Develop Reference

How to deal with floating-point accuracy when calculating the lateral surface of a frustum cone using double in Java

I'm trying to calculate the lateral surface of a frustum cone using the code below.
Input: 2 Nodes containing x, y, z values (absolut position) and radius (radius at position) all in double
What I am doing so far:
1. calculate length of frustum cone
2. calculate lateral surface of frustum cone
My Problem:
Hypothetically substraction of 2 floating point numbers with similar magnitude is problematic due to precision loss. (I didn't run into the issue)
My Question:
What can I do to improve the final result?
Possibilities I found / thought of:
- using BigDecimal (what I don't want to because of longer runtime)
- replacing (r1-r2)*(r1-r2) with r1^2 - 2*r1*r2 + r2^2
- implement a check on how close two double values are and if very close assume their difference to be 0. Would that even improve the accuracy of my final result? Doesn't the result of the substraction have a smaller absolute error than the one with 0 assumed?
public static double calculateLateralSurface(Node node1, Node node2) {
double x, y, z, length, r1, r2, lateralSurface;
// calculate length of frustum cone
x = node1.getPosX() - node2.getPosX();
y = node1.getPosY() - node2.getPosY();
z = node1.getPosZ() - node2.getPosZ();
length = Math.sqrt(x * x + y * y + z * z);
r1 = node1.getRadius();
r2 = node2.getRadius();
// calculate lateral surface of frustum cone
lateralSurface = (r1+r2)*Math.PI*Math.sqrt((r1-r2)*(r1-r2)+length*length);
return lateralSurface;
}
I hope someone can help me :)

double has more than enough accuracy for any practical and even not-so-practical use. So, unless your cone describes the field of view of an optical telescope from Earth all the way to Alpha Centauri, you should not have any precision problems.
I can point out to you that you are calculating length by taking a square root only to square it again later before using it, so eliminating that unnecessary pair of calculations might somehow improve things, but I doubt that this is your problem.
So, if your results do not look correct, then maybe a bug is to blame and not the precision of double.
In any case, why don't you provide a self-contained, compilable example, and actual input values, and actual output values, and your expected values, and we can look deeper into it.

Computer Graphics: Rotating a polygon

Purpose
I'm implementing a polygon rotation with Java AWT.
I'm already able to draw polygons on the screen, and I'd like to apply a rotation matrix manually upon my polygons coordinates (rotation is done around the lookAt point of the user).
What I've already done
In order to rotate the world, the user first clicks on the screen and then drags the mouse around to perform the rotation.
Let's note the first click point as S, the following point from the drag event as L, and the center of the screen as C.
In order to calculate the rotation angle, when first clicking the screen, I keep a vector from C to S: C-S.
Then, when a drag event occurs, I calculate the vector from C to L: C-L.
I then calculate the angle in radians between C-S to C-L, and that's what I apply on my world.
This works well, and the polygon is indeed rotation around the lookAt point.
My problem
The problem occurs when the user finishes a rotation of PI, and then the polygon is rotating backward.
e.g. When the user starts rotating, the angle starts from 0.1.... 0.2... 1.. 2.. 3.. and in value ~3.1 (I assume PI), the values are starting to go down: 3... 2.. 1.. until 0, and vice versa.
This makes sense since the radians range is [0, PI].
I assume the base vector C-S lies on the right side of X axis, and when the rotation goes down below the X axis the polygon is rotating backwards.
However, I have no idea how to keep the polygon rotating in the same direction all the time (when the user performs a full rotation around the polygon).
Edit
Angle function is:
public final double angle(Vector2D v1)
{
double vDot = this.dot(v1) / ( this.length()*v1.length() );
if( vDot < -1.0) vDot = -1.0;
if( vDot > 1.0) vDot = 1.0;
return ((double) (Math.acos( vDot )));
}

This is a problem of the arcus cosine, acos(cos(x)) is a periodic hat function moving up and down in the range of 0 to pi.
In higher dimensions that can not be avoided, as there is no preferred frame of reference, so there is no way to say that phi should really be -phi. In 2 dimensions there is a prefered orientation of the plane so that one can say what is the first and what the second vector and define a unique angle in positive orientation. Rotate the situation so that the first vector comes to lay on the positive real half axis to get the angle and correct quadrant from the coordinates of the rotated second vector.
Easiest to reconstruct is the complex picture, to compute the angle from a=a.x+i*a.y to b=b.x+i*b.y rotate b back by multiplying with the conjugate of a to get an angle from the zero angle resp. the positive real axis,
arg((a.x-i*a.y)*(b.x+i*b.y))
=arg((a.x*b.x+a.y*b.y)+i*(a.x*b.y-a.y*b.x))
=atan2( a.x*b.y-a.y*b.x , a.x*b.x+a.y*b.y )
Note that screen coordinates use the opposite orientation to the cartesian/complex plane, thus change atan2(y,x) to atan2(-y,x) to get an angle in the usual direction.

public Point rotate(Point original, Point vertex, double angle){
Point translated = new Point(original.x - vertex.x, original.y - vertex.y);
int x = (int)Math.round(translated.x * Math.cos(angle) - translated.y * Math.sin(angle));
int y = (int)Math.round(translated.x * Math.sin(angle) + translated.y * Math.cos(angle));
return new Point(vertex.x+x,vertex.y+y);
}
This is a simple rotation method that you can use to rotate a point around a given vertex.

Trying to rotate a polygon using math

I have a school assignment where I'm supposed to (among other things) rotate a polygon. I can not use any premade rotate functions, so I have an array of points. The array is set up like this:
intArray[2][amount_of_points] where intArray[0] equals the points x coordinate, and intArray[1] holds the y coordinates.
//x=pivot point x coordinate, y = pivot point y coordinate.
public int[][] rotate(int[][]matrix,int x, int y, double degrees){
double s=Math.sin(degrees);
double c=Math.cos(degrees);
for (int i=0;i<matrix.length;i++){
//translate to origin
int px=matrix[0][i]-x;
int py=matrix[1][i]-y;
//rotate
double xnew = (px*c)-(py*s);
double ynew = (px*s)+(py*c);
//translate back
px=(int)((xnew+x)+0.5);
py=(int)((ynew+y)+0.5);
matrix[0][i]=px;
matrix[1][i]=py;
}
This is my code so far, and it is definitely not working out for me. I tried to trim the code as much as I could. Any help would mean a lot!
edit: I'm not getting any errors when I run the code, no exceptions etc. The only problem is that the polygon isn't rotating the way I intend it to.
I've made a test polygon:
polyArray = new int [2][3];
polyArray[0][0]=400;
polyArray[1][0]=200;
polyArray[0][1]=300;
polyArray[1][1]=500;
polyArray[0][2]=500;
polyArray[1][2]=500;
Which I draw in a JPanel, then I run this array through the rotation method like this:
polyArray=mm.rotate(polyArray, polyArray[0][0], polyArray[1][0], Math.PI);
Using the top point as pivotpoint. The whole polygon is then deformed.

Although still not very clear on question, I feel your problem is with the loop.
matrix.length is 2. So, the code never uses these :
polyArray[0][2]=500;
polyArray[1][2]=500;
If you change the condition as below, it should work :
for (int i=0;i<matrix[0].length;i++)

Math issue: place circles around the clock

Look at the gray circles along the lines, they need to be placed equally to represent the hours, here is my code for that, where did I go wrong?
for (int i=0; i<12; i++)
{
c.drawCircle(140*(float)Math.cos((double) (i*30))+ width/2,
140*(float) Math.sin((double) (i*30)) + height/2, 1, p);
}

The angle must be provided to Math.cos in radians (between 0 and 2*Math.PI).
So, instead of
Math.cos((double) (i*30))
Use
Math.cos(i*Math.PI/6.0)
Note that I also removed the useless cast to double : multiplying an int and a double already produces a double.

Math.sin and Math.cos operate on angles in radians and not degrees. To correct, multiply by pi and divide by 180.

Minimising cumulative floating point arithmetic error

I have a 2D convex polygon in 3D space and a function to measure the area of the polygon.
public double area() {
if (vertices.size() >= 3) {
double area = 0;
Vector3 origin = vertices.get(0);
Vector3 prev = vertices.get(1).clone();
prev.sub(origin);
for (int i = 2; i < vertices.size(); i++) {
Vector3 current = vertices.get(i).clone();
current.sub(origin);
Vector3 cross = prev.cross(current);
area += cross.magnitude();
prev = current;
}
area /= 2;
return area;
} else {
return 0;
}
}
To test that this method works at all orientations of the polygon I had my program rotate it a little bit each iteration and calculate the area. Like so...
Face f = poly.getFaces().get(0);
for (int i = 0; i < f.size(); i++) {
Vector3 v = f.getVertex(i);
v.rotate(0.1f, 0.2f, 0.3f);
}
if (blah % 1000 == 0)
System.out.println(blah + ":\t" + f.area());
My method seems correct when testing with a 20x20 square. However the rotate method (a method in the Vector3 class) seems to introduce some error into the position of each vertex in the polygon, which affects the area calculation. Here is the Vector3.rotate() method
public void rotate(double xAngle, double yAngle, double zAngle) {
double oldY = y;
double oldZ = z;
y = oldY * Math.cos(xAngle) - oldZ * Math.sin(xAngle);
z = oldY * Math.sin(xAngle) + oldZ * Math.cos(xAngle);
oldZ = z;
double oldX = x;
z = oldZ * Math.cos(yAngle) - oldX * Math.sin(yAngle);
x = oldZ * Math.sin(yAngle) + oldX * Math.cos(yAngle);
oldX = x;
oldY = y;
x = oldX * Math.cos(zAngle) - oldY * Math.sin(zAngle);
y = oldX * Math.sin(zAngle) + oldY * Math.cos(zAngle);
}
Here is the output for my program in the format "iteration: area":
0: 400.0
1000: 399.9999999999981
2000: 399.99999999999744
3000: 399.9999999999959
4000: 399.9999999999924
5000: 399.9999999999912
6000: 399.99999999999187
7000: 399.9999999999892
8000: 399.9999999999868
9000: 399.99999999998664
10000: 399.99999999998386
11000: 399.99999999998283
12000: 399.99999999998215
13000: 399.9999999999805
14000: 399.99999999998016
15000: 399.99999999997897
16000: 399.9999999999782
17000: 399.99999999997715
18000: 399.99999999997726
19000: 399.9999999999769
20000: 399.99999999997584
Since this is intended to eventually be for a physics engine I would like to know how I can minimise the cumulative error since the Vector3.rotate() method will be used on a very regular basis.
Thanks!
A couple of odd notes:
The error is proportional to the amount rotated. ie. bigger rotation per iteration -> bigger error per iteration.
There is more error when passing doubles to the rotate function than when passing it floats.

You'll always have some cumulative error with repeated floating point trig operations — that's just how they work. To deal with it, you basically have two options:
Just ignore it. Note that, in your example, after 20,000 iterations(!) the area is still accurate down to 13 decimal places. That's not bad, considering that doubles can only store about 16 decimal places to begin with.
Indeed, plotting your graph, the area of your square seems to be going down more or less linearly:
This makes sense, assuming that the effective determinant of your approximate rotation matrix is about 1 − 3.417825 × 10-18, which is well within normal double precision floating point error range of one. If that's the case, the area of your square would continue a very slow exponential decay towards zero, such that you'd need about two billion (2 × 109) 7.3 × 1014 iterations to get the area down to 399. Assuming 100 iterations per second, that's about seven and a half months 230 thousand years.
Edit: When I first calculated how long it would take for the area to reach 399, it seems I made a mistake and somehow managed to overestimate the decay rate by a factor of about 400,000(!). I've corrected the mistake above.
If you still feel you don't want any cumulative error, the answer is simple: don't iterate floating point rotations. Instead, have your object store its current orientation in a member variable, and use that information to always rotate the object from its original orientation to its current one.
This is simple in 2D, since you just have to store an angle. In 3D, I'd suggest storing either a quaternion or a matrix, and occasionally rescaling it so that its norm / determinant stays approximately one (and, if you're using a matrix to represent the orientation of a rigid body, that it remains approximately orthogonal).
Of course, this approach won't eliminate cumulative error in the orientation of the object, but the rescaling does ensure that the volume, area and/or shape of the object won't be affected.

You say there is cumulative error but I don't believe there is (note how your output desn't always go down) and the rest of the error is just due to rounding and loss of precision in a float.
I did work on a 2d physics engine in university (in java) and found double to be more precise (of course it is see oracles datatype sizes
In short you will never get rid of this behaviour you just have to accept the limitations of precision
EDIT:
Now I look at your .area function there is possibly some cumulative due to
+= cross.magnitude
but I have to say that whole function looks a bit odd. Why does it need to know the previous vertices to calculate the current area?