinct for where they go approximately.

Much of this exercise is about developing that intuition for how the Crossing Lines go.

As you read through each section, do the exercise of trying to draw that interlocked form. It is one thing to read this document. It’s an entirely different thing to draw these configurations of forms a few times. You’ll have a much deeper understanding of these interlocking forms if you try to draw them a few times.

At first, drawing lightly with a pencil and eraser is okay to develop that intuition as you search for the right line and erase previous ones.

Later, when you get the hang of it, and you start to visualize the forms in space and how they interlock, try to do it directly in pen!

 

Non-Rotated Boxes

We are going to start by drawing two overlapping boxes that are not rotated relative to each other.

You can practice drawing boxes separately using this box model, which you can rotate inside your browser .

Imagine two boxes overlapping on the page with ribs that go to the same vanishing points. They are not rotated relative to each other. See Figure 3.

Construction of two overlapping boxes that are not rotated relative to each other

Figure 3: Construction of two overlapping boxes that are not rotated relative to each other.

 
These boxes can interlock in different ways because one thing we didn’t lock down is the relative distance. A box can be nearby and small or farther away and bigger, and that matters for how the boxes interlock.

To solve this, look at the overlapping boxes and imagine a spot on one rib of one box where you think it goes through a face on the other box. In Figure 4, I give two examples, two different points, and we’ll develop both cases so you can see how that influences things.

Choosing the point where a rib goes through a face on the other box

Figure 4: Choosing the point where a rib goes through a face on the other box.

 
The Crossing Lines in this simple example will be lines that are parallel to the ribs of both boxes. They will go to the same vanishing points as the ribs of both boxes. That makes this a simple example: after you have one point on the Crossing Line, you can easily draw it because it goes to one of the vanishing points.

In Figures 5 and 6, I develop the Crossing Lines, which are the intersections of two different faces of both boxes.

Development of the Crossing Lines for the first example

Figure 5: Development of the Crossing Lines for the first example where the intersection point is lower on the rib.

 
Development of the Crossing Lines for the second example

Figure 6: Development of the Crossing Lines for the second example where the intersection point is higher on the rib.

 
Figure 7 shows the final results with the lines cleaned up and hidden parts removed. As you can see, even though we started with the same two-dimensional configuration of boxes, we ended up with two different configurations of Crossing Lines. They interlock differently. Both can be correct, depending on their relative distances from us and each other on the axis that goes into your page.

Interlocking non-rotated boxes

Figure 7: Interlocking non-rotated boxes.

 

Boxes Rotated Around One Axis

This configuration of boxes is still ‘simple’ in that we can still accurately construct the Crossing Lines for these configurations.

Start with two overlapping boxes again, but rotated relative to each other along one axis. as shown in Figure 8.

Two overlapping boxes rotated relative to each other along one axis

Figure 8: Two overlapping boxes rotated relative to each other along one axis.

 
You can still approximately follow the same routine we used for the boxes that were not rotated relative to each other, with one small change: the Crossing Lines of the non-rotated boxes were parallel to some ribs of both boxes, and these Crossing Lines will now be parallel to the ribs of at least one box. In the direction of the axis of rotation, the Crossing Lines will still be parallel to some ribs in both boxes, but in the other two directions, they will be parallel to the ribs of only one of the two boxes.

See figures 9 and 10 for how to develop this type of configuration of interlocking forms:

Choosing a point where two faces of the boxes touch and developing one Crossing Line parallel to the ribs of at least one of the boxes

Figure 9: Choosing a point where two faces of the boxes touch and developing one Crossing Line parallel to the ribs of at least one of the boxes. In this case, the Crossing Line is parallel to ribs of both boxes.

 
Developing the Crossing Line between a different combination of faces of the boxes, cleaning up the lines, and removing invisible parts

Figure 10: Developing the Crossing Line between a different combination of faces of the boxes. Here, the Crossing Line is parallel to ribs of only one of the two boxes. On the right side, the interlocking forms with cleaned up lines and invisible parts removed.

 

Boxes Rotated Arbitrarily

Drawing interlocking boxes that are rotated arbitrarily is a more complex case because the Crossing Lines are generally not parallel to any of the boxes’ ribs. Unfortunately, they are not easy to construct from geometric principles like we did in the previous, simpler examples.

We can easily figure out the rough direction of the Crossing Lines and develop an instinct for where they should go precisely.

I based the drawing on a 3D model (see Figure 11). More on that below. It was a good idea to make sure I got it right for this example, but for practice, it is helpful to be able to develop it from our imagination eventually.

3d model of interlocking boxes rotated arbitrarily

Figure 11: 3D model used for reference for this example.

 
Draw two overlapping boxes that are rotated arbitrarily. See Figure 12.

Two overlapping boxes rotated arbitrarily

Figure 12: Two overlapping boxes rotated arbitrarily.

 
Again, we must first decide on a point where a rib from one box intersects with a face of the other box and then draw a line parallel to the ribs of that face in one of the faces. See Figure 13.

Choosing a point where a rib intersects with a face and drawing lines parallel to the ribs of one box, one going through the chosen point

Figure 13: Choosing a point where a rib intersects with a face and drawing lines parallel to the ribs of one box, one going through the chosen point.

 
As you can hopefully visualize, the line we drew, the one with the arrow head, is not on the face of the other cube! This is the problem: how to find that Crossing Line, the line that is in the plane of both faces.

It’s useful to draw this line because now that you’ve drawn it, you can see that it goes inside the other box or points outside of the box.

The line with arrow head that goes through the selected point and parallel to the ribs points inside the box. If you can’t see it, you can mentally rotate the arrow far to the left while still in the plane of that face. See the left part of figure 14. You’ll notice that rotating left, it moves more inwards, inside the other box, whereas if you rotate the arrow clockwise enough, you’ll realize it’s now pointing outside the other box.

This gives you a vital clue: you can start with a line through the selected point and parallel to two ribs, and this insight then tells you which direction to change the orientation of the line so that it is also in the plane of the face of the other box. See Figure 14. Unfortunately, how much it needs to be rotated depends on instinct and intuition. But at least this construction technique allows you to determine whether you need to rotate it to the left or the right.

Rotating the line in one plane so that it also lies in the other plane

Figure 14: Rotating the line in one plane so that it also lies in the other plane.

 
In Figure 15, we continue constructing the other Crossing Lines until we’re finished, and then we clean up the drawing, removing parts that are not visible.

Constructing all remaining Crossing Lines to end up with a configuration of interlocking boxes rotated arbitrarily, then cleaning up the lines and removing invisible parts

Figure 15: Constructing all remaining Crossing Lines to end up with a configuration of interlocking boxes rotated arbitrarily, then cleaning up the lines and removing invisible parts.

 
This part is initially unsatisfying: it's not easily possible to construct this line. You will have to develop an intuition for it.

The good news is that to develop your instinct for this, you can practice using the 3D model provided here .

You can rotate these interlocking boxes in your browser, and if you click on the dice icon, the configuration is changed randomly.

Try to draw these from observation and note how the Crossing Lines are oriented depending on the boxes' orientation. Try to apply the scheme presented above to these models and you'll start to be able to see them in your mind hopefully.

 

Other Primitive Forms

You can also learn to construct interlocking forms using forms that are not boxes, but it requires a slightly different approach.

The advantage of boxes is that they are constructed from flat faces. This makes the Crossing Lines linear. When you use other primitive forms, like the sphere, cylinder, and cone , this is not so due to their roundness. The Crossing Lines will be roundish too.

For these, it is useful to intuitively imagine the contour lines on both forms and where they touch. There is no easy way to construct these accurately, and it is useful to develop an instinct for it.

Figure 16 shows an example breakdown of an intersecting sphere and cylinder, drawing the contour lines to help figure out where the forms intersect, where these contour lines meet. Again, I used a 3D model. More on that below.

Interlocking sphere and cylinder

Figure 16: Interlocking sphere and cylinder, using the contour lines of both forms to try to intuit where the two forms intersect.

 
Two interlocking spheres are a special case because the Crossing Line, if there is one, is a circle, so this is again a simple configuration. See this interactive 3D model .

You can train your instincts by applying the method described and using them on the 3D models you can find here .

 

Using Non-Primitive Forms

You can move on to using forms that were created from primitive forms in some way and create interlocking forms from these.

One example is an arm from a George Bridgman illustration. You can see the demonstration of how to construct the arm here:

And you can practice it from any angle using this 3D model here:

 

 
Check Out Other Guides