Confused about degrees and OpenGL/GLUT camera movement/rotation

Question

NOTICE: I have edited the question below which is more relevant to my real issue than the text right below, you can skip this if you but I'll leave it here for historic reasons.

To see if I get this right, a float in C is the same as a value in radians right? I mean, 360º = 6.28318531 radians and I just noticed on my OpenGL app that a full rotation goes from 0.0 to 6.28, which seems to add up correctly. I just want to make sure I got that right.

I'm using a float (let's call it anglePitch) from 0.0 to 360.0 (it's easier to read in degrees and avoids casting int to float all the time) and all the code I see on the web uses some kind of DEG2RAD() macro which is defined as DEG2RAD 3.141593f / 180. In the end it would be something like this:

anglePitch += direction * 1; // direction will be 1 or -1
refY = tan(anglePitch * DEG2RAD);

This really does a full rotation but that full rotation will be when anglePitch = 180 and anglePitch * DEG2RAD = 3.14, but a full rotation should be 360|6.28. If I change the macro to any of the following:

#define DEG2RAD 3.141593f / 360
#define DEG2RAD 3.141593f / 2 / 180

It works as expected, a full rotation will happen when anglePitch = 360.

What am I missing here and what should I use to properly convert angles to radians/floats?

IMPORTANT EDIT (REAL QUESTION):
I understand now the code I see everywhere on the web about DEG2RAD, I'm just too dumb at math (yeah, I know, it's important when working with this kind of stuff). So I'm going to rephrase my question:

I have now added this to my code:

#define PI         3.141592654f
#define DEG2RAD(d) (d * PI / 180)

Now, when working the pitch/yawn angles in degrees, which are floats, once again, to avoid casting all the time, I just use the DEG2RAD macro and the degree value will be correctly converted to radians. These values will be passed to sin/cos/tan functions and will return the proper values to be used in GLUT camera.

Now the real question, where I was really confused before but couldn't explain myself better:

angleYaw += direction * ROTATE_SPEED;
refX = sin(DEG2RAD(angleYaw));
refZ = -cos(DEG2RAD(angleYaw));

This code will be executed when I press the LEFT/RIGHT keys and the camera will rotate in the Y axis accordingly. A full rotation goes from 0º to 360º.

anglePitch += direction * ROTATE_SPEED;
refY = tan(DEG2RAD(anglePitch));

This is similar code and will be executed when I press the UP/DOWN keys and the camera will rotate in the X axis. But in this situation, a full rotation goes from 0º to 180º degrees and that's what's really confusing me. I'm sure it has something to do with the tangent function but I can't get my head around it.

Is there way I could use sin/cos (as I do in the yawn code) to achieve the same rotation? What is the right way, the most simple code I can add/fix and what makes more sense to create a full pitch rotation from 0º to 360º?

Answer 1

'float' is a type, like int or double. radians and degrees are units of measure, both of which can be represented with any precision you want. i.e., there's no reason you can't have 22.5 degrees, and keep that value in a float.

a full rotation in radians is 2*pi, about 6.283, whereas a full rotation in degrees is 360. You can convert between them by dividing out the starting unit's full circle, then multiplying by the desired unit's full circle.

for example, to get from 90 degrees to radians, first divide out the degrees. 90 over 360 is 0.25 (note this value is in 'revolutions'). Now multiply that 0.25 by 6.283 to arrive at 1.571 radians.

follow up

the reason you're seeing your pitch cycle twice as fast as it should is precisely because you're using tan(pitch) to compute the Y component. What you should have is that the Y component depends on sin(pitch). i.e., try changing

refY = tan(DEG2RAD(anglePitch));

to

refY = sin(DEG2RAD(anglePitch));

a technical detail: the numbers that go into the look matrix should all be in the range of -1 to +1, and if you were to inspect the values you're feeding to refY, and run your pitch outside of -45 to +45 degrees, you'd see the problem; tan() runs off to infinity at +/-90 degrees.

also, note that casting a value from int to float in no sense converts between degrees and radians. casting just gives you the nearest equivalent value in the new storage type. for example, if you cast the integer 22 to floating point, you get 22.0f, whereas if you cast 33.3333f to type int, you'd be left with 33. when working with angles, you really should just stick with floating point, unless you're constrained by working with an embedded processor or something. this is especially important with radians, where whole number increments represent leaps of (about) 57.3 degrees.

Answer 2

360° = 2 * Pi, Pi = 3.141593…

Radians are defined by the arc length of an angle along a circle of radius 1. The circumfence of a circle is 2*r*Pi, so one full turn on a unit circle has an arc length of 2*Pi = 6.28…

The measure of angles in degrees stem from the fact, that by aligning 6 equilateral triangles you span a full turn. So we have 6 triangles, each making up a 6th of the turn, so the old babylonians divided a circle into pieces of 1/(6*6) = 1/36, and to further refine it this was subdivded by 10. That's why we ended up with 360° in a full circle. This number is arbitrarily choosen, though.

So if there are 2*Pi/360° this makes Pi/180° = 3.141593…/180° which is the conversion factor from degrees to radians. The reciprocal, 180°/Pi = 180/3.141593…

Why on earth the old OpenGL function glRotate and GLU's gluPerspective used degrees instead of radians I cannot fathom. From a mathematical point of view only radians make sense. Which I think is most beautifully demonstrated by Euler's equation

e^(i*Pi) - 1 = 0

There you have it, all the important numbers of mathematics in one single equation. What's this got to do with angles? Well:

e^(i*alpha) = cos(alpha) + i * sin(alpha), alpha is in radians!

EDIT, with respect to modified question:

Your angles being floats is all fine. Why would you even think degress being integers I cannot understand. Normally you don't have to define PI yourself, it comes predefined in math.h, usually called M_PI, M_2PI, and M_PI2 for Pi, 2*Pi and Pi/2. You also should change your macro, the way it's written now can create strange effects.

#define DEG2RAD(d) ( (d) * M_PI/180. )

GLUT has no camera at all. GLUT is a rather dumb OpenGL framework I recommend not using. You probably refer to gluLookAt.

Those obstacles out of the way let's see what you're doing there. Remember that trigonometric functions operate on the unit circle. Let the angle 0 point towards the right and angles increment counterclockwise. Then sin(a) is defined as the amount of rightwards and cos(a) and the amount of forwards to reach the point at angle a on the unit circle. This is what the refX and refZ are getting assigned to.

refY however makes no sense written that way. tan = sin/cos so as we approach n*pi/2 (i.e. 90°) it diverges to +/- infinity. At least it explains your pi/180° cyclic range, because that's the period of tan.

I was first thinking that tan may have been used to normalize the direction vector, but didn't make sense either. The factor would have been 1./sqrt(sin²(Pitch) + 1) I double checked: using tan there does the right thing.

EDIT2: I don't see where your problem is: The pitch angle is -90° to +90°, which makes perfect sense. Go get yourself a globe (of the earth): The east-west coordinates (longitude) go from -180° to +180°, the south-north coordinate (latitude) goes -90° to +90°. Think about it: Any larger coordinate range would create ambiguities.

The only good suggestion I offer you is: Grab some math text book and bend your mind around spherical coordinates! Sorry to tell you that way. Whatever you have works perfectly fine, you just need to understand sperical geometry.

You're using the terms Yaw and Pitch. Those are normally used in Euler angles. Now unfortunately Euler angles, which compelling at first, cause serious trouble later on (like gimbal lock). You should not use them at all. It may also be a good idea if you used some pencil/sticks/whatever to decompose the rotations you're intending with your hands to understand their mechanics.

---

And by the way: There are also non-integer degrees. Just hop over to http://maps.google.com to see them in action (just select some place and let http://maps.google.com give you the link to it).

Answer 3

Assuming that your ref components are intended to be used as your look-at vector, I think what you need is

refY = sin(DEG2RAD(anglePitch)); 
XZfactor = cos(DEG2RAD(anglePitch));
refX = XZfactor*sin(DEG2RAD(angleYaw));
refZ = -XZfactor*cos(DEG2RAD(angleYaw));