Before the discovery of linear perspective as an artistic technique, artists struggled for centuries with the difficult task of representing the illusion of depth in their paintings and sketches. You can clearly see from some medieval paintings (and earlier) that their best methods of the time were far from adequate. The artist and architect Brunelleschi first demonstrated its principles, architect and writer Leon Battista Alberi was the first to write down the rules, but by and large, it wasn't until the great Leonardo Da Vinci, who probably learned from Alberi, did the first poplar drawings and paintings with a clear horizon and vanishing point that the secret of representing depth accurately was finally unlocked. It was such a simple idea that it's a wonder that it wasn't thought of earlier (like a lot of things in this world).
There are many different forms of the perspective technique, each (except for one) differs by the number of points used. And as you'll see, the more points you add, the more complicated - and bizarre - the drawings become.
One Point Perspective
One point perspective is the simplest kind of perspective. Typically this is first taught to young art students in late elementary or early middle or junior high school (roughly fifth to seventh grade). It involves drawing a straight horizontal line across roughly the middle of the paper and placing a dot at around the center of the line.
The dot is the vanishing point. And the horizontal line is, well, the horizon. Typically the first thing a young artist is taught is to draw two angled lines and draw them back to the vanishing point to create a road going off into the distance.
Another thing the young artist is taught to do is to maybe add some cacti: the ones closer to the vanishing point would be smaller and the ones further away would be bigger. (A road going off into the desert seems to be a common theme for these early perspective projects, probably because the scenery is simplified which allows for greater focus on the main learning task.)
Buildings are also added to the side of the road as the lesson advances, showing how this technique works with different shapes. Here to simplify the ASCII art here we're just going to present the building.
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Hopefully this crude representation shows how you make a building, or box, in perspective. It involves drawing a rectangle and then drawing a line (most likely using a ruler or straight edge of some sort for accuracy, unless you have an amazingly steady hand) from each corner on one side back to the vanishing point. These lines are drawn very softly so they can easily be erased because they would just be orthogonal lines, or guidelines to just help with rendering the rest of the shape.
Two Point Perspective
Two point perspective involves having two vanishing points on the horizon. The perspective shifts in regards to how the viewer is looking at the shape, i.e. its corner in the case of a cube (building or a box). You draw one straight vertical line and then draw four guidelines going back to the vanishing point, two on each side from each end of the line.
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From there the rest of the shape is defined, the sidelines equidistant from each other if the viewer is looking straight at the corner, or not if you want to shift the perspective slightly to the left or to the right. Two point perspective is usually first introduced to art students at about seventh or eighth grade.
What if you're drawing a pyramid? Well, you'd simply draw two angled lines to define the rest of the shape in addition to taking part of the bottom guidelines to render the bottom of the pyramid. But how do you determine the angle of the angled lines? What are they pointing to? Sometimes you might need to add another perspective point.
Three Point Perspective
In three point perspective things begin to get a little strange. This is where you add a point either above or below the horizon. Besides the need to add a point in the case of pyramids (sometimes) usually the point (ha ha) of adding another point is to illustrate the viewer looking at the scene either from a point above or below the horizon in addition to looking at or being in a corner. Think of yourself looking down at a scene from being up at the ceiling in a corner. Or you're an ant looking up at a room from a corner at the floor. This can be quite fun to draw but a three point perspective lesson is almost always reserved for advanced students in high school (ninth or tenth grade). This is because the concept of it can be quite difficult to grasp for younger students as it involves a lot more thought into the concept of perspective as a whole - how simple shifts in the angle of viewing can alter the entire scene.
As to drawing it, add the aforementioned point either above or below the horizon line in addition to the points on the right and the left. Draw the vertical line. Now there are six guidelines to be drawn: the four in the example above with two added on either side drawing down or up to the third point.
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Because of my limitations of character usage in terms of ASCII art I cannot give you a better example here. The drawing of the shape above should be thought of as incomplete with some lines needing to be drawn but you should get the idea. If not, see this.
Four Point Perspective
Here is where things get funky. This is something that I hadn't even thought was possible when I was learning three point. Usually (and I say "usually" because here is where you step out of practical applications of perspective and into art theory, i.e. you can really add the fourth point anywhere you want) this involves adding a fourth point, the result having a point on the right, left, bottom, and top of the picture. In the case of drawing a building two more guidelines are added and the tops and bottom lines of either side of the building go back to points on the right and left and on the bottom and top.
The best way to verbally describe the effect of four point perspective - and the angle of it - is that not only are you looking at the corner of a building, but you are zooming straight towards it. At perhaps a high rate of speed. Or, the viewer is looking at a very tall building and the top of it is above eye level and the bottom is below eye level. Sound weird? It is. I'm not even going to attempt ASCII art with this one. Check this out.
Normally lessons on four point perspective aren't covered until the advanced high school art classes, perhaps not until students are seniors.
Five Point Perspective
Now for even more funkiness. Now you add a fifth point in the center. Everything looks like you've put a transparent sphere in front of your eyes. Five-point perspective drawings create 180 degrees of visual space around the viewer. Check this out. By the way, I had never heard a lecture on anything beyond four point perspective until I was a freshman and sophomore in college.
Six Point Perspective
Didn't think it was possible did you? The sixth point is added behind the viewer. Yes, I said behind. To draw this you must turn the paper over (it helps if it's at least somewhat see-through) and draw what would be behind the viewer and thusly you've completed a 360-degree drawing.
Linear Perspective Beyond Six Point - or, Adding Points Willy Nilly
And here you get from weird and funky to bizarre - and even more into art theory. You can actually have a drawing with seven points, or nine, or ten, or one thousand, the more points you have, the more you get a drawing from Dr. Seuss' worst nightmare. The most benign example (and we're going to use my favorite subject again, a building) is let's say you have an architect who likes to partake in illicit substances while designing his buildings and he or she decides to make all the window overhangs off kilter, all wrong in precisely the same manner. If this building was actually built then drawing it would require adding another perspective point in addition to the one or two or three or four you already have.
If you can imagine lots of off kilter, or broken overhangs or trimming a point would need to be added for each example. And you can add these willy nilly points to two point, three point, four point - etc. - drawings. Which means you can have a four or six point perspective drawing that doesn't look like the examples above.
This kind of perspective is totally different from all the point perspectives and applies to paintings only (if you are merely sketching shapes this won't have much relevance, if any). The only aspect it shares with the other is it is indeed a very simple concept that we all know to be true even if we don't consciously think about it. In other words, a young artist might struggle for years trying to figure out how to make his or her mountain scene he or she is painting more realistic and the young artist might get frustrated, knowing that there is something that's missing, some simple technique that would solve the problem, it's at the back of the mind. This might often be atmospheric perspective.
You ever notice how the further away things are, as you're looking far off into the distance, the more muted they appear? The more grey, or even blueish, the hills become? This is because the further away something is, the more molecules of air there are in between you and the object that diffuses the light (and therefore color) of the object. The dirtier the air the more this effect is pronounced, but no matter how clean the air is there are still those annoying oxygen and hydrogen (and other gases in normal air) molecules to contend with, making things more difficult to see the further away they are.
To make a nature scene (or really any scene) more realistic, a painter must employ this technique. Things Further Away should be muted more and more, depending on how far away from the viewer they're supposed to be. Colors should be greyed and lightened (adding a touch of blue might help, too - probably because the sky is blue and it is the most abundant color). Young painters usually aren't taught this technique until early-to-mid high school.
Atmospheric perspective is not a technique the late great painter Bob Ross often employed, if I may offer up an example of how really good paintings can be made even better. He rarely purposely muted things further away from the viewer, like mountains for example, one of his favorite subjects. Most often his technique was not to mute further off objects, but to put things in the extreme foreground in a very dark color, like his "happy little trees." Or his jolly little paths. Or some other positive emotion on objects or vegetation. Just think how much better those Bob Ross works could have been with more employment of atmospheric perspective.
No matter which perspective technique an artist uses the ultimate goal of each one is to trick the viewer with the illusion of depth. The more skilled the artist the more this deception is achieved. Also, the more skilled the artist, the more intuitive perspective becomes. Advanced artists might think it's too much trouble to bother to draw out the lines and points for the point perspectives. Some prefer to use them regardless of skill level in the interest of exact accuracy. Whether to utilize the techniques or not is up to the artist.
It's all a matter of perspective.