I credit C++ Programming Forum members, dwks and _Mike, for giving me the information I lack in my linear algebra class, or I should blame my school for teaching the lessons up to subspaces before we even reached vectors for this semester's linear algebra course and gave us a winter vacation. The source for the information is

linked here.

I credit RulerOfNothing in his very last post for giving me the inspiration to rewrite my program from scratch, and from there I finally took notice of a bug when comparing the new project with my old in Subversion.

This time, from dwks, I calculate the two points by using normal vectors and distances, along with doing the arithmetic entirely in double. I revised my Dot class by not using the java.awt.Point class, and added a few more functions to separate the logic in a more versatile way.

Here is my results:

public class Dot implements Tickable {
public double xPosition = 0;
public double yPosition = 0;
public double xTarget = 0;
public double yTarget = 0;
public double xDist = 0;
public double yDist = 0;
public void tick() {
// Setting up a vector.
xDist = xTarget - xPosition;
yDist = yTarget - yPosition;
// Normalize the vector.
double length = Math.sqrt(xDist * xDist + yDist * yDist);
xDist /= length;
yDist /= length;
}
public void update() {
if (Math.abs(xTarget - xPosition) < 1)
xDist = 0;
if (Math.abs(yTarget - yPosition) < 1)
yDist = 0;
xPosition += xDist;
yPosition += yDist;
System.out.println(xPosition + " " + yPosition + " " + xDist + " " + yDist + " " + xTarget + " " + yTarget);
}
public void render(int[] pixels, int offset, int width, int height) {
if (xPosition < 0 || xPosition >= width)
return;
if (yPosition < 0 || yPosition >= height)
return;
pixels[(((int) xPosition) + offset) + ((int) yPosition) * width] = -1;
}
public void getInput(MouseInputHandler i) {
if (xTarget == -1 || yTarget == -1)
return;
xTarget = i.X / Game.SCALE;
yTarget = i.Y / Game.SCALE;
}
}

There is no trigonometric calculations involved, due to Math.atan2() function containing many special cases, and there's one of the special cases turning into the source of my logic error, resulting in a bug, as mentioned by most people in this thread. I didn't use plain vectors, due to me lacking information in this subject. In fact, I don't even know what it's called until I read the thread mentioned at the top of this post.

There is room for improvements in this class, but I'm pretty sure I have laid a nice looking foundation that works pretty nicely. All there's left to do, is to thank the people who helped in this thread.

In short, I have my problem solved, thanks to the people mentioned at the top.