[Third Session] Learning the Basics of Perspective: How to draw a regular cube in one-point perspective

Introduction

Explanation in the video

The ‘Overview, Summary, or Conclusion’ of this article can be found at the beginning of the YouTube video, so please refer to it.

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Drawing a cube using one-point perspective (M-point method)

What is the Picture Plane?

What is the screen, its meaning, definition, and an easy-to-understand explanation

It is like the two-dimensional image that appears in the camera’s viewfinder when aimed at the three-dimensional world around us.

What is a viewfinder, its meaning, definition, and an easy-to-understand explanation

When discussing the picture plane in oil painting, it refers to the canvas.

For example, in digital painting, such a white rectangle is commonly referred to as the canvas.

Of course, it doesn’t have to be a rectangle, but it is a typical shape.

It can also be a square, an oval, or even a star shape.

From a large screen to a specific screen

Let’s imagine when we try to capture a certain scene on a screen. In this case, we may still see parts of the scene outside the screen.

In other words, we are cutting out a ‘specific frame‘ (filter, canvas) from a ‘larger frame‘ (field of view) to take a photo or paint.

This is the image when represented in a diagram.

What is the field of view angle?

What is the field of view angle, its meaning, definition, and an easy-to-understand explanation

The size of the field of view, or how wide it is, varies depending on the animal or the situation.

The human field of view spans about 200 degrees horizontally and 125 degrees vertically.

The range where text can be clearly recognized is called the ‘central field of view,’ and it’s only about 1 to 2 degrees.

The ‘effective field of view,’ where objects appear without distortion, is around 35 degrees to either side.

Beyond this 35-degree range, the resolution of vision starts to decline, causing things to appear blurry.

What is the relationship between the angle of view and the station point?

When we take a photo with a camera or paint, there is often a distance between our body and the viewfinder (or canvas).

Just as there is a distance between the subject and the canvas, there is also a distance between us and the canvas.

Few people paint with their eyes abnormally close to the canvas, and the same goes for the camera.

The point is that this distance matters.

The angle of view changes depending on the position of the vanishing point

What is important is the relationship between the angle of view and the station point.

In simple terms, the angle of view changes depending on the position of the station point.

Therefore, when painting, it is essential to first establish the ‘station point.’ This can also be rephrased as determining the distance and spatial relationship with the subject.

The level of angle of view that becomes unnatural, what is commonly referred to as perspective distortion?

When visualizing the station point, it looks like this.

The field of view angle is about 18 degrees when close to the subject and approximately 140 degrees when farther away. This is an extreme case.

We have learned earlier that the natural field of view for humans is at most 60 degrees.

A field of view of 140 degrees actually encompasses a range that would typically be invisible. Furthermore, if we attempt to compose a painting with such a field of view, the cube will become distorted.

In other words, the painting will appear unnatural, making the cube no longer look like a regular cube.

A field of view of 100 degrees can be considered our ‘visual field.’ However, we do not draw everything within our field of view.

For instance, from a wide flower field visible before us, we can focus on a single flower and capture only that on the canvas.

For example, a maximum ‘field of view’ for humans is 100 degrees (left and right).

The cube at the edges appears distorted. While we can see beyond 60 degrees, objects start to look warped at that point. Beyond 100 degrees, they may not even be visible at all.

When we paint, we usually depict something close to our central field of view.

No one would likely position Michelangelo at a diagonal right while sketching. Typically, we place the subject directly in front of us, at a distance that isn’t too close or too far.

When the spatial relationship feels natural, the composition is often constructed within a natural field of view.

It is commonly said that a range of about 40 to 60 degrees is natural.

This standard is especially useful when there are no objects in front of us, and we are constructing images from imagination.

For example, when we create a ‘specific frame (canvas)’ that fits within 60 degrees of a scene with a 100-degree field of view, it looks like this.

The key point is that the cubes within this specific frame are not significantly distorted.

Of course, it’s not necessary to depict the entire scene within a 60-degree field of view. Within this range, you can select any part to zoom in on or enlarge.

Depending on the situation, you are free to change shapes or arrange elements differently using your imagination.

The choice of where to focus is where the creator’s skill is showcased.

Relationship Between Station Point (SP) and Ground Line (GL)

What are the baseline and base plane, their meaning, definition, and an easy-to-understand explanation

When we capture an object on the screen, there is usually ground beneath it.

In a room, this would be the floor, and outdoors, it could be gravel, a flower field, or a sidewalk. This ground is referred to as the ‘ground plane (GP).’

The person standing on this ground plane is at a position known as the ‘station point (SP).

There is a distance between the station point and the object. Of course, there is also a distance between the object and the ‘specific frame,’ such as the canvas (or paper).

This means we need to distinguish between our field of view (PP1) and the specific frame or canvas (PP2) that we extract from our visual field.

In terms of realism, it feels like cutting out a larger scene and capturing it.

To put it more intuitively, it resembles framing a target with both hands and transferring that image onto the canvas.

In that sense, the distance between the canvas and the observer may not be very important.

In extreme cases, some people can paint without even looking at the paper, so the angle at which a person views the canvas isn’t particularly significant.

What matters is the angle at which they observe the object and how they wish to frame it on the canvas.

When illustrated, it looks like this.

What is the Distance Point?

What are the diagonal vanishing point (distance point) and measuring point, their meaning, definition, and an easy-to-understand explanation

It is a type of measuring point (MP).

This point is used to gauge the depth in parallel perspective. The perspective method that utilizes the measuring point (MP) is called the M-point method.

Can the ‘diagonal vanishing point’ be counted as one of the vanishing points?

In one-point perspective, the characteristic is that there is only one central vanishing point. However, if there exists a distance point, doesn’t that make it more than one?

Regarding this question, it can be understood that both the measuring point (MP) and the distance point (DP) are ‘vanishing points used solely as aids in drawing.’ Thus, while there is one main vanishing point, there are two additional sub points.

What is a diagonal line, its meaning, definition, and an easy-to-understand explanation

[Definition] Diagonal refers to a straight line connecting two non-adjacent vertices in a polygon.

In a rectangle, it connects opposite corners.

Not all diagonals in polygons are at a 45-degree angle, but in the case of a square, it will always measure at 45 degrees in practice.

The angle formed by connecting the station point (SP) and the ends of the screen (PP) is known as the field of view.

To achieve a 45-degree angle, one should stand in a position that creates this angle.

This knowledge is more of a technique for constructing something from imagination when it isn’t present in front of you, rather than a consideration for sketching from real life.

【Example】Where is the diagonal vanishing point on a 400×400 pixel canvas?

For example, consider a sheet of paper that is 400 by 400 pixels.

If you want to use this screen as a canvas, where would the distance point (DP) be placed in this case?

First, let’s consider where the station point (SP) is located. Since the SP does not overlap with the screen (PP), it must always be outside of the canvas.

For instance, if we place a 400-pixel canvas within a 1600 by 1600-pixel area, it would look like this.

The line where the screen (PP) and the ground line (GL) overlap is located here.

The point where a straight line is drawn at a 45-degree angle from the edge of the screen(PP) becomes the station point (SP).

Of course, the station point does not always fall here; this position is specific to a 45-degree angle. For angles of 60 degrees or 30 degrees, the station point will be located elsewhere.

The eye level changes according to the observer’s position. Typically, the eye level aligns with the actual horizon (the boundary between the water surface and the sky).

The Diagonal Point (DP) is located where the diagonal line from the eye level (EL) intersects with the line from the station point (SP).

In one-point perspective drawing, the eye level generally corresponds to the center of the canvas.

Of course, in some artworks, the eye level (horizon line) may be positioned above or below, but this simply means that the central part of the canvas is being cropped from the upper or lower sections.

Whether we stretch up or bend down, the horizon line (HL) remains at the center of our field of vision.

We do not include everything we see or imagine on the canvas; instead, we make selective choices.

When drawing the eye level (EL) at the center of the screen (PP), it looks like this. If it’s difficult to draw the center by hand, you can use the two diagonals.

The intersection point of the diagonals is where the vanishing point lies, so you can draw the horizon line at that height.

When you extend a straight line from the station point (SP) toward the eye level (EL), you will find the point where it intersects with EL.

This point is known as the diagonal vanishing point (DP). It’s important to note that DP is generally located outside of the screen.

Investigating the Diagonal Vanishing Point with Canvas Size

Is the method of finding it different between a square canvas and a rectangular canvas?

The previous task may have seemed complicated, but in this particular case, you can simply use double the width of the canvas size to determine the diagonal vanishing point.

Since both the width and height are the same, you only need to multiply by two.

For example, if you have a 400×400 pixel canvas, the diagonal vanishing point will be located at the edge of an 800×800 pixel area. (If the canvas is 800×800 pixels, the point will be at the edge of a 1600×1600 pixel area.)

If the A4 size is set to 72 DPI, it measures 595×842 pixels.

If the width is 595 pixels, the diagonal vanishing point (DP) would ideally be located at 1190 pixels. However, it seems that this doesn’t always work as expected.

When using an actual A4 size and extending a line from the edge of the screen to the IL, it becomes clear that the size does not match the expected double size.

If the canvas is a square of 595 by 595 pixels, then it is appropriate for the diagonal vanishing point to be at 1190 pixels.

However, if the canvas is a rectangle of 595 by 842 pixels, then this would not be appropriate.

However, even if we hypothetically consider the position of 1190 pixels as the diagonal vanishing point, a canvas of 595 by 842 pixels does not particularly distort.

The focus here is on where the diagonal vanishing point at 45 degrees is located.

It is not that using a 45-degree diagonal will cause distortion; rather, the discussion is that within the range of 40 to 60 degrees, distortion is less likely to occur.

In fact, in one-point perspective, the range of 40 to 50 degrees is often recommended.

In the case of a rectangle, is there a situation where placing the diagonal vanishing point simply at twice the width doesn’t cause distortion?

When you actually extend a line from the edge of the screen, that is, from the ground line (GL), at a 45-degree angle, the diagonal vanishing point comes at approximately 1437 pixels, which is to the left of 1190.

Consider a method to determine the position of the diagonal vanishing point in the case of a rectangle

We have tried to guess the calculation of why they come to the position of 1437 pixels, first calculating the difference between each other.

The distance from the newly created optic centre to the diagonal vanishing point is 1437/2 = 718.5 pixels.

The position created by simply doubling is 1190, so 1190/2 = 595 pixels; therefore, 718.5 – 595 = 123.5 pixels.

They also calculate the difference in ‘height’ between each other.

For a square it was 595 pixels, the same as the width, but for A4 it was 842 pixels.

Therefore, 842-595 = 247 pixels.

The difference was found to be 247 and 123.5, meaning that there is a difference of roughly half a per cent.

Therefore, when you want to approximate the vanishing point of the diagonal line for a canvas sized 595×842 (A4), you first double the width:

The width is 595 × 2 = 1190.

Then, you add the difference between the height and the width:

The difference is 842 − 595 = 247.

So, the vanishing point (DP) is:DP = 1190 + 247 = 1437.

The distance from the focal point to the vanishing point is half of this, which is 718.5 pixels.

The formula to find the diagonal vanishing point in the case of a rectangle

To simplify

1. If the height is greater than the width, then DP = 2W + (H − W).
2. If the width is greater than the height, then DP = 2H + (W − H).

【Example】The position of the diagonal vanishing point for an A4 size (595×842)

1: Create a 595 (width) × 842 (height) rectangle.

It has been shaded in gray for clarity.

2: Change the width to 1437. The top edge at eye level becomes the diagonal vanishing point.

DP = 2W + (H – W)
DP = 2(595) + (842 – 595)

【Example】The position of the diagonal vanishing point for a B4 size (729×1032)

The question is whether this calculation can be applied to canvases with other ratios. Let’s experiment with a B4 canvas.

B4 measures 729 (width) × 1032 (height). Therefore, the difference is 303.

When you add this difference to twice the width, which is 1458, you get 1761.

When visualized, it looks like this.

From here, let’s extend a line at a 45-degree angle from GL to verify. It seems to be approximately correct.

Suppose the A4 size has a width of 842 and a height of 595. We can apply the formula DP = 2H + (W – H).

This gives us DP = 2(595) + (842 – 595), which equals 1437. Since drawing a 45-degree line from GL resulted in a match, we can say it is approximately correct.

【Example】The position of the diagonal vanishing point for an A4 size (842×595)

It’s interesting that the same DP is reached when the height is 842. This means that simply swapping the height and width allows for the use of the same DP.

Of course, this method can be used even without digital software. For a square, you can prepare three A4 sheets, folding two of them in half (or cutting one in half from two sheets).

By aligning the folded A4 sheets on either side of the unfolded A4 sheet, you can determine the diagonal vanishing point.

This approach works not just for squares; you can also use a ruler or similar tools to find the diagonal vanishing point.

Once you’ve created a template for A4 size, you can reuse it. Another method could involve drawing on a larger sheet of paper and cutting it out later.

Tools like strings or perspective rulers may also come in handy.

Creating a cube using the Distance Point

(1) First, draw a square

Now, let’s get started on creating a cube. For example, let’s draw an arbitrary square like this.

(2) Extend each vertex towards the vanishing point and the diagonal vanishing point

In one-point perspective, each vertex converges towards a single vanishing point.

This means that they will converge like this. However, the question arises: how far back should we define the depth to create a proper cube?

To utilize the vanishing point of the diagonal, simply extend lines from each vertex towards the diagonal vanishing point.

For the right vertex, extend the line to the left, and for the left vertex, extend the line to the right. In this case, since the size is 400 by 400, the vanishing point (DP) will be located at the edge of an 800 by 800 square.

(3) Find the point that determines the depth

The point where the line extending to the center of perspective (CP or VP) intersects with the line extending to the diagonal vanishing point (DP) is referred to as the point that determines depth.

(4) Extend the lines from the point that determines the depth as a reference

From this “point that determines depth,” you can extend lines to complete the cube.

By drawing more precisely into the interior, it will look like this.

When the auxiliary lines are erased, it will look like this.

Of course, you can draw the square at any position to create a neat cube. It’s also possible to use just one line towards the DP.

Additionally, if you use digital software, you can set it up to automatically draw lines to the DP.

By erasing the auxiliary lines and returning the canvas size to its original dimensions, it becomes neat like this.

 Plans for the next session

Next time, we plan to discuss the “visual cone.”

This time, we will create a visual cone of 90 degrees, introducing a mechanism that allows the 45-degree visual cone to fit the screen (canvas, specific screen).

We have learned that the field of view (visual cone) is not limited to 45 degrees; a range from 40 to 60 degrees results in less distortion.

Now, we would like to address how to set up other angles. However, we will limit our discussion to one-point perspective, as we have not yet covered two-point perspective in the video.

References

Books that are easy for beginners to understand

David Chelsea「Extreme Perspective! For Artists: Learn the Secrets of Curvilinear, Cylindrical, Fisheye, Isometric, and Other Amazing Drawing Systems that Will Make Your Drawings Pop Off the Page 」

David Chelsea「Extreme Perspective! For Artists: Learn the Secrets of Curvilinear, Cylindrical, Fisheye, Isometric, and Other Amazing Drawing Systems that Will Make Your Drawings Pop Off the Page 」

The book contains many illustrations and is easy to understand. It also explains basic perspective terminology and provides a simple explanation of how to use perspective. However, it is important to note that the book focuses on ‘illustration (manga)’ rather than architectural perspective.

It is suitable as the first book to pick up for learning the basics of perspective in general.

Robbie Lee「Perspective Made Easy: A Step-by-Step Guide」

Robbie Lee「Perspective Made Easy: A Step-by-Step Guide」

This is a suitable book to pick up as the first one for learning the basics of perspective in general.

I found it to be simpler and more detailed than ‘Perspective! Learn Perspective Through Manga.’ Therefore, I especially recommend this book to beginners as their first read.

Scott Robertson「How to Draw: drawing and sketching objects and environments from your imagination」

Scott Robertson「How to Draw: drawing and sketching objects and environments from your imagination」

A book specialized in drawing, particularly focused on line art. Though somewhat complex, it provides a broad and in-depth explanation.

About the Japanese version of this article

This article is a translation of an article written in [https://souzoulog.com/2024/10/17/basic-of-perspective-3/]. For detailed references, please refer to this link.