Object rotating via world axis and not local axis OpenGL - c

So I have a function here
void rotateLocal(GLfloat deg, GLfloat x, GLfloat y, GLfloat z)
where x, y, z are the coordinates of a local axis you want to rotate around for this object. I am using (0,1,0) for testing purposes, but yet it still ONLY rotates around world y instead of local y. Here is the logic for one vertex of an object in this function:
ax = x;
ay = y;
az = z;
//normalize
length = sqrt((ax * ax) + (ay * ay) + (az * az));
ux = ax / length;
uy = ay / length;
uz = az / length;
//square these
uxS = ux * ux;
uyS = uy * uy;
uzS = uz * uz;
getx = vertex[0];
gety = vertex[1];
getz = vertex[2];
//find new vertex points using rotation matrix for local axis
vertex[0] = (getx * (uxS + cos(deg) * (1 - uxS))) + (gety * (ux * uy * (1 - cos(deg)) - uz * sin(deg))) + (getz * (uz * ux * (1 - cos(deg)) + uy * sin(deg)));
vertex[1] = (getx * (ux * uy * (1-cos(deg)) + uz * sin(deg))) + (gety * (uyS + cos(deg) * (1 - uyS))) + (getz * (uy * uz * (1 - cos(deg)) - ux * sin(deg)));
vertex[2] = (getx * (uz * ux * (1-cos(deg)) - uy * sin(deg))) + (gety * (uy * uz * (1-cos(deg)) + ux * sin(deg))) + (getz * (uzS + cos(deg) * (1-uzS)));
is there something wrong with my rotation matrix? Am I using incorrect variable somewhere?
NOTE: I don't want to use RotateGL or anything like that, I want to do the matrix math myself.

The problem is that you are expressing the axis in world coordinates. What you have to do is:
Get the local to world transformation matrix of the object you want
to rotate.
Use the inverse of this matrix to convert the world axis (in your
example (0,1,0)) to local coordinates.
Use the converted axis to compute the rotation as you are doing in
the code.
I understand that the coordinates of the object are expressed in local space.
That's all.

Related

Finding a point on the right side of a line at a given distance

I want to find the point at distance d on the right side of a line defined by P1(x1,y1) and P2(x2,y2) (the distance is calculated from the middle of the line). I came up with the following code, which works well, but I think I have made unnecessary calculations, and it can be done faster.
#define PI 3.141592653589793238462643383279502884197169399375105820974944592308
double x2, x1, y1, y2, px, py, p1x, p1y, p2x, p2y, d, ax, ay, b, dx, dy;
d = 2.0; // given distance
ax = (x1 + x2) / 2; // middle point
ay = (y1 + y2) / 2; // middle point
b = tan(atan2(y2 - y1, x2 - x1) + PI / 2); // slope of the perpendicular line
dx = (d / sqrt(1 + (b * b)));
dy = b * dx;
p1x = ax + dx;
p1y = ay + dy;
p2x = ax - dx;
p2y = ay - dy;
// cross product
if (((x2 - x1) * (p1y - y1) - (y2 - y1) * (p1x - x1)) > 0)
{
px = p1x;
py = p1y;
}
else
{
px = p2x;
py = p2y;
}
You don't need atan, b value, cross product to check orientation (moreover, b might be zero and cause division error).
Instead calculate normalized (unit length) direction vector and get right normal to it:
d = 2.0; // given distance
ax = (x1 + x2) / 2; // middle point
ay = (y1 + y2) / 2; // middle point
dx = x2 - x1;
dy = y2 - y1;
scale = d / sqrt(dx*dx + dy*dy); //distance/vector length
px = ax + dy * scale; // add normal vector to the right side of p1-p2 direction
py = ay - dx * scale; //note minus sign
For generating a 2D vector perpendicular to another, one result that falls out from a special case of the dot product is that you can swap the two components of the vector and negate one of them.
For example, let's say you have the vector d which points from p1 to p2:
dx = p2x - p1x;
dy = p2y - p1y;
And now you want to generate right which is perpendicular, it is simply:
rightx = dy;
righty = -dx;
Now, let's do a quick visual check for our definition of "on the right", in case we actually want to negate those two values...
o p2 = [2, 3]
/
o p1 = [0, 0]
Above, d is simple: [2, 3]. Intuitively, we would think of (as viewed from above) walking from p1 to p2 and looking to the right, which would mean a vector in the positive X direction and the negative Y direction. So yes, that looks fine.
Note: If your co-ordinate system is screen-based (i.e. positive Y direction is down), then the inverse is true (and you would negate both the terms in the calculation of the right vector). This is due to the handedness of the co-ordinate system being left instead of right.
Now, you can calculate the midpoint mid as either (p1 + p2) / 2 or p1 + d / 2.
midx = (p1x + p2x) / 2;
midy = (p1y + p2y) / 2;
And finally to generate p3 you start from mid and extend down the vector right by an amount height, you need to normalize that vector by dividing by its length and scale by height. Formally, the final point will be mid + right * height / length(right).
This is the only particularly expensive part of the calculation, because it needs a square root.
rdist = height / sqrt(rightx * rightx + righty * righty);
p3x = midx + rightx * rdist;
p3y = midy + righty * rdist;
Congratulations! You now have an isosceles triangle!

Rotate around a point other than origin?

This is more math-y than programming-y but how can I improve this rotation matrix to accept a point at which to rotate around? There's the axis, which specifies the nature of the rotation, but there's also the point at which the vertices are rotated around per the axis. How can I improve this matrix to accept a position? It happens to be written in C but that's not very relevant, because I'm looking for the logic.
matrix_float4x4 Rotate(const float Radians, const float X, const float Y, const float Z) {
const float
Sin = sinf(Radians),
Cos = cosf(Radians),
C = 1-Cos, SinX = Sin*X,
SinY = Sin*Y,
SinZ = Sin*Z,
CX = X*C,
CY = Y*C,
CZ = C*Z;
return (matrix_float4x4){
.columns = {
{Cos+(X*CX), (Y*CX)+SinZ, (Z*CX)-SinY, 0},
{(X*CY)-SinZ, Cos+(Y*CY), (Z*CY)+SinX, 0},
{(X*CZ)+SinY, (Y*CZ)-SinX, Cos+(Z*CZ), 0},
{0, 0, 0, 1}
}
};
}
The question is, do you want to transform the object or the world? In other words, you can rotate the object and move (translate) it to a specific position or you can rotate the world around a specific point (orbit camera).
As pointed out by 'scg': T * R != R * T
A Transformation formula for the object would be: T * C * R * -C
(T: Translate, C: Center, R: Rotation)
Now, there are a lot of operations involved. If you look at a Translation- or Rotationmatrix, there are a lot of zeros. You can eliminate those steps:
Example:
typedef struct mat4_rec {
float
a11, a12, a13, a14,
a21, a22, a23, a24,
a31, a32, a33, a34,
a41, a42, a43, a44;
} mat4_t;
mat4_t* translate(mat4_t *m, float x, float y, float z)
{
m->a14 += m->a11 * x + m->a12 * y + m->a13 * z;
m->a24 += m->a21 * x + m->a22 * y + m->a23 * z;
m->a34 += m->a31 * x + m->a32 * y + m->a33 * z;
m->a44 += m->a41 * x + m->a42 * y + m->a43 * z;
return m;
}
To translate a 4x4 Matrix (M *= T). These are a lot less operations than a full 4x4 Multiplication.
Rotation (M *= R) would look like this:
mat4_t* rotate(mat4_t *m, float rad, float x, float y, float z)
{
float s = sinf(rad);
float c = cosf(rad);
float t = 1.0f - c;
float a11 = x * x * t + c;
float a12 = x * y * t - z * s;
float a13 = x * z * t + y * s;
float a21 = y * x * t + z * s;
float a22 = y * y * t + c;
float a23 = y * z * t - x * s;
float a31 = z * x * t - y * s;
float a32 = z * y * t + x * s;
float a33 = z * z * t + c;
float m11 = m->a11;
float m12 = m->a12;
float m13 = m->a13;
float m21 = m->a21;
float m22 = m->a22;
float m23 = m->a23;
float m31 = m->a31;
float m32 = m->a32;
float m33 = m->a33;
float m41 = m->a41;
float m42 = m->a42;
float m43 = m->a43;
m->a11 = m11 * a11 + m12 * a21 + m13 * a31;
m->a12 = m11 * a12 + m12 * a22 + m13 * a32;
m->a13 = m11 * a13 + m12 * a23 + m13 * a33;
m->a21 = m21 * a11 + m22 * a21 + m23 * a31;
m->a22 = m21 * a12 + m22 * a22 + m23 * a32;
m->a23 = m21 * a13 + m22 * a23 + m23 * a33;
m->a31 = m31 * a11 + m32 * a21 + m33 * a31;
m->a32 = m31 * a12 + m32 * a22 + m33 * a32;
m->a33 = m31 * a13 + m32 * a23 + m33 * a33;
m->a41 = m41 * a11 + m42 * a21 + m43 * a31;
m->a42 = m41 * a12 + m42 * a22 + m43 * a32;
m->a43 = m41 * a13 + m42 * a23 + m43 * a33;
return m;
}
To implement the transformation: T * C * R * -C
mat4_t m; // <- set to identiy or something else
// T - move position of object by one unit along the x axis
translate(&m, 1, 0, 0);
// C - move pivot point down along the y axis
translate(&m, 0, -1, 0);
// R - 180 degress around the z axis
rotate(&m, M_PI, 0, 0, 1);
// -C - restore pivot point
translate(&m, 0, 1, 0);
// multiply matrix by any vector (M * V)

Rotate Custom Object in Animation Loop

I created some Functions, these will draw Rectangles, Circles, Hexagons etc.
One of them looks like this:
rotation = 45;
function hex(hex_sides, hex_size, hex_color){
x = ctx.canvas.width/2;
y = ctx.canvas.height/2;
ctx.save();
ctx.rotate(rotation*Math.PI/180);
ctx.translate(ctx.canvas.width/2, ctx.canvas.height/2);
ctx.moveTo(x + hex_size * Math.cos(0), y + hex_size * Math.sin(0));
ctx.restore();
for (i = 0; i < hex_sides+1; i++) {
ctx.lineTo(x + hex_size * Math.cos(i * 2 * Math.PI / hex_sides), y + hex_size * Math.sin(i * 2 * Math.PI / hex_sides));
}
ctx.strokeStyle = hex_color;
ctx.stroke();
}
Now i call the Functions to Draw the Shapes inside my animation loop.
function loop() {
ctx.clearRect(0, 0, canvas.width, canvas.height);
circle(200);
circle(220);
hex(6, 180, "#fff");
rotation += 0.4;
requestAnimationFrame(loop);
}
I'm incrementing the var rotation inside the loop but it does not rotate the whole shape but just one line of it instead. Other Shapes i cant get to rotate at all.
I think i got a wrong approach, maybe because of my confusion about .save() and .restore() or .beginPath() and .closePath().
In General the behaviour is very strange when i start to use .translate() and .rotate()
The entire Code is here.
UPDATE
It is definitely something about this line:
ctx.translate(ctx.canvas.width/2, ctx.canvas.height/2);
Somehow it does not translate correctly. The rotation now happens around the right middle side of the normal shape position but i want the rotation around its own axis.
I changed the hex() function to:
ctx.save();
ctx.translate(ctx.canvas.width/2, ctx.canvas.height/2);
ctx.rotate(rotation*Math.PI/180);
ctx.moveTo(x + hex_size * Math.cos(0), y + hex_size * Math.sin(0));
for (i = 0; i < hex_sides+1; i++) {
ctx.lineTo(x + hex_size * Math.cos(i * 2 * Math.PI / hex_sides), y + hex_size * Math.sin(i * 2 * Math.PI / hex_sides));
}
ctx.strokeStyle = hex_color;
ctx.stroke();
ctx.restore();
Again, entire Code is here.
Nevermind... figured it out, i have forgotten to substract half of the shapes width from the translation point and also the order seems to be more important then i tought.
Working Code
function hex(hex_sides, hex_size, hex_color){
x = ctx.canvas.width/2;
y = ctx.canvas.height/2;
ctx.save();
ctx.translate(x,y);
ctx.rotate(rotation*Math.PI/180);
//ctx.moveTo(x + hex_size * Math.cos(0), y + hex_size * Math.sin(0));
ctx.moveTo(0,0);
for (i = 0; i < hex_sides+1; i++) {
ctx.lineTo(x/hex_size + hex_size * Math.cos(i * 2 * Math.PI / hex_sides), y/hex_size + hex_size * Math.sin(i * 2 * Math.PI / hex_sides));
}
ctx.strokeStyle = hex_color;
ctx.stroke();
ctx.restore();
}

C: Improving performance of function with heavy sin() usage

I have a C function that computes the values of 4 sines based on time elapsed. Using gprof, I figured that this function uses 100% (100.7% to be exact lol) of the CPU time.
void
update_sines(void)
{
clock_gettime(CLOCK_MONOTONIC, &spec);
s = spec.tv_sec;
ms = spec.tv_nsec * 0.0000001;
etime = concatenate((long)s, ms);
int k;
for (k = 0; k < 799; ++k)
{
double A1 = 145 * sin((RAND1 * k + etime) * 0.00333) + RAND5; // Amplitude
double A2 = 100 * sin((RAND2 * k + etime) * 0.00333) + RAND4; // Amplitude
double A3 = 168 * sin((RAND3 * k + etime) * 0.00333) + RAND3; // Amplitude
double A4 = 136 * sin((RAND4 * k + etime) * 0.00333) + RAND2; // Amplitude
double B1 = 3 + RAND1 + (sin((RAND5 * k) * etime) * 0.00216); // Period
double B2 = 3 + RAND2 + (sin((RAND4 * k) * etime) * 0.002); // Period
double B3 = 3 + RAND3 + (sin((RAND3 * k) * etime) * 0.00245); // Period
double B4 = 3 + RAND4 + (sin((RAND2 * k) * etime) * 0.002); // Period
double x = k; // Current x
double C1 = 0.6 * etime; // X axis move
double C2 = 0.9 * etime; // X axis move
double C3 = 1.2 * etime; // X axis move
double C4 = 0.8 * etime + 200; // X axis move
double D1 = RAND1 + sin(RAND1 * x * 0.00166) * 4; // Y axis move
double D2 = RAND2 + sin(RAND2 * x * 0.002) * 4; // Y axis move
double D3 = RAND3 + cos(RAND3 * x * 0.0025) * 4; // Y axis move
double D4 = RAND4 + sin(RAND4 * x * 0.002) * 4; // Y axis move
sine1[k] = A1 * sin((B1 * x + C1) * 0.0025) + D1;
sine2[k] = A2 * sin((B2 * x + C2) * 0.00333) + D2 + 100;
sine3[k] = A3 * cos((B3 * x + C3) * 0.002) + D3 + 50;
sine4[k] = A4 * sin((B4 * x + C4) * 0.00333) + D4 + 100;
}
}
And this is the output from gprof:
Flat profile:
Each sample counts as 0.01 seconds.
% cumulative self self total
time seconds seconds calls Ts/call Ts/call name
100.07 0.04 0.04
I'm currently getting a frame rate of roughly 30-31 fps using this. Now I figure there as to be a more efficient way to do this.
As you noticed I already changed all the divisions to multiplications but that had very little effect on performance.
How could I increase the performance of this math heavy function?
Besides all the other advice given in other answers, here is a pure algorithmic optimization.
In most cases, you're computing something of the form sin(k * a + b), where a and b are constants, and k is a loop variable. If you were also to compute cos(k * a + b), then you could use a 2D rotation matrix to form a recurrence relationship (in matrix form):
|cos(k*a + b)| = |cos(a) -sin(a)| * |cos((k-1)*a + b)|
|sin(k*a + b)| |sin(a) cos(a)| |sin((k-1)*a + b)|
In other words, you can calculate the value for the current iteration in terms of the value from the previous iteration. Thus, you only need to to do the full trig calculation for k == 0, but the rest can be calculated via this recurrence (once you have calculated cos(a) and sin(a), which are constants). So you eliminate 75% of the trig function calls (it's not clear the same trick can be pulled for the final set of trig calls).
If you don't need all that precision, create a lookup for the sin() values you need, so if 1 degree is enough, use double sin_lookup[360], etc.. And possibly float sin_lookup[360] if float precision is sufficient.
Also, as noted in comments, at a certain point as per Keith, "You might also consider using linear interpolation between lookup values, which should give you substantially better accuracy (a reasonably continuous function rather than a step function) at a fairly small cost in performance"
EDIT: also consider changing the hardcoded A1,A2,A3,A4 pattern to arrays of size[4], and looping from 0 to 3 - should allow vectorization on many platforms and allow parrellism without needing to manage threads
EDIT2: some code and results
(Coded in C++ just to make comparisons easy between precisions, calcs are the same in C)
class simple_trig
{
public:
simple_trig(size_t prec) : precision(prec)
{
static const double PI=3.141592653589793;
const double dprec=(double)prec;
const double quotient=(2.0*PI)/dprec;
rev_quotient=dprec/(2.0*PI);
values.reserve(prec);
for (int i=0; i < precision; ++i)
{
values[i]=::sin(quotient*(double)i);
}
}
double sin(double x) const
{
double cvt=x*rev_quotient;
int index=(int)cvt;
double delta=cvt-(double)index;
int lookup1=index%precision;
int lookup2=(index+1)%precision;
return values[lookup1]*(1.0-delta)+values[lookup2]*delta;
}
double cos(double x) const
{
double cvt=x*rev_quotient;
int index=(int)cvt;
double delta=cvt-(double)index;
int lookup1=(index+precision/4)%precision;
int lookup2=(index+precision/4+1)%precision;
return values[lookup1]*(1.0-delta)+values[lookup2]*delta;
}
private:
const size_t precision;
double rev_quotient;
std::vector<double> values;
};
Examples Low is 100, Med is 1000 and High is 10,000
X=0 Sin=0 Sin Low=0 Sin Med=0 Sin High=0
X=0 Cos=1 Cos Low=1 Cos Med=1 Cos High=1
X=0.5 Sin=0.479426 Sin Low=0.479389 Sin Med=0.479423 Sin High=0.479426
X=0.5 Cos=0.877583 Cos Low=0.877512 Cos Med=0.877578 Cos High=0.877583
X=1.33333 Sin=0.971938 Sin Low=0.971607 Sin Med=0.971935 Sin High=0.971938
X=1.33333 Cos=0.235238 Cos Low=0.235162 Cos Med=0.235237 Cos High=0.235238
X=2.25 Sin=0.778073 Sin Low=0.777834 Sin Med=0.778072 Sin High=0.778073
X=2.25 Cos=-0.628174 Cos Low=-0.627986 Cos Med=-0.628173 Cos High=-0.628174
X=3.2 Sin=-0.0583741 Sin Low=-0.0583689 Sin Med=-0.0583739 Sin High=-0.0583741
X=3.2 Cos=-0.998295 Cos Low=-0.998166 Cos Med=-0.998291 Cos High=-0.998295
X=4.16667 Sin=-0.854753 Sin Low=-0.854387 Sin Med=-0.854751 Sin High=-0.854753
X=4.16667 Cos=-0.519036 Cos Low=-0.518818 Cos Med=-0.519034 Cos High=-0.519036
X=5.14286 Sin=-0.90877 Sin Low=-0.908542 Sin Med=-0.908766 Sin High=-0.90877
X=5.14286 Cos=0.417296 Cos Low=0.417195 Cos Med=0.417294 Cos High=0.417296
X=6.125 Sin=-0.157526 Sin Low=-0.157449 Sin Med=-0.157526 Sin High=-0.157526
X=6.125 Cos=0.987515 Cos Low=0.987028 Cos Med=0.987512 Cos High=0.987515
X=7.11111 Sin=0.73653 Sin Low=0.736316 Sin Med=0.736527 Sin High=0.73653
X=7.11111 Cos=0.676405 Cos Low=0.676213 Cos Med=0.676403 Cos High=0.676405
X=8.1 Sin=0.96989 Sin Low=0.969741 Sin Med=0.969887 Sin High=0.96989
X=8.1 Cos=-0.243544 Cos Low=-0.24351 Cos Med=-0.243544 Cos High=-0.243544
X=9.09091 Sin=0.327701 Sin Low=0.327558 Sin Med=0.3277 Sin High=0.327701
X=9.09091 Cos=-0.944782 Cos Low=-0.944381 Cos Med=-0.944779 Cos High=-0.944782
X=10.0833 Sin=-0.611975 Sin Low=-0.611673 Sin Med=-0.611973 Sin High=-0.611975
X=10.0833 Cos=-0.790877 Cos Low=-0.790488 Cos Med=-0.790875 Cos High=-0.790877
It seems to me that sine1, sine2, sine3 and sine4 arrays are completely independent from eachother. So you are basically running a single for loop for 4 different arrays which have no dependency.
Spawn 4 threads, 1 for each, so you have 4 for loops running at the same time. On multicore machine this should speed up your function dramatically. As a matter of fact, it should be a perfect 4x speedup (+- ...).
Actually combining the use of threads (consider this with OpenMP) and the use of a table for the sin is a good idea. If possible use float instead of double and, depending on the platform, you could also use simd instructions, but the later would make the use of threads unnecessary.
Cheers
Here is a C++ snippet to use the rotation matrix suggested in the accepted answer.
float a = 0.343;
float b = 2.3232;
float sina{};
float cosa{};
sincosf(a, &sina, &cosa);
float resSin{};
float resCos{};
for (int k = 0; k < 5; k++) {
if (k == 0) {
sincosf(b, &resSin, &resCos);
} else {
float newResCos, newResSin;
newResCos = cosa * resCos - sina * resSin;
newResSin = sina * resCos + cosa * resSin;
resCos = newResCos;
resSin = newResSin;
}
}

Animating a sine wave infinitely

I need a function to animate a sine wave infinitely over time. The sine wave moves to the left.
My sine wave is built using the following equation:
A * sin(B * x + C) + D
Now to animate the sine wave as if it is moving to the left, I simply increase C by 1 everytime I refresh the screen. Now this is all fine and dandy for a few minutes but I need to have that animation run for hours. I can't just have an integer build up 60 times a second forever. How does someone deal with this? Do I just try to find a point where the sine wave crosses 0 and then restart the animation from 0?
I just need to have the logic of something like this explained.
EDIT #1
I forgot to mention that there's a randomized component to my sine. The sine is not continuously the same. A and D are sinusoidal functions tied to that integer at the moment. The sine needs to look random with varying periods and amplitudes.
EDIT #2
Edited see Edit 3
EDIT #3
#Potatoswatter I tried implementing your technique but I don't think I'm getting it. Here's what I got:
static double i = 0;
i = i + (MPI / 2);
if ( i >= 800 * (MPI / 2) ) i -= 800 * (MPI / 2);
for (k = 0; k < 800; ++k)
{
double A1 = 145 * sin((rand1 * (k - 400) + i) / 300) + rand3; // Amplitude
double A2 = 100 * sin((rand2 * (k - 400) + i) / 300) + rand2; // Amplitude
double A3 = 168 * sin((rand3 * (k - 400) + i) / 300) + rand1; // Amplitude
double B1 = 3 + rand1 + (sin((rand3 * k) * i) / (500 * rand1)); // Period
double B2 = 3 + rand2 + (sin((rand2 * k) * i) / 500); // Period
double B3 = 3 + rand3 + (sin((rand1 * k) * i) / (500 * rand3)); // Period
double x = k; // Current x
double C1 = 10 * i; // X axis move
double C2 = 11 * i; // X axis move
double C3 = 12 * i; // X axis move
double D1 = rand1 + sin(rand1 * x / 600) * 4; // Y axis move
double D2 = rand2 + sin(rand2 * x / 500) * 4; // Y axis move
double D3 = rand3 + cos(rand3 * x / 400) * 4; // Y axis move
sine1[k] = (double)A1 * sin((B1 * x + C1) / 400) + D1;
sine2[k] = (double)A2 * sin((B2 * x + C2) / 300) + D2 + 100;
sine3[k] = (double)A3 * cos((B3 * x + C3) / 500) + D3 + 50;
}
How do I modify this to make it work?
Halp!
Sine has a period of 2 pi, meaning that sin(x) = sin(x + 2 * M_PI), for any x.
So, you could just increase C by, say, pi/n where n is any integer, as you refresh the screen, and after 2n refreshes, reset C (to 0, or whatever).
Edit for clarity: the integer n is not meant to change over time.
Instead, pick some n, for example, let's say n = 10. Now, every frame, increase x by pi / 10. After 20 frames, you have increased x by a total of 20 * pi / 10 = 2 * pi. Since sin(x + 2 * pi) = sin(x), you may as well just reset your sin(...) input to just x, and start the process over.
sin is periodic, with a period of 2π. Therefore, if the argument is greater than 2π, you can subtract 2 * M_PI from it and get the same answer.
Instead of using a single variable k to compute all waves of various speeds, use three variables double k1, k2, k3, and keep them bound in the range from 0 to 2π.
if ( k2 >= 2 * M_PI ) k2 -= 2 * M_PI;
They may be individually updated by adding some value each frame. If the increment may be more than 2π then subtracting a single 2π won't bring them back into range, but you can use fmod() instead.
I decided to change my course of action. I just drive i with the system's monotonic clock like so:
struct timespec spec;
int ms;
time_t s;
static unsigned long long etime = 0;
clock_gettime(CLOCK_MONOTONIC, &spec);
s = spec.tv_sec;
ms = spec.tv_nsec / 10000000;
etime = concatenate((long)s, ms);
Then I simply changed i to etime in my sine equations. Here's the concatenating function I used for this purpose:
unsigned concatenate(unsigned x, unsigned y) {
x = x * 100;
return x + y;
}

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