## Cubes in one-point perspective

This post is a reply to Jamie’s comment on Drawn to Math:

Would anyone know how to construct a cube in one point perspective?

This is an excellent question because it isolates an important, simplified situation that enables one to study key aspects of perspective drawing.

(If you are completely new to perspective drawing, I’d suggest working through the Summer Fun problem set on pages 21-22 of the June, 2013 issue of the Girls’ Angle Bulletin.)

What is the “point” in a one-point perspective drawing of a cube?

The “point” in “one-point perspective” refers to a vanishing point. So before we draw cubes, let’s review the concept of vanishing point.

Vanishing points correspond to sets of parallel lines in the following way. Imagine a bunch of parallel lines in space. Assume that these lines are not parallel to the drawing canvas. A perspective drawing of these lines will look like a bunch of spokes emanating from a common point. This common point on the canvas is the vanishing point associated to the parallel lines in space. Each parallel line is drawn as a ray emanating from the vanishing point.

Where should the vanishing point be on the canvas?

If we fix a set of parallel lines, the exact location of their vanishing point depends on from where exactly we are meant to view the drawing. Perspective drawings are designed to be viewed from a specific location. The further the observer moves away from this ideal vantage point, the more distorted the drawing will seem. When you make a perspective drawing, you must decide where exactly you want an observer to place her eye when viewing your drawing.

The famous painting The Ambassadors by Hans Holbein features a highly distorted skull. It appears distorted because that skull is a perspective drawing designed for viewers off to the right of the painting, whereas the rest of the painting is designed to be viewed from a standard location directly in front.

Only after you have decided where you want the observer’s eye to be can you determine the location of vanishing points.  To find the vanishing point associated to a set of parallel lines, place your eye at the specific viewing spot you want your drawing to be viewed from. Then, look at the canvas in the direction of the parallel lines. The point on the canvas directly in front of your eye is the vanishing point. In other words, the vanishing point for a set of parallel lines is where the one (of these lines) that passes through your observing eye intersects the drawing canvas. If you imagine following a point travelling along another of these parallel lines into the distance with your eye, your line of sight will converge to the special line that passes through your eye. In this way, we can see that drawings of such lines are rays emanating from the vanishing point.

Parallel lines that are parallel to the plane of the drawing canvas have no vanishing point on the canvas. When you look in their direction, your line of sight won’t even intersect the drawing canvas, even if the drawing canvas were an infinite plane. (I’m assuming that we’re drawing on a flat canvas.)

Back to the cube

A cube has 12 edges. These can be arranged into 3 groups of 4 mutually parallel edges. These 3 groups potentially define 3 vanishing points. If the edges in 2 of these groups are parallel to the canvas, then only the 3rd group will yield a vanishing point. This, then, is the setup for a one-point perspective drawing of a cube. A one-point perspective drawing of a cube arises when you make a perspective drawing of a cube that has 8 edges parallel to the canvas (or, equivalently, two faces parallel to the canvas).

In a cube, edges meet at right angles. When 8 of the edges of the cube are parallel to the drawing canvas, the other 4 will be perpendicular to it. The vanishing point for these 4 edges is therefore located right where the viewer of the drawing looks when looking straight onto the canvas. I’ll refer to this point as V for the rest of this blog post.

Let’s begin drawing.

Let’s start by placing a dot on the canvas directly in front of the observer’s eye to mark the point V.

Next consider the front face of the cube. This face forms the base of a square pyramid whose apex is the observer’s eye. The drawing canvas intersects this pyramid in a plane parallel to the front face. Using properties of similar figures, we can deduce that the front face gets projected onto a square. Let’s now draw in this square. (If the front face of the cube is in front of the drawing canvas, then the sides of the square pyramid must be extended to meet the canvas.)

The 4 edges of the cube perpendicular to the canvas are drawn as line segments that extend from each corner of the square (that represents the front face) toward their vanishing point, which is V. This brings up the next problem: How long should these line segments be?

The answer is that it depends on how far away from the canvas the observer’s eye is supposed to be. If the observer is to view the drawing from far away, then the 4 perpendicular edges will be represented by short line segments.  But if the observer is supposed to view the drawing from up close, then the 4 perpendicular edges will be represented by line segments that extend almost all the way to the vanishing point. This fact is illustrated below.

The black square represents the front face of a cube. Close one eye and situate your other eye directly over the vanishing point. When you are very far away from the image, both the green and the blue squares will appear to be too far from the black square in depth to depict the back face of a proper cube. As you move your open eye closer to the canvas (but always staying directly over the vanishing point), there will be a place where the black and green squares do look like the front and back of a perfect cube, while the blue square appears far in the distance. Move your open eye even closer to the drawing, and the apparent distance between the black and green squares will appear to shrink, and at some point, the black and blue squares will appear to form the front and back faces of a perfect cube.

Suppose you have decided that the viewer should view the drawing from a distance D away from the canvas. Then the line segments representing the 4 edges perpendicular to the drawing canvas must be a very precise length to properly depict a cube. How do we figure out how long to draw these line segments?

I’ll first explain one method to compute this in the case of a cube that is upright and not rotated relative to the ground. With such a cube, we can refer to its bottom face. Consider one of the diagonals of the bottom face. For definiteness, let’s pick the diagonal that points into the distance to the right. In the drawing, the diagonal will extend toward some vanishing point V’ different from V. Where is V’? To find V’, we put our eye at the observation point, which is exactly a distance D directly in front of V, and we look in the direction of the diagonal. The diagonal bisects the right angles in the base of the cube, so to turn from the direction perpendicular to the canvas to the direction of the diagonal, one must turn right by 45 degrees. We will then be staring directly at V’.

The observer’s eye, the point V’, and the point V form the vertices of a 45-90-45 isosceles right triangle with V the right-angled vertex. Therefore V’ is located exactly a distance D to the right of V, as shown in the diagram below.

When we draw in the diagonal, it intersects the line connecting V to the lower right corner of the square depicting the front face precisely where we must draw the lower right corner of the square that depicts the back face. This intersection is indicated by the red arrow in the diagram above. That’s how we can determine the exact lengths of the 4 line segments representing the 4 edges of the cube perpendicular to the drawing canvas. The rest of the cube can now be drawn in, as shown below. If you close one eye and place your open eye exactly the distance D directly over point V, you’ll be rewarded with the illusion of a perfect cube.

To handle cubes that are not upright (but still in one-point perspective), you can rotate everything about the axis that passes through the observer’s eye and V until the cube becomes upright, draw that, and then rotate everything (including the drawing canvas) back to the original position.

There are other ways to construct one-point perspective drawings of cubes and we encourage you to invent your own. In fact, we have avoided giving a step-by-step procedure or lots of diagrams in the hope that you’ll create your own.

Conclusion

Even though the one-point perspective represents the simplest kind of perspective drawing of a cube,  it already reveals the critical importance of designing the drawing for a specific observer location and the value of the vanishing point concept. Elaborate cityscapes can be constructed in one-point perspective. Joel Babb‘s drawing for the cover of Volume 6, Number 6 of the Girls’ Angle Bulletin is essentially a 1-point perspective drawing. For an example of constructing a 2-point perspective drawing of a perfect cube, check out Rowena‘s article in the April, 2010 issue of the Girls’ Angle Bulletin.

If you make a one-point perspective drawing of several cubes of different sizes, we’d love to see it. Feel free to scan it in and send it to us at girlsangle@gmail.com. When a drawing shows more than one cube, it becomes crucial to consistently design the drawing for a specific observation point. If you don’t, you’ll end up with a drawing of bricks of different shapes instead of a drawing of cubes.

Thank you for your question, Jamie!