/sci/ - Science & Math

>>9201494
What I do know is that you can translate lines by rotating their vanishing points along the center of vision, but I don't see an easy way of calculating this position in relation to another line.

>>9201494
I realize that certain compass construction techniques may be impossible, i.e. squaring the circle, trisecting an angle, but I really think this could help out a lot of artists that I know.

>>9201494
To clarify most tutorials will leave the picture plane completely perpendicular to the ground, and it distorts the image quite a bit.
If I can rotate a cube, then I can build a world in relation to the camera, and remove this distortion.
Unfortunately, this is quite difficult for a person without much STEM experience.

>>9201494
Am I making any sense to sci?

>can't perfectly accurately represent 3-D in a 2-D plane
wow its almost like you're missing a dimension

>>9202632
Yes, but your missing the point.
I can represent that projection that much better by understanding it.
Isn't that the point of /sci/?

>>9202695
Also I fucked up the picture.

I think I can help resolve a few of your issues. I am going to focus on objects that are sitting on or parallel to the ground at any rotation around a vertical axis (so no banked/tilted objects).

For starters, three-point perspective is always angled up or down. If it isn't angled up or down, there is no third vanishing point and vertical lines will be perfectly parallel, i.e. you have two-point perspective. The distortion you are noticing is because you are using a very large field of view - indeed, your boy Scotty Robertson recommends not using greater than a 60 degree cone of vision for three-point perspective. Additionally, everything falling outside the triangle formed by your vanishing points is going to be distorted as shit. This is why three-point perspective drawings that cross the horizon look distorted.

I think most of your problems stem from the placement of your vanishing points. The left and right vanishing points will always be placed on the horizon. For 45/45 three-point perspective, if the width between these points is equal to the width of the circle that represents your cone of vision, then you have a 90 degree lateral field of vision - you could stand on a street corner and see both perpindicular streets vanish into the distance (pic related). If the width between these vanishing points is much greater than the width of your field of vision, your field of vision will be much less than 90 degrees.

Cont'd (1/3)

>>9203831
The vertical field of vision is defined in a similar fashion. If the distance between the horizon and your top/bottom vanishing point is equivalent to the height of the cirle representing your cone of vision, you have a 90 degree vertical field of vision. Increase the distance between the horizon and the top/bottom vanishing point to reduce this FoV. Unless you are drawing something pretty crazy, you will generally not have more than one of the three-point perspective vanishing points on or inside the circle. The smaller your FoV is compared to the triangle formed by your vanishing points, the less distorted and more isometric your work will look.

Drawing objects at rotations other than 45 degrees in three-point perspective will be pretty much identical to drawing objects at rotations other than 45 degrees in two-point perspective. The only difference is that the vertical lines are no longer vertical. Suppose you want a 60 degree FoV for a 30/60 three-point perspective drawing. The width between your vanishing points will need to be twice the width of the circle representing your cone of vision. Why? Because trigonometry. Pages 24-25 of Robertson's book discusses this, but the math is fairly simple. Let x be your desired field of view, let r be the radius of this FoV and let y be one of the angles you want your object rotated at (e.g., 45, 30). The equation for the width between vanishing points on the horizon is then given by:

[eqn]Distance = (r/tan(x/2)) * (1/tan(y) + tan(y))[/eqn]

>>9203835
I'm assuming this thread is yours: >>>/ic/3154106 . In your diagram, the top of your cone of vision is the horizon, and the bottom is looking straight down. This will result in distortion since much of the image falls outside of the triangle formed by the vanishing points, a consequence of your 90 degree FoV. If you want to draw an image of somebody looking down at a cube at a 45 degree angle, with the horizon at the top of their FoV, just:
>Make sure the LVP and RVP are quite a bit wider than your FoV, and are on the horizon
>Make the bottom vanishing point very far below the bottom of your FoV
>To make a perspective grid, draw a fuckload of rays from these points

Anyway, I don't draw or draft, so take this all with a grain of salt. Robertson's book was pretty light on the details, as you indicated, and Loomis just says to make sure that the vanishing points on the horizon are far apart. If any of this looks wrong, feel free to say so.

>>9201494
I appreciate the help, but unfortunately, it's not entirely what I need to solve.
The horizon is an infinite plane, and in most drafting instruction it doesn't move from the 0 degree horizontal line because you are always looking directly forwards.
This is a bit confusing, but in the first picture you posted you're still looking directly forwards, but your cone of vision allows you to see upwards.
I want everything to be independent of the horizon relative to the camera so that I can set up a scene in much the same way you can in a 3d program.
My mistake was trying too early to make a geometric proof before I had thoroughly defined the problem.
A geometric proof needs to be made on the picture plane for every orientation of a cube.
In relation to a 90 degree cone of vision projected onto a piece of paper.
That way I can draft it in much the same way you would manipulate objects in a 3d program.
I'm going over a bit of linear algebra to work out how this happens exactly, then based on that proof making a method of drafting via a geometric proof.
I think I can solve this on my own, and it will be in much more detail when I do.
Frankly, I think that it's something I've overthought, and a smart high-schooler could probably finish it.
It is not the same problem you you detailed, however.

>>9201494

Perhaps 4 point or 5 point perspective is what you are looking for?

>>9205430

Once you are in 4+ point perspective you need to use curved lines to represent lines that are straight in the third dimension. While this may seem odd, it is the only real solution to projection.

To minimise the distortion of perspective drawing, you need to keep the vanishing points far away from your drawing. (This is true for any number of points except 1 point perspective.) Your whole drawing should be less than a quarter the distance from one vanishing point to the other. Note that many of the drawings on this page, including the two I posted, violate this rule.

>>9205642
I might actually be too autistic to explain this properly.
This is something that I might actually have to show you once I'm done for people to understand what I mean.
I've tried, but every time it goes right over people's heads.
This isn't meant to insult people, but I don't see how I can explain it any further.
I'm sorry.

>>9205665

Have you ever tried using a Camera Obscura? You'll see the same distortions with real life objects, and it may help you formulate what it is that you are trying to do.

>>9205692
I know what I'm trying to do.
I've even modeled a picture plane, cube, and point representing a camera in blender.

>>9205692
There's a problem with using a Camera Obscura, or 4 point, or 5 point perspective for that matter.
You've generated one view that perfectly replicates a human's sight.
One view only.
Using the three point perspective that I've thought of, you can create in my opinion the most convincing picture plane possible.
The only difference between my three point perspective, and the three point perspective that others would use is that you mark three vanishing points for EACH object you want projected onto the picture plane.
The objects also do not have to be PARALLEL to the ground.
The ground does not have to be parallel with the viewer's center of vision either.
You can look up or down, and you move the ground instead of moving your drawing up or down on the picture plane.

>>9205744

In perspective drawing, the first vanishing points you draw represent North, South, East, West, Up, or Down.

To point your "camera" in different directions, you move these points. This will move all the objects that have lines that are parallel to N. S. E. W. U. D.

If you have an object that is NOT parallel to those lines, you would start by giving it it's own set of vanishing points. If this object is a cube, then the new vanishing points would be the same distance from each other as the basic ones. If the object is like a diagonal street, it's vanishing points will still be on the horizon. If it is like the leaning tower of Pizza, you'd have vanishing points in the air. You'd need some trigonometry to calculate exact angles and distances, but most artists would just eyeball it.

>>9205692
I know what I'm doing.
The problem with 4 point or 5 point perspective is that it generates one view only.
You can't look around the image, and still have it look convincing.
Three point perspective, therefore, the best approximation to a person looking through a picture plane.
A vanishing point is a point on the picture plane that represents a set of lines all parallel to each other.
One point perspective, Two Point perspective, and Three point perspective all follow the same rules governing how you find vanishing points.
The only difference is that one point perspective limits the objects being viewed, so that they are all parallel to one line.
Two point perspective has the problem of all objects being restricted to the ground plane.
The ground plane is always represented as an infinite plane parallel to a person's center of vision.
Every representation of three point perspective that I've seen is in fact restricted to the ground plane.

>>9205795
This is exactly what I've been trying to explain.
However I would like to figure out a method of finding those vanishing points in the air with a compass and straightedge.

>>9205800
I now see why proofs have to be so damn formal.
I want a geometric proof of this.

I see why proofs have to be so damn formal.
>>9205795
I want a geometric proof of this, so that I can draft it.
https://en.wikipedia.org/wiki/3D_projection#Perspective_projection

>>9205800
>However I would like to figure out a method of finding those vanishing points in the air with a compass and straightedge.

Ah ha! I think I understand now. But I am afraid I don't know the answer beyond "Use trigonometry". I think this sort of thing is why people buy 3D graphics programs.

BTW, When you say "Look around the image" are you trying to extend your field of view beyond 90° or so? There is really no solution to that other then drawing on the inside of a sphere. It's the same issue as why 2d maps can't show the earth without distortion.

BTW2, I'm wrong above about the rotated object having "the new vanishing points the same distance from each other as the basic ones." If you rotate a cube on the plane of the ground, one of the vanishing points would shoot off to infinity and reappear on the other side of the page.

>>9205385
Infinity doesn’t exist because: infinity is an imaginary number

>>9205847
You can determine the point where it approaches infinity on the picture plane.

>>9205847
>Infinity doesn’t exist because: infinity is an imaginary number

Okay, the last Real number then. ;P

>>9205843
I'm not trying to go beyond 90°.
When I say 90° cone of vision, it's actually 45° in either direction.

>>9205867
>I'm not trying to go beyond 90°.
>When I say 90° cone of vision, it's actually 45° in either direction.

In photography that would be called a 90° field of view, and would be a wide angle lens.

I have a question for you now: How far away does your current drawing setup allow your vanishing points to be? What would you do if you calculated that a vanishing point needed to be 10 or 50 feet away?

>>9205987
The field of view I usually use is 60° just because it was recommended as being close to the human eye.
Remember that you don't have to actually draw the vanishing point, it just needs to have a position defined.
Usually I would just draw a line pointed at that vanishing point.
You can use a protractor and work out the angle you need.

>>9201494
https://handprint.com/HP/WCL/perspect4.html

i found this very helpful. imagine the vertex of a cube overlapping your viewpoint. the three edges radiating outward, intersecting the picture plane, will be where the vanishing points are for all other lines in space that are parallel to those edges.

i found erik olson's perspective series very informative for vanishing points of sloped lines (ie not directed toward the horizon line, which is just an informal construction for artists' use anyway). you should realize that the method for identifying sloping vanishing points is really akin to using a spherical coordinate system (two angles), from which a tedious version of three-point perspective can be derived - that is, starting from two-point perspective, learning sloping vps, and then building three-point in this stupid way from there might be instructive for you.

afaik three-point construction is exceptionally tedious. i believe you actually have to deal with vanishing lines that "pop out" of the page. you won't have time to build these systems from the ground up for every project you do anyway. just use sketchup and trace.

>>9206020
>just use sketchup and trace.
No.

I knew someone had to have thought of this before, thank you.
This has been bothering me for days.

>>9206071
no problem. and i made a mistake - you'd only have to deal with the vanishing lines that "pop out" if you want to find vanishing points of sloped lines in 3pp... which is just even more complex than the basic 3pp model. at least when i tried doing it myself, though i might be wrong.

why not use a computer or just eyeball it? what sort of illustrations are you doing? tmk there aren't many people, if any at all, who'd go to this trouble. craig mullins uses software.

>>9206085
It's a bit hard to describe.
I see a lot of tutorials that have you "draw a box" or feel things out.
I have a feeling that if you improve your frame of reference beyond what you think is necessary at the time, certain things don't require as much practice because you aren't guessing at that point, it's a very educated guess.
I think you can also more effectively do things when you have a working mental model of how they are.
Did you know there are Japanese kids that are raised on a the abacus instead of calculators?
They have a working visual model of the abacus, and can then use it as if it were in front of them.
With practice I think visual things can be integrated mentally, and used in this manor.
For most illustrations maybe I wouldn't use this, but imagine practicing this enough to be able to draw without software as if not more accurate?
How much time would that save Craig Mullins?

https://www.youtube.com/watch?v=3g63WR_PelY

>>9206134
hmm i see what you mean. good fundamentals are good, certainly. i guess if it makes you comfortable to approach art then that's fine - ime though i've used it as an excuse to avoid actually practicing. don't make the same mistake.

and honestly, if you just practice drawing cubes enough you'll be able to intuit 1 2 and 3pp from your head. this is expected practice in industrial design and related fields. you should be able to apply this intuitive perspective without necessarily needing autistic math, but you do you. i'm just guessing here, but mullins probably sketches things out and confirms things more precisely later on. there's a painting he did of the white house, and i remember he said in an interview he modelled it. so i guess there's flexibility there, too - finding just the right angle and composition and all without having to work too hard.

>>9206168
*the capitol

Why not draw a mirror image beyond the horizon?

>>9206177
w..what is that anon?

>>9206183
Autism 2.0

It's some compass and straight edgy way of creating perspective. Most don't even see it though

>>9206191
I mean I actually did see it, but jesus.
I feel like I'm looking at god's sketchpad.

>>9206200
Tnx anon! That's a huge compliment