<p>For example, a vertical plane (geometrical entity) seen from the front of an orthographic system will shrink on the horizontal axis due to rotation around the vertical axis until it becomes invisible, but at middle point of rotation, the projection will be longer then half of the actual length.
<a href="https://i.sstatic.net/b3l24.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/b3l24.jpg" alt="enter image description here"></a></p>

<ol>
<li>How come is the projection shrinking in the first place?</li>
</ol>

<p>Top angle rotation implies movement in 2 axis, in this case upwards and to the left. However, because from the front angle the up vector becomes backwards vector, the movement on that axis will not be visible in projection. That leaves only movement to the left to be projected, therefore the plane will shrink horizontally.
<a href="https://i.sstatic.net/Ovcuq.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Ovcuq.jpg" alt="enter image description here"></a></p>

<ol start="2">
<li>At what rate is the shrinking?</li>
</ol>

<p>The change in size happens gradually (not linearly), first slower, then faster, or vice-versa, because the trajectory is an arc, not a straight line.</p>

<p>2.1. How to geometrically describe an arc/circle?</p>

<p>An arc can be simplified by using equal segments. Every segment has a certain angle, depending on it’s length, and the next segment inherits the angle and is rotated still by the same degree.</p>

<p><a href="https://i.sstatic.net/MbQbT.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/MbQbT.jpg" alt="enter image description here"></a></p>

<p>2.2. Why does the trajectory matters?</p>

<p>The closer the plane is to 90 degrees (to the projection plane) the smaller the projection, and the closer it is to being parallel, the closer to actual length. If every projected segment has a different, increasing angle, the length will grow bigger with every increment of the arc. This is the projection of the rate of shrinking..</p>

<p><a href="https://i.sstatic.net/LO7Dl.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LO7Dl.jpg" alt="enter image description here"></a></p>

<p>If the trajectory would be a straight line at 45 degrees, then at 45 degrees the projection would be at half the actual length. However, this means the plane would have to shrink.</p>

<p><a href="https://i.sstatic.net/qVFFL.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qVFFL.jpg" alt="enter image description here"></a></p>

<p>In conclusion I would like to add that this is a matter of rotation and it is valid in perspective as well. However, in perspective translation also marks the scaling of an object. I think foreshortening refers only to translation, not to rotation. Nevertheless, a projection can change size according to rotation as well.</p>

CC BY-SA 4.0
2019-06-24T06:24:03.537Z
I will have to be honest: I did not understand the question, I did not understand the answer, and I did not understand how they are connected. The question is clearly about projection, the answer is about trajectory... Will you please update both the question and the answer to make things more clear?
virolino
CC BY-SA 4.0
2019-06-24T06:27:05.397Z
Also: a plane (or its projection) does not get smaller when you rotate it in any way, because by definition it is infinite. Of course, unless it degenerates to a line, in the case of perpendicularity. And how do you connect 45 deg to 1/2? Do you have some mathematical formula to prove it? My recollection of trigonometry would lead to sqrt(2) / 2 = 1/ sqrt(2).
virolino
CC BY-SA 4.0
2019-06-24T06:55:58.807Z
@virolino Thank you for challenging my question. I can’t say it’s “plain surface”, because all matter has volume. It’s like when you say that in perspective projections that are farther away are getting smaller. This wouldn’t be true for an infinite plane either, but for a plain object it’s actually measurable. 
I think I see the confusion, but I don’t know how to clarify it.
3Dtoaster
CC BY-SA 4.0
2019-06-24T07:01:32.873Z
You actually added to the confusion :) A plane has volume?!
virolino
CC BY-SA 4.0
2019-06-24T07:12:02.283Z
@virolino Graphics can have pretty subtle concepts… 
3Dtoaster
CC BY-SA 4.0
2019-06-24T07:14:16.023Z
Your concepts defy mathematics so far :(
virolino
CC BY-SA 4.0
2019-06-24T07:43:29.500Z
A link to [orthographic projections](https://en.wikipedia.org/wiki/Orthographic_projection) might help some to understand your question answer more.
Chris Rogers
CC BY-SA 4.0
2019-06-24T07:45:10.780Z
I had to really get my head around what you were trying to put across for a bit, thinking the whole hypothesis was bunkum like @virolino but you are very right.  Great share.
Chris Rogers
CC BY-SA 4.0
2019-06-24T09:03:58.867Z
Let me try to grasp the concept as well: your question seems to be "why is a plane, seen at a 45 degree angle in orthographic projection, not (apparently) half of the width it would have when seen perpendicularly (its absolute width)?"? Then I can try to clarify both posts.
Joachim
CC BY-SA 4.0
2019-06-24T10:29:01.893Z
@Joachim You got it. If you can point out how I can develop on the answer, I’d be glad to hear it.
3Dtoaster
CC BY-SA 4.0
2019-06-25T11:31:05.010Z
I am still in complete dark: is the question + answer about why sin45=cos45=0.707 and not 0.5?
virolino
CC BY-SA 4.0
2019-07-23T23:02:33.237Z
@virolino - Yes... that is exactly the question, just asked and "answered" in math antagonistic terms. OP: draw a circle radius 1 at the origin, draw a line bisecting the x and y axes (45 degrees), now extend lines from where the bisecting line intersects the circle to the axes. Measure those lines, they will equal .707)   It all comes down to Pythagoras' theory.
rebusB
CC BY-SA 4.0
2019-06-24T05:30:38.883Z
2019-06-24T05:30:38.883Z
3Dtoaster
<p>We have a plane of certain dimensions X and Y, and are looking at it perpendicularly by means of orthographic projection. The plane rotates 45 degrees on the X or Y axis.</p>

<p>Given this situation, I would expect that the plane, once it has rotated, would appear to have half its original dimension opposite the direction of rotation (rotation on X-axis would change the apparent length of Y); or that the plane shrunk by 50%.   </p>

<p>In short, I would expect a <em>linear transition</em>. Why isn't this the case?</p>

CC BY-SA 4.0
2019-06-24T12:43:47.407Z
Having edited the question, I realize this is actually -topic for A&C: it's a purely geometrical/mathematical question. Unless you have a direct practical application for this, I think it needs to be moved to [math](https://math.stackexchange.com).
Joachim
CC BY-SA 4.0
2019-06-24T12:48:06.933Z
@Joachim You can use it to draw better, obviously, or try to extract proportions from photos using this theory. Thanks for editing.
3Dtoaster
CC BY-SA 4.0
2019-06-24T13:04:07.087Z
If you edit one of those examples - as directly relating to that practical application - into both your question and answer, that would be helpful.
Joachim
CC BY-SA 4.0
2019-06-24T13:15:32.657Z
@Joachim I can't help you for now. Hopefully I'll do another question related to this one that deals with applications. But it's a lot of difficult content.
3Dtoaster
CC BY-SA 4.0
2019-06-24T13:27:41.427Z
I'm voting to close this question as off-topic because it is purely theoretical, and not related to either arts or crafts.
Joachim
<drawing>
Why isn’t a orthographically projected plane at 45 degrees seemingly half its actual length?
3Dtoaster
# Why isn’t a orthographically projected plane at 45 degrees seemingly half its actual length?

(Asked by: username_0 on 2019-06-24)

We have a plane of certain dimensions X and Y, and are looking at it perpendicularly by means of orthographic projection. The plane rotates 45 degrees on the X or Y axis.

Given this situation, I would expect that the plane, once it has rotated, would appear to have half its original dimension opposite the direction of rotation (rotation on X-axis would change the apparent length of Y); or that the plane shrunk by 50%.

In short, I would expect a *linear transition*. Why isn't this the case?

**Tags:** drawing

> **Comments:**
> **username_1**: Having edited the question, I realize this is actually -topic for A&C: it's a purely geometrical/mathematical question. Unless you have a direct practical application for this, I think it needs to be moved to [math](https://math.stackexchange.com).  
> **username_0**: @username_1 You can use it to draw better, obviously, or try to extract proportions from photos using this theory. Thanks for editing.  
> **username_1**: If you edit one of those examples - as directly relating to that practical application - into both your question and answer, that would be helpful.  
> **username_0**: @username_1 I can't help you for now. Hopefully I'll do another question related to this one that deals with applications. But it's a lot of difficult content.  
> **username_1**: I'm voting to close this question as off-topic because it is purely theoretical, and not related to either arts or crafts.  

---

## 1 answers

### Answer by username_0 on 2019-06-24. Score: 2 | Accepted answer

For example, a vertical plane (geometrical entity) seen from the front of an orthographic system will shrink on the horizontal axis due to rotation around the vertical axis until it becomes invisible, but at middle point of rotation, the projection will be longer then half of the actual length.
[![enter image description here](https://i.sstatic.net/b3l24.jpg)](https://i.sstatic.net/b3l24.jpg)

1. How come is the projection shrinking in the first place?

Top angle rotation implies movement in 2 axis, in this case upwards and to the left. However, because from the front angle the up vector becomes backwards vector, the movement on that axis will not be visible in projection. That leaves only movement to the left to be projected, therefore the plane will shrink horizontally.
[![enter image description here](https://i.sstatic.net/Ovcuq.jpg)](https://i.sstatic.net/Ovcuq.jpg)

2. At what rate is the shrinking?

The change in size happens gradually (not linearly), first slower, then faster, or vice-versa, because the trajectory is an arc, not a straight line.

2.1. How to geometrically describe an arc/circle?

An arc can be simplified by using equal segments. Every segment has a certain angle, depending on it’s length, and the next segment inherits the angle and is rotated still by the same degree.

[![enter image description here](https://i.sstatic.net/MbQbT.jpg)](https://i.sstatic.net/MbQbT.jpg)

2.2. Why does the trajectory matters?

The closer the plane is to 90 degrees (to the projection plane) the smaller the projection, and the closer it is to being parallel, the closer to actual length. If every projected segment has a different, increasing angle, the length will grow bigger with every increment of the arc. This is the projection of the rate of shrinking..

[![enter image description here](https://i.sstatic.net/LO7Dl.jpg)](https://i.sstatic.net/LO7Dl.jpg)

If the trajectory would be a straight line at 45 degrees, then at 45 degrees the projection would be at half the actual length. However, this means the plane would have to shrink.

[![enter image description here](https://i.sstatic.net/qVFFL.jpg)](https://i.sstatic.net/qVFFL.jpg)

In conclusion I would like to add that this is a matter of rotation and it is valid in perspective as well. However, in perspective translation also marks the scaling of an object. I think foreshortening refers only to translation, not to rotation. Nevertheless, a projection can change size according to rotation as well.

> **Comments:**
> **username_2**: I will have to be honest: I did not understand the question, I did not understand the answer, and I did not understand how they are connected. The question is clearly about projection, the answer is about trajectory... Will you please update both the question and the answer to make things more clear?  
> **username_2**: Also: a plane (or its projection) does not get smaller when you rotate it in any way, because by definition it is infinite. Of course, unless it degenerates to a line, in the case of perpendicularity. And how do you connect 45 deg to 1/2? Do you have some mathematical formula to prove it? My recollection of trigonometry would lead to sqrt(2) / 2 = 1/ sqrt(2).  
> **username_0**: @username_2 Thank you for challenging my question. I can’t say it’s “plain surface”, because all matter has volume. It’s like when you say that in perspective projections that are farther away are getting smaller. This wouldn’t be true for an infinite plane either, but for a plain object it’s actually measurable.  I think I see the confusion, but I don’t know how to clarify it.  
> **username_2**: You actually added to the confusion :) A plane has volume?!  
> **username_0**: @username_2 Graphics can have pretty subtle concepts…   
> **username_2**: Your concepts defy mathematics so far :(  
> **username_3**: A link to [orthographic projections](https://en.wikipedia.org/wiki/Orthographic_projection) might help some to understand your question answer more.  
> **username_3**: I had to really get my head around what you were trying to put across for a bit, thinking the whole hypothesis was bunkum like @username_2 but you are very right.  Great share.  
> **username_1**: Let me try to grasp the concept as well: your question seems to be "why is a plane, seen at a 45 degree angle in orthographic projection, not (apparently) half of the width it would have when seen perpendicularly (its absolute width)?"? Then I can try to clarify both posts.  
> **username_0**: @username_1 You got it. If you can point out how I can develop on the answer, I’d be glad to hear it.  
> **username_2**: I am still in complete dark: is the question + answer about why sin45=cos45=0.707 and not 0.5?  
> **username_4**: @username_2 - Yes... that is exactly the question, just asked and "answered" in math antagonistic terms. OP: draw a circle radius 1 at the origin, draw a line bisecting the x and y axes (45 degrees), now extend lines from where the bisecting line intersects the circle to the axes. Measure those lines, they will equal .707)   It all comes down to Pythagoras' theory.  

---

