Formula for the location of vanishing points

@burvantill tan is short for "tangent," which is a trigonometric function. It has a different meaning in this context than in the context of composition, where tangent means things that are touching on the edge. Scientific calculators should have a "tan" button and you can even find a tangent right in the Google search bar. Try typing "tan(45)" into Google and it will bring up a little calculator that tells you the answer. The number you put in the tan() parentheses is an angle. In the vanishing point formula, the angle you use is half of the field of view. You have to make sure the calculator is set to degrees if you are entering the angle in degrees. In the Google calculator there is a slider that lets you switch from radians to degrees.
I know using the formula is complicated : ) That's why everyone just eyeballs it. This is more than you need to know, but the tangent of an angle is equal to, when the angle is one of the angles in a right triangle, the length of the side of the triangle opposite from the angle divided by the length of the side of the triangle adjacent to the angle.

@smceccarelli Good idea! Having an accurate grid would make it easy to place things where you want in the picture. I think what you're asking is if you want to draw a perfect square in perspective, and you draw one side of the square (you can arbitrarily decide how long to make the first side since you are choosing the size of the square), then how do you know how long to make the next side of the square so that it actually is a square and not a rectangle? That is a good question and I'm sure there is a way to figure that out. I'll think about it. Once you have one perfect square drawn you should be able to expand it to a grid of any size you want by drawing lines through the midpoint of the sides like Jake does with telephone poles or fence posts in the perspective course.

@dlarmantrout Exactly! The perfect square is the key to correct perspective in line with the chosen field of view and normally I wing it. The key is the position of the diagonal vanishing point (DVP), the vanishing point of the diagonal of the square (or the 45° lines).
Here is a scan from "Vanishing Point"  an excellent book on perspective by Jason CheesemanMeyer, that talks about the relation between fieldofview and DVP. It's quite obvious with onepoint perspective, but not so trivial with 2point perspective (I'm not sure I ever learnt how to do it, and if I did I forgot...)Given that you have a formula for the second vanishing point (i.e. the 90° line to the first), it should be relatively easy to work out a formula for the vanishing point of the 45° line)...

@burvantill I'd be lucky to get C's in math! In my world, that actually qualifies you as a Math Whiz.

My head just exploded.
Thank you for answering my question. I'm going to have to reread this many times, but I like learning new things so I'm going to try it out. I'll need to get one of those doohickies, a protractor??

@eli LOL! Thanx! My son would disagree with you. He hates it when I help him with his math homework, because he always seems to know more and ends up teaching me. Lol! If I master this technique it will impress the hell out of him.

@burvantill Don´t worry  this is a disease, not an asset. It´s called “incurable nerdyness”. It keeps your mind from just relaxing and doing great art, is a big enemy of instinct and innate skills (which it fundamentally mistrusts) and I’m pretty sure it´s responsible for the fact that I have such a hard time writing fiction...
Believe me, you don’t need maths to do great perspective drawings! It´s just the peculiar way people who have this disease go about things...

Thank you

I'm also one of those "not so great at math but still nerdy enough to want to try this" so my question is: in your formula where w = width of paper, and the v's are the vanishing points, what is your measurement unit (metric?) or can you use any unit as long as you are consistent? Or have I so completely misunderstood the formula that my question doesn't even make sense?

@smceccarelli Yep, this was all due to my incurable nerdyness making me want to find the exact right answer. If I were teaching a class on drawing in perspective and someone asked how do know where to put the vanishing points, I would probably say... "There is a way to figure out the exact spot, but TRUST ME, you would rather just do it by feel." With that said, I am trying to a work out a formula for the diagonal vanishing point. And then maybe the 3rd vanishing point for 3point perspective.

@demotlj You can use any units as long as you are consistent. The w is the width of the paper and v1 and v2 are the distance from the center point of the horizon line to each vanishing point (v1 on the left and v2 on the right)