The measurable construct of perspective was developed in the 1500s by Brunelleschi. It is an incredibly useful geometric construct, but using it well doesn't rely on using it perfectly. Before Brunelleschi established the exact mathematical principles of perspective, artists were relying simply on observation to convey depth in their paintings.
Architectural Graphics by Francis Ching explains in detail how to construct perspective to render three dimensional objects or spaces. It is a great resource, but it's easy to get bogged down in the technicalities. In contrast, this narrative is intended to provide a general understanding of perspective that can be applied in the field.
As you look at objects or spaces, take a moment to consider what visual cues are conveying a sense of depth. In general, you will be able to observe four major cues: Overlapping objects, diminishing sizes, foreshortening, and convergence of parallel lines. These are the same cues you can use to enhance your drawing.
The most obvious cue for spatial depth is when objects in the foreground overlap (and block a portion) of objects in the background, as illustrated below. The person is in front of the tree, which is in front of the foothills, which are in front of the mountain:
You will also note that objects which are closer to you look bigger than the same sized objects that are farther from you. In the diagram below, the only difference in the people is their size. The larger people appear much closer, than the smaller people:
Foreshortening is the minimizing of a visible surface as it tips away from you. Let’ say your friend is waving to you. Consider how her hand looks: you can see the full palm and all five fingers all at once - like the hand that you would trace in kindergarten to make a rooster.
Now, imagine that your friend aims her arm and fingers directly at you. Her hand no longer resembles poultry. Instead, it will look like five little circles.
In this second scenario, her palm of her hand has been foreshortened. This visual phenomenon happens all around us. The closer a surface aligns with the axis of your view, the more foreshortened it will appear.
Now try this effect with a piece of paper. After holding it vertically in front of you, lay it flat. If you get it lined up with your line of sight line, you will only see a thin horizontal line for the paper. The take away is that the more the surface tips away from you, the shorter it looks on the page. Eventually, the piece of paper becomes just a line:
An amazing thing about living out West is driving across enormous valleys between distant mountains. Recently I drove through the Utah's West Desert on the Loneliest Highway in America.
Although the highway was the same width at my location as it was twenty miles away, the parallel lines appeared to converge to one point, in a long "A" shape. But, the edges of the highway don’t actually converge - this is just a construct our brains to understand depth of space. Convergence of parallel lines toward one point is called one point perspective:
You can also see parallel lines converge with one point perspective in the room you are sitting in:
The sketch below of a courtyard within the Blue Mosque is constructed from "one point perspective," which is generally the way we see spaces:
But converging lines are more universal than one point perspective. Objects (blocks, toys or buildings) tend to have two point perspective as shown below:
The sketch below of Jan Van Eyck plaza in Brugge illustrates how buildings can be represented with two point perspective.
And when you mix spaces with objects, you can observe multi-point perspective.
The sketch below of a walkway and buildings in Barcelona is constructed from multi-point perspective:
The easiest way to understand one, two, and multi-point perspective is to understand where the lines converge. The most common place for lines to converge is the horizon.
Finding the horizon can be obvious if you are in a vast landscape or looking out at the ocean, but if you are in an urban setting, the horizon is difficult to see. To find the horizon, I use this exercise: I imagine the entire world filling up with water – right up to my eyeballs.
Every time you start a sketch, take the time to imagine filling your space with water to help visualize the horizon line. Once your environment is flooded, take note of what is under water, and what is above water. And what is close to the surface, or far from the surface?
You are looking up at objects which are "above the water", and looking down at objects which are "below the water level."
The flatter the lines appear in your subject (and composition), the closer they are to the horizon. The steeper the lines appear, the farther (above or below) they are from the horizon.
Once you have identified the horizon you can adjust where the horizon line occurs on your page. Is it high on the page with lots of information below it? Or is it low on the page with lots of information above it.
If you place the horizon line low on your composition, then most of your lines will tend to slope down toward the horizon line:
If you place the horizon line high on your composition, then most of the lines will slope up toward the horizon:
To summarize: Perspective isn't reality. It is just the way our brain converts information from our eyes. You don't have to be a slave to it - it is simply a tool for reference. You can chose to use it as accurately or as loosely as you please.
Remember that many, many artists were conveying space and depth and space without any understanding of "perspective."
Spend time observing and understanding your subject before your put pen to paper. Keep in mind that sketching is a way of seeing and understanding. And never be concerned about whether or not you know how to draw this or that, if you already knew how to draw your subject then you wouldn't be sitting there trying to sketch it.
Finally, it is important to remember that your sketch will be full of little mistakes - and these are the details that will give your sketch character and personality.