java - 3D 坐标旋转缺乏透视 2D 精度

Question

所以我编写了一个程序来绘制和显示 3D 立方体，使用等 Axis 测图中使用的这些简单的转换公式:

x2 = x*cos(30) - y*cos(30)
y2 = x*sin(30) + y*sin(30) + z

坐标转换很好，一切都清晰可见。

问题是旋转，大角度旋转会弄乱所有坐标并给我一个完整的形状。并且多次以小角度旋转(即 1000 次 1 度旋转或更多)会减小立方体的大小。

public void rotateX(double dg) //cube is shrinking along y and z  
{
    y = (y*Math.cos(dg)-z*Math.sin(dg));  
    z = (y*Math.sin(dg)+z*Math.cos(dg));  
}
public void rotateY(double dg) //cube is shrinking along x and z  
{  
    x = x*Math.cos(dg)-z*Math.sin(dg);  
    z = x*Math.sin(dg)+z*Math.cos(dg);  
}  
public void rotateZ(double dg) //cube is shrinking along x and y  
{  
    x = x*Math.cos(dg)-y*Math.sin(dg);  
    y = x*Math.sin(dg)+y*Math.cos(dg);  
}

如何解决多次使用后 cos 和 sin 精度不足的问题？

这是用 3 个单独的类编写的完整代码:
主类:

import java.awt.*;
import javax.swing.*;
import java.util.Random;
public class Frame extends JFrame
{
    private Random rnd = new Random();
    private cubeGUI cube;
    public Frame()
    {
        super();
    }

    public void paint(Graphics g)
    {
        cube = new cubeGUI(75,300.0,300.0);
        cube.convertall();
        double dg = 0.5; // The Smaller the degree, the less the error after long rotations.
        int sl = 5;
        int turns, axe;  
        while (1 == 1)
        {
            turns = rnd.nextInt(200)-100;
            axe = rnd.nextInt(3);
            for(int i = 0; i<turns; i++)
            {
                switch (axe)
                {
                    case 0: cube.rotatx(dg); break;
                    case 1: cube.rotaty(dg); break;
                    case 2: cube.rotatz(dg); break;
                }
                g.clearRect(0,0,600,600);
                g.drawLine(cube.a.x2,cube.a.y2,cube.b.x2,cube.b.y2);
                g.drawLine(cube.a.x2,cube.a.y2,cube.c.x2,cube.c.y2);
                g.drawLine(cube.c.x2,cube.c.y2,cube.d.x2,cube.d.y2);
                g.drawLine(cube.b.x2,cube.b.y2,cube.d.x2,cube.d.y2);
                g.drawLine(cube.e.x2,cube.e.y2,cube.f.x2,cube.f.y2);
                g.drawLine(cube.e.x2,cube.e.y2,cube.g.x2,cube.g.y2);
                g.drawLine(cube.g.x2,cube.g.y2,cube.h.x2,cube.h.y2);
                g.drawLine(cube.f.x2,cube.f.y2,cube.h.x2,cube.h.y2);
                g.drawLine(cube.a.x2,cube.a.y2,cube.e.x2,cube.e.y2);
                g.drawLine(cube.b.x2,cube.b.y2,cube.f.x2,cube.f.y2);
                g.drawLine(cube.c.x2,cube.c.y2,cube.g.x2,cube.g.y2);
                g.drawLine(cube.d.x2,cube.d.y2,cube.h.x2,cube.h.y2);
                try
                {
                    Thread.sleep(sl); //Rotation Speed, In relation with Angle of rotation.
                } catch(InterruptedException ex)
                {
                    Thread.currentThread().interrupt();
                }
            }
        }
}
public static void main(String[] args)
{
    Frame cube = new Frame();
        cube.setSize(600,600);
        cube.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        cube.setVisible(true);
    }
}

立方体类:

public class cubeGUI
{
    public Point center,a,b,c,d,e,f,g,h;
    private double x, y;
    public cubeGUI(int m, double x, double y)
    {
        this.x = x;
        this.y = y;
        a = new Point(-m,-m,-m);
        b = new Point(m,-m,-m);
        c = new Point(-m,m,-m);
        d = new Point(m,m,-m);
        e = new Point(-m,-m,m);
        f = new Point(m,-m,m);
        g = new Point(-m,m,m);
        h = new Point(m,m,m);
    }
    public void rotatx(double dg)
    {
        a.rotateX(Math.toRadians(dg));
        b.rotateX(Math.toRadians(dg));
        c.rotateX(Math.toRadians(dg));
        d.rotateX(Math.toRadians(dg));
        e.rotateX(Math.toRadians(dg));
        f.rotateX(Math.toRadians(dg));
        g.rotateX(Math.toRadians(dg));
        h.rotateX(Math.toRadians(dg));
        convertall();
    }
    public void rotaty(double dg)
    {
        a.rotateY(Math.toRadians(dg));
        b.rotateY(Math.toRadians(dg));
        c.rotateY(Math.toRadians(dg));
        d.rotateY(Math.toRadians(dg));
        e.rotateY(Math.toRadians(dg));
        f.rotateY(Math.toRadians(dg));
        g.rotateY(Math.toRadians(dg));
        h.rotateY(Math.toRadians(dg));
        convertall();
    }
    public void rotatz(double dg)
    {
        a.rotateZ(Math.toRadians(dg));
        b.rotateZ(Math.toRadians(dg));
        c.rotateZ(Math.toRadians(dg));
        d.rotateZ(Math.toRadians(dg));
        e.rotateZ(Math.toRadians(dg));
        f.rotateZ(Math.toRadians(dg));
        g.rotateZ(Math.toRadians(dg));
        h.rotateZ(Math.toRadians(dg));
        convertall();
    }
    public void convertall()
    {
        a.convert(x,y);
        b.convert(x,y);
        c.convert(x,y);
        d.convert(x,y);
        e.convert(x,y);
        f.convert(x,y);
        g.convert(x,y);
        h.convert(x,y);
    }
}

点类(计算所有坐标):

public class Point
{
    private double x, y, z, F;
    public int x2, y2;
    public Point(double a, double b, double c)
    {
        x = a;
        y = b;
        z = c;
    }
    public int getX()
    {
        return (int)x;
    }
    public int getY()
    {
        return (int)y;
    }
    public int getZ()
    {
        return (int)z;
    }
    public void rotateX(double dg) //cube is shrinking along y and z
    {
        y = (y*Math.cos(dg)-z*Math.sin(dg));
        z = (y*Math.sin(dg)+z*Math.cos(dg));
    }
    public void rotateY(double dg) //cube is shrinking along x and z
    {
        x = x*Math.cos(dg)-z*Math.sin(dg);
        z = x*Math.sin(dg)+z*Math.cos(dg);
    }
    public void rotateZ(double dg) //cube is shrinking along x and y
    {
        x = x*Math.cos(dg)-y*Math.sin(dg);
        y = x*Math.sin(dg)+y*Math.cos(dg);
    }
    public void convert(double xx, double yy)
    {
        x2 = (int)(-(Math.cos(Math.toRadians(30))*x - Math.cos(Math.toRadians(30))*y) + xx);
        y2 = (int)(-(Math.sin(Math.toRadians(30))*x + Math.sin(Math.toRadians(30))*y + z) + yy);
    }
    public String toString()
    {
        return ("Y = " + y + ", Z = " + z);
    }
}

Answer 1

通常的方法是将立方体表示为点配置和当前变换。旋转时，更新变换，但不更新点本身。仅当需要点坐标时(用于渲染、显示坐标值等)，才应将变换应用于点。点本身不应被修改。

这将消除按顺序应用多次旋转时累积的错误。然而，重要的是变换矩阵应保持为旋转(行列式 1)。否则，转换仍会引入随机伪影(缩放、倾斜或其他扭曲)。因此，在应用每次旋转之后，应该对变换矩阵进行重新归一化，以使其保持纯变换。标准化可以像将每个条目除以行列式一样简单。