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I have no formal technical drawing background, and I was just wondering if there was a mathematical way of determining the correct length of lines in 3D perspective, e.g., if I want to draw a cube in relation to a vanishing point, how can I get the receding lines to be the correct length? Judging by eye gets me close but I need to learn this precisely.
I assume this is fairly straightforward (for architecture, etc) but have been unable to find the answer anywhere.
I tend to first and always recommend the Francis D.K. Ching series of books, which were pivotal in my own technical drawing and later architectural education - I think the two volumes most relevant to this discussion are Design Drawing and Architectural Graphics, which I recall having a pretty in depth chapter on constructing perspectives correctly. I loved all the Francis D.K. Ching books enough that I re-purchased most of them years layer as resource books, and still have most of them to this day.
In Architectural illustration, perspective construction is not just about vanishing points, but a number of specific methods for building highly accurate perspective representations of designs whose plan, section and elevations are already worked out to a pretty significant degree, so the perspectives have to match and be dimensionally consistent with those.
Here are a few images from the Francis D.K. Ching book Architectural Graphics so you can see what I mean about why this is a phenomenal resource book, not just a textbook:
Constructing measured perspective:
Measured diagonals and horizontals:
VP for angled lines & shadows:
Paraline example of shadowcasting (various forms):
Measured shadowcasting plan to elevation:
Shadowcasting in perspective 01:
Shadowcasting in perspective 02:
Shadowcasting in perspective 03:
I think those are the techniques you are looking for.
The Francis D.K. Ching books are all available as e-books, both Kindle and E-Pub, and so can be used as ongoing references on your tablet as you apply the techniques.
If the Francis D.K. Ching books are too old-school for you, (1970's vintage, but revised every years since, as they're so widely used in architectural education) there's a more recently written book (loosely associated with ArtStation) called How to Draw by Scott Robertson which touches on many of the same techniques, but with less laser-focus on architecture - it's aimed at folks wanting to break into art direction, video game design and so on. Here's an image as example of Scott's instruction on perspective construction:
Constructing a basic perspective grid:
I am also a big fan (once you have some of the basic perspective illustrative techniques) of the Micheal E Doyle book "Colour Drawing" to learn how to take your illustrations to the next level - this is another textbook from my architecural education which I loved enough to buy as an ongoing resource book: here are a few images to show why:
Layout and perspective construction of vignette:
Basic shadowcasting and occlusion of vignette:
Exterior vignette with atmospheric perspective:
Interior perspective with carefully applied colour warmth / distance emphasis:
Exterior perspective using colour warmth to invite viewer into space:
Hope this helps.
Before going to advanced perspective drawing techniques you should learn the most basic thing: how to construct an image from rectangular projections with lines of sight. I believe user joojaa refers just this when he commented "with a ruler". You have a top wiew and right or left side view and there are marked the station point(=the observer) and the image plane.
It's used here:
Vanishing points and other drawing shortcuts such as crossing diagonals for a centerpoints are mathematical consequences. They are useful in practical drawing, but the basic construction with lines of sight doesn't need them.
Advanced drawing guides especially try to learn how to use those shortcuts to be able to draw faster but still have consistent perspective. An artist can well start from the vanishing points for a certain artistic impression. Some 2D drawing programs have perspective grid. The user sets it up for the wanted impression.
Yes, there is. The "mathematics" involved with perspective is called Projective Geometry. The formulas are solved by diagrams as with most all geometry problems.
There are different kinds of 3D perspectives such as isometric and orthographic, etc.
Search key terms such as "3D projections," "vanishing point," "one-point perspective," "two-point perspective" for precise diagrams rather than mathematical and/or geometric formulas.
Iso Perspective sketch pads are also commercially available with light guide lines to help you align things in your composition.
Valuable Beginners Tip.
Here's a little tip that quite a few people neglect to point out. The "horizon" is always horizontal and it corresponds to your eye level. Vanishing points are located at/on the "horizon." If anything appears above the "horizon," it is above your eye level. You are shorter than it is. If anything appears below the "horizon," it is below your eye level. You are taller than it is. You can't see the top of any flat object that appears above your eye level—it is also above the "horizon."
