<p>I have a floor plan to which I would like to draw the room. Then I want to put a piano in there. I have the dimensions for everything. But I couldn't really realize it. So I drew it with vanishing points, but only roughly estimated the dimensions. This is my result:</p>
<p><a href="https://i.sstatic.net/CbIJ5GJr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CbIJ5GJr.png" alt="Pencil drawing with freehand perspective." /></a></p>
<p>After a bit of research, I found <a href="https://handprint.com/HP/WCL/perspect3.html" rel="nofollow noreferrer">this book by Bruce MacEvoy</a>. It explains the whole thing in very technical terms. It took me a few hours of reading, but now I understand it. I would now like to share this with you.</p>
<p>First, we construct a crosshair from the horizon line and the center of view. The point of intersection is the vanishing point to which the gaze is directed.</p>
<p><a href="https://i.sstatic.net/cwZN0N9g.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cwZN0N9g.png" alt="Crosshair." /></a></p>
<p>One important concept is the 90-degree field of vision. This results in other things. Firstly, the viewing height (i.e. eye level above the floor) and the viewing distance to the (virtual) image plane are the same. The viewing height is marked by the point V.</p>
<p>In addition, an angle of 90° is included between the two vanishing points. I want to create a perspective with a 30°/60° rotation so that it is not exactly symmetrical. I draw from viewpoint V at an angle of 30° to the horizon and mark the first vanishing point VP₁ there. Then I do the same with 90° and construct the vanishing point VP₂.</p>
<p><a href="https://i.sstatic.net/19QP9Nc3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/19QP9Nc3.png" alt="Perspective construction." /></a></p>
<p>The next step is to construct the so-called measuring points. This is the crucial concept. To do this, draw a circle with VP₁ in the center of V to the horizon. The point of intersection is then MP₁.</p>
<p><a href="https://i.sstatic.net/niSX2DPN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/niSX2DPN.png" alt="Construction." /></a></p>
<p>Do the same on the other side to obtain MP₂.</p>
<p><a href="https://i.sstatic.net/Cb13w6rk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Cb13w6rk.png" alt="Construction." /></a></p>
<p>Now we have prepared the frame and can start with the actual drawing. I am drawing a simple cuboid here that extends to infinity. I can already add a scale to the front edge. The other lines go to the respective vanishing points. However, I can't add a scale there yet.</p>
<p><a href="https://i.sstatic.net/26YZqxrM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/26YZqxrM.png" alt="Construction." /></a></p>
<p>I can then use this scale to create the lines for an even grid to the vanishing points. Here we have nice even spacing.</p>
<p><a href="https://i.sstatic.net/fHVZe06t.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fHVZe06t.png" alt="" /></a></p>
<p>I define the front edge as the image plane. I can therefore create a scale there. The trick now is to create another scale in exactly this image plane, with which you can work in the other direction. I place this horizontally, but you can choose this freely. The important thing is that it is in the image plane and not part of the projected three-dimensional structure. As everything is just lines in the picture, this distinction is difficult. But the black lines describe the three-dimensional space, while the blue and red lines only describe the flat image.</p>
<p><a href="https://i.sstatic.net/2fYFKusM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2fYFKusM.png" alt="" /></a></p>
<p>With the uniform red scale (from the image plane), I can now create lines to the respective measuring points. We want to divide the line that goes to VP₂. Therefore, you also have to take MP₂. I create red lines to the black line that I want to divide. It is also important that I use the bottom black line, as I have also placed the scale at the bottom corner.</p>
<p><a href="https://i.sstatic.net/2fFToCQM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2fFToCQM.png" alt="" /></a></p>
<p>The intersections of the red and black lines show where the subdivisions must be. Then I can draw normal vertical lines, which again come from three-dimensional space.</p>
<p><a href="https://i.sstatic.net/modIUMDs.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/modIUMDs.png" alt="" /></a></p>
<p>You can already see that the distances between the lines are getting smaller and smaller. That's the way it has to be.</p>
<p><a href="https://i.sstatic.net/TML1CdNJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TML1CdNJ.png" alt="" /></a></p>
<p>I can then use the same principle to create the scene with the grid and the piano. Now the piano is centered in front of the window. The vertical bars for the grille are the same distance apart. It looks much more harmonious.</p>
<p><a href="https://i.sstatic.net/TtNkTNJj.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TtNkTNJj.png" alt="" /></a></p>
<p>The drawback now, however, is that the auxiliary lines take up considerably more space than the actual drawing. I now like to draw on A4, but the actual drawing tends to be in A6 format. I could get A3 drawing paper now, then the problem would be a little less severe. Or I could scan this template, crop it and then print it out larger. Then I can trace the edges and have it in the size I want.</p>
<p>With that I managed to do an improved version of the initial drawing:</p>
<p><a href="https://i.sstatic.net/oJN24n4A.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oJN24n4A.png" alt="" /></a></p>

CC BY-SA 4.0
2024-12-23T16:10:01.833Z
2024-12-29T17:28:58.430Z
2024-12-29T17:28:58.430Z
Martin Ueding
<p>What you are looking for is a perspective ruler/scale.</p>
<p>When I was taking an architectural drafting class back in the 80's, the instructor had such a device. To our surprise, it was made by a local mathematician. (I will see if I can find it and post a picture).</p>
<p>As far as I have searched online, I have not been able to find a commercially available product.</p>
<p>Note: There is a question here on the Stack Exchange on how to calculate the distances in perspective: <a href="https://math.stackexchange.com/questions/613725/a-formula-for-perspective-measurement">A formula for Perspective measurement</a>.</p>

CC BY-SA 4.0
2024-12-23T16:33:46.167Z
2024-12-23T16:33:46.167Z
agarza
<p>This is a short cut to OP's answer for more casual use.</p>
<ol>
<li>Place a line in your drawing that represents an outer edge of the shape that will have regular divisions across its length as desired (eg. one of the ladder legs).</li>
<li>Extend guidelines from the start and end of this line to your horizon/eye line. This will locate the vanishing point (VP) for the division lines.</li>
<li>Somewhere convenient along those first two VP lines draw a vertical reference line. Mark the vertical line with the number of desired divisions, equally spaced.</li>
<li>Extend reference lines from the VP through each division mark and on to the shape's edge. The intersecting points will divide the shape equally in perspective (eg. mark where the rungs will be).</li>
</ol>

CC BY-SA 4.0
2024-12-29T17:32:55.657Z
I don't quite understand how to construct this from your description. How do I pick the point on the horizon that becomes the VP? It seems that the VPs are different for both ends?
Martin Ueding
CC BY-SA 4.0
2024-12-29T20:59:28.980Z
Good point. That is arbitrary based on how you want to orient the object based on how much the perpendiculars point out of the drawing plane. As long as they get to the eye line somewhere. This is just a quick way of getting the correct spacing in perspective in a drawing, not analytically constructing a perspective from floor plan.
rebusB
CC BY-SA 4.0
2024-12-28T16:54:19.003Z
2024-12-28T17:07:39.997Z
2024-12-28T17:07:39.997Z
rebusB
<p>Say I want to draw a ladder with even spaced rungs. In flat perspective this is straightforward to do. I can just space them equally and it looks realistic:</p>
<p><a href="https://i.sstatic.net/pQUuLdfg.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pQUuLdfg.jpg" alt="Ladder in flat perspective." /></a></p>
<p>When using linear perspective with a vanishing point, I can construct the sides of the ladders. But how exactly do I have to space the rungs to make it look realistic?</p>
<p><a href="https://i.sstatic.net/yV4jxr0w.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yV4jxr0w.jpg" alt="Ladder with vanishing point." /></a></p>
<p>I can guess, but I'd like to know how to construct this in a drawing.</p>
<p>I know how to linear perspective is constructed in general. We have the evenly spaced rungs on the bottom, going to the right. There is a virtual observer and there is a canvas that is tilted. The intersection of the rays from the eye with the canvas indicates the position that it will be on the drawing.</p>
<p><a href="https://i.sstatic.net/UD9PDJbE.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UD9PDJbE.jpg" alt="Drawing of the construction of linear perspective." /></a></p>
<p>One can apply trigonometry here and calculate this. But I don't necessarily know what the position of the canvas is, what focal length I want and so on. But since it is linear, the placement of two rungs together with the vanishing point must be enough to construct all the other ones.</p>
<p>Is there some straightforward method to construct this without having to basically construct a ray tracing algorithm on the side? Or is there some named concept that I could look up?</p>
<p><strong>Added later</strong>: After I formulated the question, <a href="https://crafts.stackexchange.com/questions/3316/how-to-measure-the-distance-to-horizon-line-one-point-perspective">this other question</a> was suggested. My last image already has the right idea, I just need to rotate it a bit and make this tilted “canvas” line the “picture plane” or “vantage line” and have the ladder with even rungs in the lower part of the image?</p>

<linear-perspective>
Construct spacing in perspective
Martin Ueding
# Construct spacing in perspective

(Asked by: username_0 on 2024-12-22)

Say I want to draw a ladder with even spaced rungs. In flat perspective this is straightforward to do. I can just space them equally and it looks realistic:

[![Ladder in flat perspective.](https://i.sstatic.net/pQUuLdfg.jpg)](https://i.sstatic.net/pQUuLdfg.jpg)

When using linear perspective with a vanishing point, I can construct the sides of the ladders. But how exactly do I have to space the rungs to make it look realistic?

[![Ladder with vanishing point.](https://i.sstatic.net/yV4jxr0w.jpg)](https://i.sstatic.net/yV4jxr0w.jpg)

I can guess, but I'd like to know how to construct this in a drawing.

I know how to linear perspective is constructed in general. We have the evenly spaced rungs on the bottom, going to the right. There is a virtual observer and there is a canvas that is tilted. The intersection of the rays from the eye with the canvas indicates the position that it will be on the drawing.

[![Drawing of the construction of linear perspective.](https://i.sstatic.net/UD9PDJbE.jpg)](https://i.sstatic.net/UD9PDJbE.jpg)

One can apply trigonometry here and calculate this. But I don't necessarily know what the position of the canvas is, what focal length I want and so on. But since it is linear, the placement of two rungs together with the vanishing point must be enough to construct all the other ones.

Is there some straightforward method to construct this without having to basically construct a ray tracing algorithm on the side? Or is there some named concept that I could look up?

**Added later**: After I formulated the question, [this other question](https://crafts.stackexchange.com/questions/3316/how-to-measure-the-distance-to-horizon-line-one-point-perspective) was suggested. My last image already has the right idea, I just need to rotate it a bit and make this tilted “canvas” line the “picture plane” or “vantage line” and have the ladder with even rungs in the lower part of the image?

**Tags:** linear-perspective


---

## 3 answers

### Answer by username_0 on 2024-12-23. Score: 5

I have a floor plan to which I would like to draw the room. Then I want to put a piano in there. I have the dimensions for everything. But I couldn't really realize it. So I drew it with vanishing points, but only roughly estimated the dimensions. This is my result:

[![Pencil drawing with freehand perspective.](https://i.sstatic.net/CbIJ5GJr.png)](https://i.sstatic.net/CbIJ5GJr.png)

After a bit of research, I found [this book by Bruce MacEvoy](https://handprint.com/HP/WCL/perspect3.html). It explains the whole thing in very technical terms. It took me a few hours of reading, but now I understand it. I would now like to share this with you.

First, we construct a crosshair from the horizon line and the center of view. The point of intersection is the vanishing point to which the gaze is directed.

[![Crosshair.](https://i.sstatic.net/cwZN0N9g.png)](https://i.sstatic.net/cwZN0N9g.png)

One important concept is the 90-degree field of vision. This results in other things. Firstly, the viewing height (i.e. eye level above the floor) and the viewing distance to the (virtual) image plane are the same. The viewing height is marked by the point V.

In addition, an angle of 90° is included between the two vanishing points. I want to create a perspective with a 30°/60° rotation so that it is not exactly symmetrical. I draw from viewpoint V at an angle of 30° to the horizon and mark the first vanishing point VP₁ there. Then I do the same with 90° and construct the vanishing point VP₂.

[![Perspective construction.](https://i.sstatic.net/19QP9Nc3.png)](https://i.sstatic.net/19QP9Nc3.png)

The next step is to construct the so-called measuring points. This is the crucial concept. To do this, draw a circle with VP₁ in the center of V to the horizon. The point of intersection is then MP₁.

[![Construction.](https://i.sstatic.net/niSX2DPN.png)](https://i.sstatic.net/niSX2DPN.png)

Do the same on the other side to obtain MP₂.

[![Construction.](https://i.sstatic.net/Cb13w6rk.png)](https://i.sstatic.net/Cb13w6rk.png)

Now we have prepared the frame and can start with the actual drawing. I am drawing a simple cuboid here that extends to infinity. I can already add a scale to the front edge. The other lines go to the respective vanishing points. However, I can't add a scale there yet.

[![Construction.](https://i.sstatic.net/26YZqxrM.png)](https://i.sstatic.net/26YZqxrM.png)

I can then use this scale to create the lines for an even grid to the vanishing points. Here we have nice even spacing.

[![](https://i.sstatic.net/fHVZe06t.png)](https://i.sstatic.net/fHVZe06t.png)

I define the front edge as the image plane. I can therefore create a scale there. The trick now is to create another scale in exactly this image plane, with which you can work in the other direction. I place this horizontally, but you can choose this freely. The important thing is that it is in the image plane and not part of the projected three-dimensional structure. As everything is just lines in the picture, this distinction is difficult. But the black lines describe the three-dimensional space, while the blue and red lines only describe the flat image.

[![](https://i.sstatic.net/2fYFKusM.png)](https://i.sstatic.net/2fYFKusM.png)

With the uniform red scale (from the image plane), I can now create lines to the respective measuring points. We want to divide the line that goes to VP₂. Therefore, you also have to take MP₂. I create red lines to the black line that I want to divide. It is also important that I use the bottom black line, as I have also placed the scale at the bottom corner.

[![](https://i.sstatic.net/2fFToCQM.png)](https://i.sstatic.net/2fFToCQM.png)

The intersections of the red and black lines show where the subdivisions must be. Then I can draw normal vertical lines, which again come from three-dimensional space.

[![](https://i.sstatic.net/modIUMDs.png)](https://i.sstatic.net/modIUMDs.png)

You can already see that the distances between the lines are getting smaller and smaller. That's the way it has to be.

[![](https://i.sstatic.net/TML1CdNJ.png)](https://i.sstatic.net/TML1CdNJ.png)

I can then use the same principle to create the scene with the grid and the piano. Now the piano is centered in front of the window. The vertical bars for the grille are the same distance apart. It looks much more harmonious.

[![](https://i.sstatic.net/TtNkTNJj.png)](https://i.sstatic.net/TtNkTNJj.png)

The drawback now, however, is that the auxiliary lines take up considerably more space than the actual drawing. I now like to draw on A4, but the actual drawing tends to be in A6 format. I could get A3 drawing paper now, then the problem would be a little less severe. Or I could scan this template, crop it and then print it out larger. Then I can trace the edges and have it in the size I want.

With that I managed to do an improved version of the initial drawing:

[![](https://i.sstatic.net/oJN24n4A.png)](https://i.sstatic.net/oJN24n4A.png)


---

### Answer by username_1 on 2024-12-23. Score: 1

What you are looking for is a perspective ruler/scale.

When I was taking an architectural drafting class back in the 80's, the instructor had such a device. To our surprise, it was made by a local mathematician. (I will see if I can find it and post a picture).

As far as I have searched online, I have not been able to find a commercially available product.

Note: There is a question here on the Stack Exchange on how to calculate the distances in perspective: [A formula for Perspective measurement](https://math.stackexchange.com/questions/613725/a-formula-for-perspective-measurement).


---

### Answer by username_2 on 2024-12-28. Score: 0

This is a short cut to OP's answer for more casual use.

1. Place a line in your drawing that represents an outer edge of the shape that will have regular divisions across its length as desired (eg. one of the ladder legs).
2. Extend guidelines from the start and end of this line to your horizon/eye line. This will locate the vanishing point (VP) for the division lines.
3. Somewhere convenient along those first two VP lines draw a vertical reference line. Mark the vertical line with the number of desired divisions, equally spaced.
4. Extend reference lines from the VP through each division mark and on to the shape's edge. The intersecting points will divide the shape equally in perspective (eg. mark where the rungs will be).

> **Comments:**
> **username_0**: I don't quite understand how to construct this from your description. How do I pick the point on the horizon that becomes the VP? It seems that the VPs are different for both ends?  
> **username_2**: Good point. That is arbitrary based on how you want to orient the object based on how much the perpendiculars point out of the drawing plane. As long as they get to the eye line somewhere. This is just a quick way of getting the correct spacing in perspective in a drawing, not analytically constructing a perspective from floor plan.  

---

