I’m about to write a more in depth tutorial about projection/perspective drawing. My plan is to create something that gives insight how viewpoint, “paper” and vanishing points really works.
Are there something you have questions about?
Is there a need for this tutorial?
I know that I would be interested in reading it. Perspective is one of the most confusing things I have ever tried to understand, with conflicting methods that only work well in limited scopes. I still don’t have a perfect grasp of perspective, but I am trying to understand it.
I have come up with some solutions (reinvented the wheel) to some of these problems, but I am still scratching by head about some things. I am curious to how you present your answers, because I know my explanations are dificult for people to understand.
The list of questions that I had, and which I assume bug other people:
-Locating vanishing points, how do we know where they go? One, two and 3 point systems. Trying to locate the 3rd vanishing point is rather tricky because it moves. This drove me nuts. The closer the object is to the horizon, the more vertical the objects and the farther the 3rd vp gets from the horizon.
-How to actualy draw a square, and a cube. This is my simplest explanation, it explains the location of the 3rd vanishing point, drawing a square and a cube: http://reflective-sentinal.deviantart.com/art/Perspective-Simplified-draw-a-cube-329293520
-Assuming my diagram is correct (which it certainly might not be, or I may be interperating it incorectly), why there is twice the angular distance verticaly than there is horizonataly. I don’t know that it matters, but it … bugs me. I probably have something wrong here somewhere: http://reflective-sentinal.deviantart.com/#/d5g7tg3
More complicated stuff:
-First example: Two 6 sided dice on a table, rotated randomly. The vanishing points on the horizon move according to plan view, while the z axis remains shared. We can compare the size of the dice by looking at the z axis, and transfering the scale measurement between the dice like a 1 point perspective system.
-Second example: Take the same two dice and roll them. As the dice tumble through the air, none of the axes line up, making it rather hard to compare locations of vanishing points or compare sizes of the cubes. I don’t have a way to simplify this one… probably one of those things to throw into a 3d program for practicality, but there should be an answer somewhere.
Thank you for your reply.
I believe that perspective drawing have been mystified by strange methods and inaccurate simplifications. Please try to look at my method with fresh eyes.
This is my method of perspective drawing:
https://docs.google.com/folder/d/0B9re-66sWTmwSjZlT1FFbFRjY3M/edit?docId=0B9re-66sWTmwZ1dMT0plVXVZS28
I use this method to verify my conclusions about how perspective works. The method is too slow to do something more complicated (like the two dices) with and it can’t be a part of a creative process.
The conclusions is a different matter: As you can see, it’s possible to draw perspective without using vanishing points (vp) or a horizon. In real life vp’s are really helpful and you only need to know why they exists to be able to estimate the placement of them (I write “estimate” because sometimes they will be really far away).
If I have managed to explain how perspective works in this PDF properly you will be able to make a conclusion about where you should place the vp’s. I don’t want to give the answer away yet because I want to know how well it works 
You will notice how the viewers eye, the frame (picture plane, paper or canvas) and the object placements, rotation and tilt are equally important.
This is an experiment with the method and may be too messy
https://docs.google.com/file/d/0B9re-66sWTmwc0dvQkNzX2c3b2M/edit
Andrew Loomis book Successful drawing holds a number of interesting conclusions about perspective but are in some cases wrong and confusing. I recommend it non the less until I find (or create) something better.
The 3rd vp will never move. It will behave just like all other vp’s. It does not depend on the objects location relative to the horizon. It depends on the papers location and tilt. I have to assume that we work on a flat surface.
I can’t find any proof in this method and I’m finding it strange.
i find this diagram even stranger… I guess you are trying to figure out how to make a projection that reflects the eye? If you try my method you will find that there is openings to make projections to a curved surface. Out of scope for me.
Try to use my projection method and then try find shortcuts for your needs. It’s not that complicated.
Use two semi-transparent papers. It is possible to let the cubes share the z-axis during the construction and then rotate to paper around the eye-center point for one of the cubes.
It seems like you guys are trying to reinvent the wheel here.
Established linear perspective techniques have been distilled, through the centuries, down to a very efficient science.
Most everything you need to know about linear perspective drawing can be found here (also located above in the Art Tutorial
sticky):
http://handprint.com/HP/WCL/tech10.html#top
I prefer starting out object construction using this method:
opposed to either one of you two guys alternative methods, which seem a bit confusing, atm.
Hurmm, either I have failed to explain my method correctly or you have not read it 
If I reinvent the wheel we have been using a square formed wheel until now. I know my method is correct and I understand it perfectly.
The method you are using is overly simplified and actually incorrect in some areas. It tries to give the reader a step-by-step method and it doesn’t learn you anything about what really happens. You just end up with an OK result. If you try to add a 3rd VP to it it will fail miserably.
This is how the method can be used to reverse an image of a cube created with Blender.
The camera lens was 35mm and sensor size 32x18mm.
https://docs.google.com/file/d/0B9re-66sWTmwR3pKMVhoMGsyWG8/edit
Image: https://docs.google.com/file/d/0B9re-66sWTmwdmJKWTMxOEpnU2M/edit
Sorry for not responding quicker.
None of these methods are exclusive. The method that Quadart prefers is similar to receptors method, and includes a method for finding the vanishing points on the horizon, but lacks a way of linking the true height line to the width (its not a big deal to put it in, but its still missing.) receptors method includes a method to do that. Neither method includes a way to draw a square in perspective without drawing it in plan view or reproducing it off a square that was in plan view. My method solves this problem, and it allows for curvilinear placement of the 3rd vanishing point. Which defines the cube.
Quadart.
Yes we are reinventing the wheel. My pet project has been driving perspective using geometry (lines and circles). With a focus on drawing a single cube. I have been able to reach where I am today by looking at tutorials and examples, then testing them to see if they actually work. Quite often some of the major features do not hold up to testing. Its hard to follow a method if the math is shaky.
The website you linked to is a fantastic if very dense reference. It explains some things very well, but it can be hard to find an explanation of a term without trying to read the entire thing again. I have tried to explain what I understand, and fall into the same trap.
receptor
The geometry of your two methods is identical. You have both chosen to draw different lines, which is why they appear to be different. My method is also geometrically compatible with you method, even if I prefer not to use plan views most of the time because of difficulties translating from the plan view to the perspective system.
The 3rd vanishing point will always move. Here is the geometry that proves it: http://reflective-sentinal.deviantart.com/gallery/33441339#/d4n7ujj Additional example, look up at a tall building, the sides converge, yet when you look at the base of the building (horizon) the sides are parallel. There is always a smooth curve between these two states. Its not usually much of a problem in an image, but it sometimes is.
I define my centre of view by the front most corner of my square, thus that point defines the relationship to the 3rd vanishing point.
The proof for a square and a cube:
Its a graphic geometric proof. The only instrument you need to check the angles is a 45-45-90 triangle.
The 2 vanishing points on the horizon are the intersections with the circle. Draw a line from one of the VP to the circle then to the other VP. Measure the angle at the circle, and measure the angle to the opposing nadir/zenith (noon or 6 o’clock position position). Then try a different point on the circle and see if your answers change. Edit the diagonals of all the squares will point back to where the 45 degree angle crosses the horizon.end edit
After locating the 3rd vanishing point, you then need to find the square on the remaining 2 sides of the cube.
Regarding the angular measurements of vision. Its a pretty simple diagram, but I am pretty confused about it. As I understand it, the only time I will ever see a truly 90 degree angle in perspective is when I am perpendicular to it. Thus the 90 degree angle at the top of the circle must be above me, and 90 degrees from the horizon. But I know that the distance between the vanishing points on the horizon is also 90 degrees. Thus I am confused: radius=90 degrees and 2 x radius=90 degrees. The math seems to be broken but the geometry works very well. I don’t worry about it too much.
Re 2 dice tumbling. I think its a bit more complicated than that, although that would look close enough.
receptor–As you should know, the diagram I presented is a 2-pt perspective scheme and adding a nadir or zenith point to it is out of the question without proper modification, though it wouldn’t change much. I used the 2-pt vp scheme to coincide with your 2 pt Perspective drawing diagram (which is self explanatory and actually seems overly simplistic on first view), as the line of site is perpendicular to the vertical edges of your cube and parallel to the ground plane.
I learned linear perspective drawing years before the computer and internet and doing technical (complex machinery) or architectural illustrations required hand drawing using drafting implements and an in-depth knowledge of perspective. You say the overly simple (yet the initial phase of the process) introduces error in the diagram I showed. Im curious as to what you think those errors are, outside of the fact that linear perspective (planar projection) introduces distortion by its own geometric nature.
Your view-1 diagrams got me scratching my head.
I still say your creating a square wheel, until proven otherwise. 
Im also curious as to whether youve ever hand drawn a complex, multi-vp scene, with shadows using your method.
DanielBrown: Sorry, I can’t make sens of the methods you are using. It would be of great help if you clarify how the logic of this really works. I need to see how you’r setting the distance between the eye, paper and object. Start a separate threads about the different problems you are dealing with. This is getting confusing and I’m feeling sorry for the other readers as well as me
Yes, those two setups gives the same results and the problems encountered in my first example are solvable by the method you present here. It does have the station point and the paper plane. With proper understanding you can get far with it. I guess that you also would be able to find the extra vp’s if you tilt the object. I would underline the importance of the placement of the station point relative to the beholder eye. It makes it easier to understand the planar projections distortions and why things looks wired when feeding it with extreme parameters.
The method is perfect for doing fast visualizations of mechanical drawings. When I was young i did a lot of these types of drawings just for fun. I have been using this method before but it didn’t give me the freedom I needed to be able to make drawings from odd angles. I became fooled by the sets of rules that 2 point perspective uses and tried to apply them when looking up or down. I guess that I’m a bit hostile to 2 point perspective since then But I agree that the diagram is correct… but limited. This method is one of the useful out there.
Which diagram and where is it confusing? I will happily try to explain. You may actually have found one of the errors I made;-)
It makes no sense to make large complex projection drawings since the computer came to our aid
When i freehand draw I use all possible shortcuts I know of but now I understand the underlying meaning of them and it helps hugely.
So my point is more that it may help people understand how perspective works and what’s important.
The link you provided is full with facts about perspective. But it’s also intimidating in its denseness and mass.
I don’t know how other people works but I don’t get insights by huge amount of information. I get insights by experimentation and reflections of facts. I believe my method to be a flexible, trustworthy and understandable base for experiments. It doesn’t make any simplification and lets the user verify on multiple points. I have no intention of it being efficient.
I have great respect for your knowledge and if you comment more about it I would be happy. I really need an extra pair of eyes on it.
Sorry, I know I’ve been a bit magniloquent about ‘my method’ but it is my baby
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