Notes about linear perspective drawing

I think that perspective is a useful tool for understanding. It may help to accept some shapes and angles that may seem strange. On the other hand, I have seen drawings that abuses the perspective rules so much that it is painful. Everything must be based on observation and reflection on what you see. If it looks wrong, it probably is wrong.

There is no reason to get obsessive about perspective drawing. It should be treated like a tool of understanding. It must not be an obstacle to creativity, rhythm, signature or beauty.
Knowledge is easy to learn. Intuition is harder and must be based on knowledge and observations. There should be no magic to it, no mystic rites.
Every method of projecting three dimensions on to two dimensions are inherently problematic. There really is no perfect way. In our culture, we have standardized linear perspective as the most accepted method and may find other methods strange and hard to read.
Linear perspective drawing is like solving a mathematical problem with lines and angles. It's a way of squashing three dimensions onto a two dimensional surface.

Handprint is a good source to learn more.

Learn and let go.

This is the standard method of perspective drawing when you need to measure distance in 3D-space. It is mentioned in Drawing for all it's worth by Andrew Loomis It is a simple setup and easy to use. It is however difficult to use when you need 3 points and it requires you to place vanishing points. The blue circle is the Cone of Vision.

This is a developed method where you can pin-point the coordinates for each point. It can be quite complicated to use and it requires a layout of the object in two axis.  It is a combination of two simple setups, one from above and one from the side. 

This is a step-by-step on the full method. Some details are left as an exercise for the reader ;-)

If a plane is parallel with another, they will share the same horizon. I'm using the term horizon loosely since it should be known that it's a concept. In the real world, most man made objects will have planes sharing the same horizon as the ground plane, such as table tops, boxes on tabletops, houses etc.

Here is an interesting consequence of a right angled box and it's vanishing points. The opposite horizon will be perpendicular to a line from the VP, passing thru the center of the image. This can be useful if you want to be a bit more arbitrary about the placement of the vanishing points but you still want som help placing the last one.

One thing you should know about is how the placement of the third point will affect the Cone of Vision. As you can see it will get smaller when you place the third point closer. If you want the perspective to be sane, you will need to keep the VPs further away as you look down. 

Here is a sequence about how to place the third point and how to get the Cone of Vision's circle. 

The three Vanishing Points of a right cube can be visualized like the corner of a gigantic aligned version of the same cube. The corner point is your eye. I'm not to happy about this description and I hope you don't get too confused about it. It will however help you to confirm your grasp when you get it.

This is the corner unfolded to show how it can be used to measure angles.

This is a method to tilt an established perspective with 10° and 45°. Notice that the two top VPs will get further away form each other until they will be infinite far away at a tilt of 90°. That would make those edges parallel with the paper plane and that would be a one point perspective.