I know there are lots of questions and answers about this topic or related but i've beenn trying for 2 hours and still haven't benn able to figure it.
I would like to get a function that looks like this:
public static Vector rotateVector(Vector v, Vector axis, double angle){
}
Where the axis is a unit vector that defines the plane of rotation (the vector v rotates towards the vector axis if angle is positive)
I have already taken a look at rotation matrices but haven't been able to implement those to the above function
Rotating (x, y, z) counter clockwise around unit vector (u, v, w) by angle theta produces a vector (xPrime, yPrime, zPrime):
double xPrime = u*(u*x + v*y + w*z)*(1d - Math.cos(theta))
+ x*Math.cos(theta)
+ (-w*y + v*z)*Math.sin(theta);
double yPrime = v*(u*x + v*y + w*z)*(1d - Math.cos(theta))
+ y*Math.cos(theta)
+ (w*x - u*z)*Math.sin(theta);
double zPrime = w*(u*x + v*y + w*z)*(1d - Math.cos(theta))
+ z*Math.cos(theta)
+ (-v*x + u*y)*Math.sin(theta);
Source here.
Got it, thanks #Chris K. Here is the java function:
public static Vector rotateVectorCC(Vector vec, Vector axis, double theta){
double x, y, z;
double u, v, w;
x=vec.getX();y=vec.getY();z=vec.getZ();
u=axis.getX();v=axis.getY();w=axis.getZ();
double xPrime = u*(u*x + v*y + w*z)*(1d - Math.cos(theta))
+ x*Math.cos(theta)
+ (-w*y + v*z)*Math.sin(theta);
double yPrime = v*(u*x + v*y + w*z)*(1d - Math.cos(theta))
+ y*Math.cos(theta)
+ (w*x - u*z)*Math.sin(theta);
double zPrime = w*(u*x + v*y + w*z)*(1d - Math.cos(theta))
+ z*Math.cos(theta)
+ (-v*x + u*y)*Math.sin(theta);
return new Vector(xPrime, yPrime, zPrime);
}
However, I will keep the check on Chris' answer.
This is the correct way to rotate a vector.
private Vector rotateZ(Vector vector,double angle) { // angle in radians
//normalize(vector); // No need to normalize, vector is already ok...
float x1 = (float)(vector.x * Math.cos(angle) - vector.y * Math.sin(angle));
float y1 = (float)(vector.x * Math.sin(angle) + vector.y * Math.cos(angle)) ;
return new Vector(x1, y1);
}
If you want the rotation for x,y and z axis then you should use rotation matrices all at once.
NewVector = [Rotation_X][Rotation_Y][Rotation_Z]*OldVector
Here Rotation_X,Rotation_Y and Rotation_Z are 3x3 matrices. (You can see http://mathworld.wolfram.com/RotationMatrix.html)
The order of multiplication depends on the problem but i guess you want only one-axis rotation (i.e. the other 2 matrices become identity matrices)
So just putting an if-block you can set the correct matrix, and leave the rest as identity matrices.
Hope this helps.
Related
I wrote a code that should turn a point around another point counterclockwise. But it does not work correctly.
public boolean contains(double x, double y) {
double ox = this.x.get() + (this.width.get()/2);
double oy = this.y.get() + (this.height.get()/2);
double theta = rotate.get() - (rotate.get() * 2);
double px1 = Math.cos(theta) * (x-ox) - Math.sin(theta) * (y-oy) + ox;
double py1 = Math.sin(theta) * (x-ox) + Math.cos(theta) * (y-oy) + oy;
return shape.contains(px1, py1);
}
x, y - are the coordinates of the point to be rotated.
ox,oy - is the coordinates of the point around which you want to rotate.
rotate.get() - angle to rotate
Update: Changes in the code that solved the problem, who can come in handy:
double px1 = Math.cos(Math.toRadians(theta)) * (x-ox) - Math.sin(Math.toRadians(theta)) * (y-oy) + ox;
double py1 = Math.sin(Math.toRadians(theta)) * (x-ox) + Math.cos(Math.toRadians(theta)) * (y-oy) + oy;
Please check, if your rotate.get() will give you a degrees value (e.g. 45°) or a radians value (e.g. 0.5*pi). Math.sin() and Math.cos() will only accept radians.
To convert them you could use something like angle = Math.toRadians(45)
Although this is answered, another simple way to get this done is using the built-in method of Rotate class. This way you dont need to worry about the Math stuff ;)
Rotate r = new Rotate();
r.setPivotX(ox);
r.setPivotY(oy);
r.setAngle(angleInDegrees);
Point2D point = r.transform(new Point2D(x, y));
I'm programming a software renderer in Java, and am trying to use Z-buffering for the depth calculation of each pixel. However, it appears to work inconsistently. For example, with the Utah teapot example model, the handle will draw perhaps half depending on how I rotate it.
My z-buffer algorithm:
for(int i = 0; i < m_triangles.size(); i++)
{
if(triangleIsBackfacing(m_triangles.get(i))) continue; //Backface culling
for(int y = minY(m_triangles.get(i)); y < maxY(m_triangles.get(i)); y++)
{
if((y + getHeight()/2 < 0) || (y + getHeight()/2 >= getHeight())) continue; //getHeight/2 and getWidth/2 is for moving the model to the centre of the screen
for(int x = minX(m_triangles.get(i)); x < maxX(m_triangles.get(i)); x++)
{
if((x + getWidth()/2 < 0) || (x + getWidth()/2 >= getWidth())) continue;
rayOrigin = new Point2D(x, y);
if(pointWithinTriangle(m_triangles.get(i), rayOrigin))
{
zDepth = zValueOfPoint(m_triangles.get(i), rayOrigin);
if(zDepth > zbuffer[x + getWidth()/2][y + getHeight()/2])
{
zbuffer[x + getWidth()/2][y + getHeight()/2] = zDepth;
colour[x + getWidth()/2][y + getHeight()/2] = m_triangles.get(i).getColour();
g2.setColor(m_triangles.get(i).getColour());
drawDot(g2, rayOrigin);
}
}
}
}
}
Method for calculating the z value of a point, given a triangle and the ray origin:
private double zValueOfPoint(Triangle triangle, Point2D rayOrigin)
{
Vector3D surfaceNormal = getNormal(triangle);
double A = surfaceNormal.x;
double B = surfaceNormal.y;
double C = surfaceNormal.z;
double d = -(A * triangle.getV1().x + B * triangle.getV1().y + C * triangle.getV1().z);
double rayZ = -(A * rayOrigin.x + B * rayOrigin.y + d) / C;
return rayZ;
}
Method for calculating if the ray origin is within a projected triangle:
private boolean pointWithinTriangle(Triangle triangle, Point2D rayOrigin)
{
Vector2D v0 = new Vector2D(triangle.getV3().projectPoint(modelViewer), triangle.getV1().projectPoint(modelViewer));
Vector2D v1 = new Vector2D(triangle.getV2().projectPoint(modelViewer), triangle.getV1().projectPoint(modelViewer));
Vector2D v2 = new Vector2D(rayOrigin, triangle.getV1().projectPoint(modelViewer));
double d00 = v0.dotProduct(v0);
double d01 = v0.dotProduct(v1);
double d02 = v0.dotProduct(v2);
double d11 = v1.dotProduct(v1);
double d12 = v1.dotProduct(v2);
double invDenom = 1.0 / (d00 * d11 - d01 * d01);
double u = (d11 * d02 - d01 * d12) * invDenom;
double v = (d00 * d12 - d01 * d02) * invDenom;
// Check if point is in triangle
if((u >= 0) && (v >= 0) && ((u + v) <= 1))
{
return true;
}
return false;
}
Method for calculating surface normal of a triangle:
private Vector3D getNormal(Triangle triangle)
{
Vector3D v1 = new Vector3D(triangle.getV1(), triangle.getV2());
Vector3D v2 = new Vector3D(triangle.getV3(), triangle.getV2());
return v1.crossProduct(v2);
}
Example of the incorrectly drawn teapot:
What am I doing wrong? I feel like it must be some small thing. Given that the triangles draw at all, I doubt it's the pointWithinTriangle method. Backface culling also appears to work correctly, so I doubt it's that. The most likely culprit to me is the zValueOfPoint method, but I don't know enough to know what's wrong with it.
My zValueOfPoint method was not working correctly. I'm unsure why :( however, I changed to a slightly different method of calculating the value of a point in a plane, found here: http://forum.devmaster.net/t/interpolation-on-a-3d-triangle-using-normals/20610/5
To make the answer here complete, we have the equation of a plane:
A * x + B * y + C * z + D = 0
Where A, B, and C are the surface normal x/y/z values, and D is -(Ax0 + By0 + Cz0).
x0, y0, and z0 are taken from one of the vertices of the triangle. x, y, and z are the coordinates of the point where the ray intersects the plane. x and y are known values (rayOrigin.x, rayOrigin.y) but z is the depth which we need to calculate. From the above equation we derive:
z = -A / C * x - B / C * y - D
Then, copied from the above link, we do:
"Note that for every step in the x-direction, z increments by -A / C, and likewise it increments by -B / C for every step in the y-direction.
So these are the gradients we're looking for to perform linear interpolation. In the plane equation (A, B, C) is the normal vector of the plane.
It can easily be computed with a cross product.
Now that we have the gradients, let's call them dz/dx (which is -A / C) and dz/dy (which is -B / C), we can easily compute z everywhere on the triangle.
We know the z value in all three vertex positions.
Let's call the one of the first vertex z0, and it's position coordinates (x0, y0). Then a generic z value of a point (x, y) can be computed as:"
z = z0 + dz/dx * (x - x0) + dz/dy * (y - y0)
This found the Z value correctly and fixed my code. The new zValueOfPoint method is:
private double zValueOfPoint(Triangle triangle, Point2D rayOrigin)
{
Vector3D surfaceNormal = getNormal(triangle);
double A = surfaceNormal.x;
double B = surfaceNormal.y;
double C = surfaceNormal.z;
double dzdx = -A / C;
double dzdy = -B / C;
double rayZ = triangle.getV1().z * modelViewer.getModelScale() + dzdx * (rayOrigin.x - triangle.getV1().projectPoint(modelViewer).x) + dzdy * (rayOrigin.y - triangle.getV1().projectPoint(modelViewer).y);
return rayZ;
}
We can optimize this by only calculating most of it once, and then adding dz/dx to get the z value for the next pixel, or dz/dy for the pixel below (with the y-axis going down). This means that we cut down on calculations per polygon significantly.
this must be really slow
so much redundant computations per iteration/pixel just to iterate its coordinates. You should compute the 3 projected vertexes and iterate between them instead look here:
triangle/convex polygon rasterization
I dislike your zValueOfPoint function
can not find any use of x,y coordinates from the main loops in it so how it can compute the Z value correctly ?
Or it just computes the average Z value per whole triangle ? or am I missing something? (not a JAVA coder myself) in anyway it seems that this is your main problem.
if you Z-value is wrongly computed then Z-Buffer can not work properly. To test that look at the depth buffer as image after rendering if it is not shaded teapot but some incoherent or constant mess instead then it is clear ...
Z buffer implementation
That looks OK
[Hints]
You have too much times terms like x + getWidth()/2 why not compute them just once to some variable? I know modern compilers should do it anyway but the code would be also more readable and shorter... at least for me
I'm just trying to code a nice looking physics game.
The ball collision looks nice but if the balls are colliding too slow, they "stick" in each other. I have no clue why they do.
Here's my collision function:
private void checkForCollision(ArrayList<Ball> balls) {
for (int i = 0; i < balls.size(); i++) {
Ball ball = balls.get(i);
if (ball != this && ball.intersects(this)) {
this.collide(ball, false);
}
}
}
public boolean intersects(Ball b) {
double dx = Math.abs(b.posX - posX);
double dy = Math.abs(b.posY - posY);
double d = Math.sqrt(dx * dx + dy * dy);
return d <= (radius + b.radius);
}
private void collide(Ball ball, boolean b) {
double m1 = this.radius;
double m2 = ball.radius;
double v1 = this.motionX;
double v2 = ball.motionX;
double vx = (m1 - m2) * v1 / (m1 + m2) + 2 * m2 * v2 / (m1 + m2);
v1 = this.motionY;
v2 = ball.motionY;
double vy = (m1 - m2) * v1 / (m1 + m2) + 2 * m2 * v2 / (m1 + m2);
if (!b)
ball.collide(this, true);
System.out.println(vx + " " + vy);
motionX = vx * BOUNCEOBJECT;
motionY = vy * BOUNCEOBJECT;
}
But this is what happens when they collide with a low speed:
So do you have an idea?
EDIT:
The update of Alnitak works very nice... but one problem is still there... if i add gravity like this:
public void physic() {
motionY += GRAVITY; // <= this part (GRAVITY is set to 0.3D)
checkForCollision(screen.balls);
keyMove();
bounceWalls();
posX += motionX;
posY += motionY;
}
They still move into each other. I think this is the wrong way to add gravity, or isn't it?
And I think I did something wrong with the collision formula, because they don't fall right:
!
and then they slowly sink into the ground.
EDIT:
found an AMAZING tutorial: http://www.ntu.edu.sg/home/ehchua/programming/java/J8a_GameIntro-BouncingBalls.html
This is a common problem that happens because sometimes the delta-v of the bouncing ball is insufficient to take it back out of the collision zone.
So the collision routine reverses the direction again, taking it back inside the other ball, ad-infinitum.
You should add a sufficient offset (in the direction of the collision force) to the position of the ball to ensure that the newly calculated positions are no longer colliding.
Alternatively, check whether the balls would collide once you add the new motion values:
public boolean intersects(Ball b) {
double dx = b.posX - (posX + motionX); // no need for Math.abs()
double dy = b.posY - (posY - motionY);
double d = dx * dx + dy * dy; // no need for Math.sqrt()
return d < (radius + b.radius) * (radius + b.radius);
}
but you should also change ball.intersects(this) to intersects(ball).
They may appear to collide slightly too early, but on a fast moving ball it probably won't be visible.
(m1 - m2) * v1 / (m1 + m2) + 2 * m2 * v2 / (m1 + m2);
This has an integer value 2. Please make it 2.0f or 2.0d then check it out. It must be the problem for small speeds. Becuse integer constant autocasts multiplied doubles.
If this does not work, then Alnitak 's answer would be helpful.
If you need real nice physics, you should use the force then convert it to velocity then convert it to displacement . Look at integrator techniques like Runge Kutta and Euler Integration
Force-->acceleration-->velocity-->displacement
if collision occurs, just update the force then the rest will be flowing.
----> http://codeflow.org/entries/2010/aug/28/integration-by-example-euler-vs-verlet-vs-runge-kutta/ <-----
http://www.forums.evilmana.com/game-programming-theory/euler-vs-verlet-vs-rk4-physics/
http://www.newagepublishers.com/samplechapter/001579.pdf
http://cwx.prenhall.com/bookbind/pubbooks/walker2/
Verlet integration is a point between Runge-Kutta-4 and Euler Integration preferably for molecular dynamics (a good example for bouncing balls if you ommit the electrical fields and bonds)
Just found an AMAZING tutorial:
http://www.ntu.edu.sg/home/ehchua/programming/java/J8a_GameIntro-BouncingBalls.html
Alright, so I got a bit of movement code and I'm thinking I'm going to need to manually input when to go up/down a slope. All I got to work with is the slope's normal, and vector, and My current and previous position, and my yaw.
Is there a better way to rotate whether I go up or down the slope based on my yaw?
Vector3f move = new Vector3f(0,0,0);
move.x = (float)-Math.cos(Math.toRadians(yaw));
move.z = (float)-Math.sin(Math.toRadians(yaw));
System.out.println("slopeNormal.z: " + slopeNormal.z + "move.z: " + move.z);
move.normalise();
float vx = (float) (Math.sqrt(Math.pow(move.y, 2) + Math.pow(move.z, 2)) * move.x);
float vy = (float) (Math.sqrt(Math.pow(move.x, 2) + Math.pow(move.z, 2)) * move.y);
float vz = - vx * slopeNormal.x - vy * slopeNormal.y;
move.scale(movementSpeed * delta);
if(vz < 0)
move.y -= slopeVec.y * 1.5f;
if(vz > 0)
move.y += slopeVec.y * 1.5f;
Vector3f.add(pos, move, pos);
Edit: updated code.
First off, the following is incorrect:
move.x = (float)-Math.toDegrees(Math.cos(Math.toRadians(yaw)));
move.z = (float)-Math.toDegrees(Math.sin(Math.toRadians(yaw)));
Math.toDegrees converts an angle in radians to one in degrees, but the results of Math.cos and Math.sin are not angles.
Assume zero yaw is in the positive x-direction... and define vx, vy, vz = rate of motion along 3 axes, s = speed, and slope normal = nx, ny, nz where nx^2 + ny^2 + nz^2 = 1. So nx = ny = 0, nz = 1 would be flat.
First, I define x', y' = axes relative to the flat ground (motion is constrained to ground). Then (the following is not valid Java, but I'm enclosing it in code format anyway):
vx' = cos(yaw) * s
vy' = sin(yaw) * s
Then I need to rotate from x', y' coordinates to real-world coordinates. That is done using the slope normal:
vx = sqrt(vy^2 + vz^2) vx'
vy = sqrt(vx^2 + vz^2) vy'
vz = - vx' nx - vy' ny
A check on this transformation: vx^2 + vy^2 + vz^2 must equal vx'^2 + vy'^2 = s^2. I think this works out.
So to answer your question: up or down? vz > 0 is up, vz < 0 is down.
I have this method for rotating points in 3D using quaternions, but it seems not to work properly:
public static ArrayList<Float> rotation3D(ArrayList<Float> points, double angle, int xi, int yi, int zi)
{
ArrayList<Float> newPoints = new ArrayList<>();
for (int i=0;i<points.size();i+=3)
{
float x_old = points.get(i);
float y_old = points.get(i+1);
float z_old = points.get(i+2);
double w = Math.cos(angle/2.0);
double x = xi*Math.sin(angle/2.0);
double y = yi*Math.sin(angle/2.0);
double z = zi*Math.sin(angle/2.0);
float x_new = (float) ((1 - 2*y*y -2*z*z)*x_old + (2*x*y + 2*w*z)*y_old + (2*x*z-2*w*y)*z_old);
float y_new = (float) ((2*x*y - 2*w*z)*x_old + (1 - 2*x*x - 2*z*z)*y_old + (2*y*z + 2*w*x)*z_old);
float z_new = (float) ((2*x*z + 2*w*y)*x_old + (2*y*z - 2*w*x)*y_old + (1 - 2*x*x - 2*y*y)*z_old);
newPoints.add(x_new);
newPoints.add(y_new);
newPoints.add(z_new);
}
return newPoints;
}
If i make this call rotation3D(list, Math.toRadians(90), 0, 1, 0); where points is (0,0,10), the output is (-10.0, 0.0, 2.220446E-15), but it should be (-10,0,0), right? Could someone take a look at my code and tell me if is there somethig wrong?
Here are 4 screens that represent the initial position of my object, and 3 rotations with -90 degrees (the object is not properly painted, that's a GL issue, that i will work on later):
I haven't studied the code but what you get from it is correct: Assuming a left-handed coordinate system, when you rotate the point (0,0,10) 90 degrees around the y-axis (i.e. (0,1,0)) you end up with (-10,0,0).
If your coordinate system is right-handed I think you have to reverse the sign of the angle.