How to measure angles: the guess-check cycle

The best way to draw an angle from life is this Guess-Check cycle:

1-Guess and draw
2-Check and correct
3-go back to 1 if needed

There are many ways to transfer the angle directly, and I'll teach you how, but you should avoid it because your purpose will be to eventually train yourself not to even need measuring. The guess-check cycle reinforces your training each time that you measure an angle. It gives you instant feedback and correction (a pleasant feeling if you got it, a bit of work and frustration if not). That is the best way to learn. By using a checking method that is accurate and reliable you'll develop a guessing ability that is accurate and reliable.

First we'll discuss guessing, then we'll discuss a couple of methods of checking, and also how to simply transfer angles without guessing, but really, guessing is better (and much faster!) than transfer in the long run, so do make an effort to guess.

Part 1: Guessing.

Measuring the angle of the line that connects the shoulder joint to the wrist joint

So, you have a straight line in front of you, that you want to copy. Say it is an imaginary line joining two points on that handsome model on the picture above: say the line joining the wrist to the shoulder. Now, close one eye, take your pencil and raise it to the line you want to copy, aligning it with the two extreme points, in this case the points being the centers of the balls of the wrist and shoulder. (tip: actually, rather than aligning it on top of the points it is usually better to make it parallel to the line that joins them but giving it a slight vertical displacement so that you can still see the points - if you cover them you don't see what you are doing anymore - the picture above shows that displacement, the pencil is a bit below the line being measured). Very important: keep that pencil on a vertical plane perpendicular to your line of sight, that is, don't let it point away from you (you'll have that tendency when trying to copy perspective lines -fight it!)

Now your pencil and the vertical define and angle. Observe that angle in front of you, put it in your visual memory and draw it on your drawing surface. Now check it.

Part 2: Checking

Few books teach you how to measure angles, and those that do are wonderfully vague. Not having an accurate way of checking angles makes triangulation unreliable, so you have to have confidence in your measuring. Over time I have developed many methods - using one or two pencils - to make angle measuring reliable. Here are a few; later we'll talk about some advanced ones, for very acute angles, and some tricks envolving muscle memory.Anyway, here are a few basic grips I came up with.

A) The vertical grip:

Vertical grip

This is a trivial one and only works if drawing on a vertical surface (say, an easel or a small pad you can raise easily when checking angles), but it is very good and fast in that situation. Just raise your pencil again to the line on the model, as you did in the guessing phase, and slide it across the vertical plane, back to your drawing pad, without disturbing the angle. It is pretty easy to do if your pad is vertical. Just use yor shoulder to move, elbow and wrist held fixed. At first you can even keep your little finger on the pad while you measure and then slide across the pad with the little finger serving as a tactile reference of your distance to the imaginary vertical plane.

Transfering the angle without guessing:

If you don't feel like guessing today: you can use the vertical grip to simply transfer the angle, like this: Place a point where you want the line to start  on your pad. Measure the angle and slide the pencil to your pad. Align the pencil (without changing the angle, just move horizontally and vertically as needed) with the dot you made, and let the tip of the pencil touch the paper to make another dot. Join the dots, they will define a line with the corect slope. For safety, check again anyway. Or, you can just trace the lign directly by sliding the pencil along its own length. Again, check it afterwards.

B) The Cross-Grip (works on any drawing surface, not just vertical)

Cross-grip

Checking with the Cross-Grip

I call this one cross-grip: Take two pencils, one crossed over the other. The thumb presses from one side and the soft part of the tips of the other fingers press from the other side, resulting on a firm grip (round pencils are tougher at first - use pencils that have flattened sides). You will find (after dropping some pencils :)) that by moving the thumb you can make those two pencils change angle as you please (if you find this hard then at first you can help the placement with the other hand). Now call the pencil against the thumb the measuring pencil and the pencil that lies against the other fingers the reference pencil. Keep the reference pencil vertical and make the measuring pencil parallel to the line you want to measure. Now the two pencils describe the angle you want and their position is firm in your hand; you can transport that angle to your horizontal, vertical, or tilted drawing surface (or even a surface across the street!) without losing that angle. Now to check your guess, just align the reference pencil with the vertical of your drawing pad, and the measuring pencil should align with your guess. If not, observe the error, correct your guess and repeat.

Transfering angles without guessing, by using the cross-grip:

If you don't feel like guessing today (shame on you) you can just transfer the angle directly: bring the two pencils to rest on your pad. The pencil closer to the pad is the measuring pencil (no accident: that's why I told you to hold the reference pencil against the thumb and not the other way around), so just place it againt the pad (check again that the reference pencil is correctly aligned with the vertical of the pad) and hold it there. Take your reference pencil on your other hand and draw the line by using the measuring pencil as a ruler. Or just draw two points of that line if you don't want to mess up your drawing. Voila, you transfered your angle.

C) The compass grip

Compass grip

In this grip, you use two pencils as a compass (not a magnetic compass, a circle drawing compass!). Now, this grip works in a different way from the others, so pay attention: You take two pencils. Now one pencil - call it the reference pencil- must remain vertical and align with one point of the line you are measuring (for example the wrist on our model) and the other pencil -measuring pencil- tilts in such a way as to align with another point of the line (say, the shoulder of the model). Not that no pencil is actually aligned with the line whose angle you want to measure! No matter.

Now bring the pencils to your drawing and align the reference pencil with the vertical of the pad and make it touch the line you guessed previously. Then the measuring pencil must also touch that line. If not, correct and check again.

Transfering without guessing: this is a very good grip for direct transfer. The fact that you align the pencil tips with the line on the model gives you greater precision. It is also an easy grip to hold and keep stable. Also, very easy to mark the paper: after measuring the angle just bring the pencils to the pad, align the vertical and place the vertical pencil tip where you want the first point to be. Now let both pencils touch the pad simultaneously, and you have two points that, when joined, describe the correct angle.



Coming soon: The chop-stick grip, the double-vertical grip, and the twins grip. (not kidding)

Why measuring angles is a good idea

Why should you measure angles instead of distances or proportions? Here are a couple of common methods of taking measurements for a life drawing. We'll discuss their properties and see why measuring angles instead might be a good idea.

1) Proportional measuring.

The most common method. First choose some distance as a unit, say, the size of the head. Then measure how many units there are from, say, the head to the navel. As all students have found, even with considerable practice this doesn't work all that well. Some problems are:

a) All measurements are only precise up to about half the size of the unit you chose. Every measurement will be "oh, a little bit more than 2 heads", or "about 3 and a half heads". A little bit more here and a little bit less there and soon nothing fits together.

b) As Robert Fawcett (a proponent of sight-size) pointed out, you don't want to do algebra in your head while drawing. You want to use your eyes and compare what you draw to what you see. Since you have to make a conversion of your unit in order to fit the drawing on the paper, you are constantly doing distracting translating work: "So, it's (about) three units...and a bit more than a half... on the model, and each unit on my paper is the size of this reference here, so I'll draw this reference size three times...(and about a half and a bit more)"

c) This more or less works when the measuring is on the same (vertical or horizontal) axis. But few things will be on that axis. Either you will have to measure angles anyway and distances along those angles (and , then why measure distances at all, since, as we shall see, angles are enough) or you will effectively have to draw in an imaginary grid, in order to relate horizontal to vertical distances. Failure to do this properly results in squashed up or elongated figures (a commonly observed problem).

2) Sight-Size.

In the sight-size method you set things up so that the model looks the same size as the picture, so all you have to do is compare by eyesight and make alignments. Sightsize solves most of the problems of the proportional method. One common objection to sight-size is the fact that most setups look cumbersome and slow (and boring!) to use, but that is an accidental feature of a certain atelier culture usually associated with sightsize. Such cumbersome setups are a type of (very accurate) sightsize, but you can use very useful and light setups for sightsize, the choice is yours. There still remains a real objection, though:

a) Sight-size is nice...when you can have it! Which outside of the studio isn't always the case. You have to place the paper vertically, place the subject at the right distance for the size of the drawing you want. Sometimes that is just impossible or requires you to do a lot of walking (or driving) to get closer or further away from a big target (and then you lost the spot you wanted) or if you want to draw a head life-size you have to place the drawing next to it and walk back and forth all day long to make changes and spot differences. You can always spot a sight-sizer by the tracks he leaves on the carpet. Sorry, not for me. Sight-size substitutes a lot of thought-work for a lot of footwork. :) I want to draw from whatever spot I choose, I want to draw spontaneously, in any scale, with no complicated setups, and with accuracy.

3) ANGLES: Here is the beautiful thing about angles:

a) Angles are always sightsize! From any distance! This is beacause angles are invariant for a change of scale.
Take a rectangle, for instance. Suppose the angle along the diagonal is 80 degrees (a tall rectangle, maybe a building seen head-on). Now move away from it, or move closer. The rectangle will look twice as big, or twice as small, or two thirds ("and a bit!") bigger, but what happens to that angle? It stays the same! Angles are always sight-size! All you have to do, at any scale you choose to draw, is check visually if your angles are equal to the ones on the model. You get the advantages of sight-size mehtods but you can stand at any spot!

b) No need to relate measurements in two different axis as a special problem. Just check angles to the vertical and obtain the distance measurements automatically. (How? Triangulation does it. With a simple drawing method, not by calculation. We'll see how ahead)

c) Measuring angles is easy to do (after some required training), requires no special apparatus, and no special limiting set-up. This is the hardest part, actually, but there are many tricks that I've developed over time and that I'll try to teach in this series of posts, after dealing with triangulation itself. Some of the ways of measuring angles can be used to draw on a vertical plane, some can be used to draw on your lap (great for drawing impromptu on the field and so on) and some are memory/observation based, not requiring any visible act of measurement (great for drawing without calling attention to yourself - you don't want to raise an arm to take measurement when drawing someone on the subway!)

d) Measuring angles does not have the "and a bit more" problem of the proportional method. The error you make in each angle measurement is not a fraction of some arbitrary unit of measurement taken from the model, it is a fixed error that only depends on your ability at angle measurement - so, it should become smaller with training, and has no arbitrary, model-dependent lower bound. Judicious choosing of the angles to measure should improve it even more (as surveyors well know).

What is triangulation

Triangulation is based on the side-angle-angle theorem: Knowing one side and two angles of a triangle determines the triangle completely (in practice, you can calculate the other two sides and the remaining angle).

This has numerous applications, for example in surveying. Take this illustration of Liu Hui's (3rd century chinese mathematician) problem of the island at sea. Suppose there is a mountain that is inaccessible to you (say, it's on an island) and you want to know its height. How do you go about it? I'll quote from Liu Hui:

Q:Now surveying a sea island, set up two three zhang poles at one thousand steps apart, let the two poles and the island in a straight line. Step back from the front post 123 steps, with eye on ground level, the tip of the pole is on a straight line with the peak of island. Step back 127 steps from the rear pole, eye on ground level also aligns with the tip of pole and tip of island. What is the height of the island, and what is the distance to the pole ?

By aligning the mountain top with the top of the pole, the surveyor determines the angle of the triangle in the picture at the vertex he is standing on (internal or external angle, either one determines the other). He does this for both vertices. Also, he knows the size of the base of the triangle, because he walked one thousand paces from one vertex of the base to the other (and his feet still hurt). So, the side-angle-angle theorem says that you can calculate all other sides and angles, and, in particular, you can know the height of the triangle, and that equals the height of the mountain.

The actual details of the calculation are not important for our purposes, because we will not be using triangulation to calculate anything, just to locate points by a mechanical and intuitive process that uses triangulation implicitly.

Oh, by the way, quoting Liu Hui again:

A: The height of the island is four li and 55 steps, and it is 120 li and 50 steps from the pole.

Hope that puts your mind at rest :). Next we'll see how to apply this to life drawing.

Berlengas by triangles

An example of a picture drawn from life wholly through triangulation - meaning that no distances at all were measured: only angles.

Note: For those in the know, this was made on the Berlenga islands (on the main island, from the old fortification), off the coast of Portugal. That is the scientic illustrator Pedro Salgado, and his lovely family, on the bridge. The next day there were bikini models lying on that bridge. :) I wish I had made a drawing of them but my boat was leaving and it was all I could do not to trample over them on my rush to get back to port. :).

Triangulation for life drawing

Triangulation is as old as geometry itself and its uses are without end. However triangulation is strangely absent from books on drawing, so much so that for a long while I thought (though I found it extremely unlikely!) that I was the first to use it for such purpose! (Oh, vanity!) Alas, there is nothing new under the sun, and recently I found it explicitely used and named as such in a 1971 book by Douglas Graves, though he didn't go into much detail (and a little while later found a couple of other sources on the web, though they used it in a more two-dimensional context, basically for drawing from photo reference - and 3D poses other challenges).

Claims to originality being thus shattered, it still remains true that I thought about it at lenght and used it and taught to colleagues for a few years, so I think I have something useful and maybe new to stress on its practical uses and caveats, so in a coming series of posts I will try to expound on triangulation - how and why to use it in drawing from life. I hope you may find this useful.

Just because...

...a blog on drawing needs some pretty things around! :)

The center must hold!

The biggest problem with most draughtsmen is that they keep changing their referential. They align the nose with the nipple, then find the hand should be more to the left by comparison to the navel, but then the wrist has to go further up, then this screws up the left eye, and so on. One can only hope that at some point this process will converge to a stable solution, but I don't know if it does or how long it takes, I just know that there is a better way:

Make an alignment of which you are sure, say, that the nose is aligned with the right...navel :)... or something...anything, as long as it is not too small a detail. Make sure of that alignment and stick to it. Compare all else to that one alignment, by checking angles, measures, and alignments. Do not deviate from it. Ever. Sacrifice everything to it. The center must hold! You can always win the day as long as some order can be had.