在 OpenGL 中绘制绕 Y 轴旋转的 Cog

Question

如标题所述：我想创建一个 3D 齿轮，它有 10 个齿，围绕其中心旋转（就像齿轮一样）。齿轮有方齿，为了简单起见，齿之间的边是平的——这个齿轮没有曲线。

齿轮一侧的可视化效果。请注意，角度并非 100% 完美。

根据上图，每个嵌齿轮齿都必须是 8 边多边形，而齿之间的每一边都必须是 4 边多边形。然而，现在，牙齿绘图执行以下操作：

尽管没有应用初始旋转，但应该面向相机的侧齿旋转到地板上。
牙齿的宽度是高度的两倍，尽管缩放和顶点创建是使用统一数字完成的（它在 x、y、z 方向上按 s 缩放，并使用 0 或 0.5 作为顶点坐标）。

完全可重现的示例：

import java.awt.Dimension;

import javax.swing.JFrame;

import com.jogamp.opengl.GL;
import com.jogamp.opengl.GL2;
import com.jogamp.opengl.GLAutoDrawable;
import com.jogamp.opengl.GLEventListener;
import com.jogamp.opengl.awt.GLJPanel;
import com.jogamp.opengl.glu.GLU;
import com.jogamp.opengl.util.FPSAnimator;

public class SpinCog3D implements GLEventListener {

    JFrame jf;
    GLJPanel gljpanel;
    Dimension dim = new Dimension(800, 600);
    FPSAnimator animator;

    float rotation;
    float speed;

    // set up the OpenGL Panel within a JFrame
    public SpinCog3D() {
        jf = new JFrame();
        gljpanel = new GLJPanel();
        gljpanel.addGLEventListener(this);
        gljpanel.requestFocusInWindow();
        jf.getContentPane().add(gljpanel);
        jf.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        jf.setVisible(true);
        jf.setPreferredSize(dim);
        jf.pack();
        animator = new FPSAnimator(gljpanel, 20);
        rotation = 0.0f;
        speed = 0.1f;
        animator.start();
    }

    public static void main(String[] args) {
        new SpinCog3D();
    }

    public void init(GLAutoDrawable dr) {
        GL2 gl2 = dr.getGL().getGL2();
        GLU glu = new GLU();
        gl2.glClearColor(1.0f, 1.0f, 1.0f, 0.0f);
        gl2.glEnable(GL2.GL_DEPTH_TEST);

        gl2.glMatrixMode(GL2.GL_PROJECTION);
        gl2.glLoadIdentity();
        glu.gluPerspective(60.0, 1.0, 100.0, 800.0);

    }

    public void display(GLAutoDrawable dr) {

        GL2 gl2 = dr.getGL().getGL2();
        GLU glu = new GLU();

        gl2.glMatrixMode(GL2.GL_MODELVIEW);
        gl2.glLoadIdentity();
        glu.gluLookAt(0.0, 200.0, 500.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);

        gl2.glClear(GL.GL_COLOR_BUFFER_BIT | GL.GL_DEPTH_BUFFER_BIT);

        // Draw 1 Cog Tooth + 1 Side
        drawTooth(gl2, 100.0, 1.0f, 0.0f, 0.0f, 0.0, 0.0, 1.0, 0.0, 0);
        drawSide(gl2, 100.0, 0.0f, 1.0f, 0.0f, 0.0, 0.0, 1.0, 0.0, 10);

        // Draw Floor
        sideRotatedColorScaledFloor(gl2, 300.0, 0.0f, 0.0f, 0.0f, 90.0, 1.0, 0.0, 0.0, 0.0);

        gl2.glFlush();
        rotation += speed;
        if (rotation > 360.9f)
            rotation = 0.0f;
    }

    // draw a single side with a set color and orientation
    private void drawTooth(GL2 gl2, double s, float r, float g, float b, double a, double ax, double ay,
                                        double az, double zoffset) {
        gl2.glPushMatrix();
        gl2.glRotated(a, ax, ay, az);
        gl2.glColor3f(r, g, b);
        gl2.glTranslated(0.0, 0.0, zoffset);
        gl2.glScaled(s, s, s);

        tooth(gl2);
        gl2.glPopMatrix();
    }

    private void drawSide(GL2 gl2, double s, float r, float g, float b, double a, double ax, double ay,
                           double az, double zoffset) {
        gl2.glPushMatrix();
        gl2.glRotated(a, ax, ay, az);
        gl2.glColor3f(r, g, b);
        gl2.glTranslated(0.0, 0.0, zoffset);
        gl2.glScaled(s, s, s);

        side(gl2);
        gl2.glPopMatrix();
    }

    private void sideRotatedColorScaledFloor(GL2 gl2, double s, float r, float g, float b, double a, double ax, double ay,
                                        double az, double zoffset) {
        gl2.glPushMatrix();
        gl2.glRotated(a, ax, ay, az);
        gl2.glColor3f(r, g, b);
        gl2.glTranslated(0.0, 0.0, zoffset);
        gl2.glScaled(s, s, s);

        side(gl2);
        gl2.glPopMatrix();
    }

    private void tooth(GL2 gl2) {
        gl2.glBegin(GL2.GL_POLYGON);
        gl2.glVertex3d(-0.5, -0.5, 0.0);
        gl2.glVertex3d(-0.5, 0.5, 0.0);
        gl2.glVertex3d(0.5, 0.5, 0.0);
        gl2.glVertex3d(0.5, -0.5, 0.0);
        gl2.glVertex3d(-0.5, -0.5, 0.5);
        gl2.glVertex3d(-0.5, 0.5, 0.5);
        gl2.glVertex3d(0.5, 0.5, 0.5);
        gl2.glVertex3d(0.5, -0.5, 0.5);
        gl2.glEnd();
    }

    private void side(GL2 gl2) {
        gl2.glBegin(GL2.GL_POLYGON);
        gl2.glVertex3d(-0.5, -0.5, 0.0);
        gl2.glVertex3d(-0.5, 0.5, 0.0);
        gl2.glVertex3d(0.5, 0.5, 0.0);
        gl2.glVertex3d(0.5, -0.5, 0.0);
        gl2.glEnd();
    }
}

是的，我知道 glBegin() 和 glEnd() 已被永远弃用，但这不是重点。如何让我的牙齿正确拔牙并与两侧正确对齐？

Answer 1

tooth drawing does all sorts of weird things...

不是很好的问题描述。我敢打赌你的网格有很多间隙和重叠，因为你使用的是硬编码形状，没有对 Cog 的实际尺寸进行任何修正。就像必须有多大的牙齿才能适合 n 个牙齿才能准确覆盖整个圆圈...

改用参数圆方程如何？

这将摆脱矩阵混乱，我认为会更简单。

所以我会简单地将齿轮分成三角形切片，每个切片都有几个边缘点。因此，将它们与 QUADs 连接起来，并将其放入 for 循环中执行所有切片。

我不在 JAVA 中编写代码，但这里使用旧 GL api 的小 C++ 示例，没有花哨的 C++ 内容，因此您应该能够轻松移植它：

void glCog(float r0,float r1,float r2,float w,int n)    // shaft/inner/outer radiuses, width, tooths
    {
    int i;
    float a,da,x,y,c,s;
    float p[6][3],q[6][3];  // slice points
    // set z for slice points
    a=-0.5*w; for (i=0;i<3;i++){ p[i][2]=a; q[i][2]=a; }
    a=+0.5*w; for (i=3;i<6;i++){ p[i][2]=a; q[i][2]=a; }
    // init first slice
    q[0][0]= r0; q[5][0]= r0;
    q[0][1]=0.0; q[5][1]=0.0;
    q[1][0]= r1; q[4][0]= r1;
    q[1][1]=0.0; q[4][1]=0.0;
    q[2][0]= r2; q[3][0]= r2;
    q[2][1]=0.0; q[3][1]=0.0;
    // divide circle to 2*n slices
    da=2.0*M_PI/float(4*n);
    glBegin(GL_QUADS);
    for (a=0.0,i=0;i<=n;i++)
        {
        // points on circles at angle a
        c=cos(a); s=sin(a); a+=da;
        x=r0*c; y=r0*s; p[0][0]=x; p[5][0]=x;
                        p[0][1]=y; p[5][1]=y;
        x=r1*c; y=r1*s; p[1][0]=x; p[4][0]=x;
                        p[1][1]=y; p[4][1]=y;
        x=r2*c; y=r2*s; p[2][0]=x; p[3][0]=x;
                        p[2][1]=y; p[3][1]=y;
        // render tooth
        c=cos(a); s=sin(a); a+=da;
        glNormal3f(0.0,0.0,-1.0);   // -Z base
        glVertex3fv(p[0]);
        glVertex3fv(p[2]);
        glVertex3fv(q[2]);
        glVertex3fv(q[0]);
        glNormal3f(0.0,0.0,+1.0);   // +Z base
        glVertex3fv(p[3]);
        glVertex3fv(p[5]);
        glVertex3fv(q[5]);
        glVertex3fv(q[3]);
        glNormal3f(-c,-s,0.0);      // shaft circumference side
        glVertex3fv(p[5]);
        glVertex3fv(p[0]);
        glVertex3fv(q[0]);
        glVertex3fv(q[5]);
        glNormal3f(c,s,0.0);        // outter circumference side
        glVertex3fv(p[2]);
        glVertex3fv(p[3]);
        glVertex3fv(q[3]);
        glVertex3fv(q[2]);
        glNormal3f(-s,c,0.0);
        glVertex3fv(p[4]);
        glVertex3fv(p[3]);
        glVertex3fv(p[2]);
        glVertex3fv(p[1]);
        glNormal3f(s,-c,0.0);
        glVertex3fv(q[1]);
        glVertex3fv(q[2]);
        glVertex3fv(q[3]);
        glVertex3fv(q[4]);

        // points on circles at angle a
        c=cos(a); s=sin(a); a+=da;
        x=r0*c; y=r0*s; q[0][0]=x; q[5][0]=x;
                        q[0][1]=y; q[5][1]=y;
        x=r1*c; y=r1*s; q[1][0]=x; q[4][0]=x;
                        q[1][1]=y; q[4][1]=y;
        x=r2*c; y=r2*s; q[2][0]=x; q[3][0]=x;
                        q[2][1]=y; q[3][1]=y;
        // render gap
        c=cos(a); s=sin(a); a+=da;
        glNormal3f(0.0,0.0,-1.0);   // -Z base
        glVertex3fv(q[0]);
        glVertex3fv(q[1]);
        glVertex3fv(p[1]);
        glVertex3fv(p[0]);
        glNormal3f(0.0,0.0,+1.0);   // +Z base
        glVertex3fv(q[4]);
        glVertex3fv(q[5]);
        glVertex3fv(p[5]);
        glVertex3fv(p[4]);
        glNormal3f(-c,-s,0.0);      // shaft circumference side
        glVertex3fv(q[5]);
        glVertex3fv(q[0]);
        glVertex3fv(p[0]);
        glVertex3fv(p[5]);
        glNormal3f(c,s,0.0);        // outter circumference side
        glVertex3fv(q[1]);
        glVertex3fv(q[4]);
        glVertex3fv(p[4]);
        glVertex3fv(p[1]);
        }
    glEnd();
    }

这里是 glCog(0.1,0.5,0.6,0.1,10); 的预览：

这里是 glCog(0.2,0.5,0.52,0.2,50); 的预览：

但是要注意牙齿并不是完全长方形的。齿数越少误差越大。如果你想要完全矩形的牙齿，你需要平移最外面的点而不是旋转（或更正它们计算的角度）

使用翻译来解决这个问题：

void glCog(float r0,float r1,float r2,float w,int n)    // shaft/inner/outer radiuses, width, tooths
    {
    int i,j;
    float a,da,dr,x,y,c,s;
    float p[6][3],q[6][3];  // slice points
    // divide circle to 2*n slices
    da=2.0*M_PI/float(4*n);
    dr=r2-r1;
    // set z for slice points
    a=-0.5*w; for (i=0;i<3;i++){ p[i][2]=a; q[i][2]=a; }
    a=+0.5*w; for (i=3;i<6;i++){ p[i][2]=a; q[i][2]=a; }
    // init first slice
    q[0][0]= r0; q[5][0]= r0;
    q[0][1]=0.0; q[5][1]=0.0;
    q[1][0]= r1; q[4][0]= r1;
    q[1][1]=0.0; q[4][1]=0.0;
    x=r1+dr*cos(-da); y=dr*sin(-da);
    q[2][0]=  x; q[3][0]=  x;
    q[2][1]=  y; q[3][1]=  y;
    glBegin(GL_QUADS);
    for (a=0.0,i=0;i<=n;i++)
        {
        // points on circles at angle a
        c=cos(a); s=sin(a);
        x=r0*c; y=r0*s; p[0][0]=x; p[5][0]=x;
                        p[0][1]=y; p[5][1]=y;
        x=r1*c; y=r1*s; p[1][0]=x; p[4][0]=x;
                        p[1][1]=y; p[4][1]=y;
        c=cos(a-da); s=sin(a-da); a+=da;
        x+=dr*c;y+=dr*s;p[2][0]=x; p[3][0]=x;
                        p[2][1]=y; p[3][1]=y;
        c=cos(a); s=sin(a); a+=da;
        // render tooth
        glNormal3f(0.0,0.0,-1.0);   // -Z base
        glVertex3fv(p[0]);
        glVertex3fv(p[1]);
        glVertex3fv(q[1]);
        glVertex3fv(q[0]);
        glVertex3fv(p[1]);
        glVertex3fv(p[2]);
        glVertex3fv(q[2]);
        glVertex3fv(q[1]);
        glNormal3f(0.0,0.0,+1.0);   // +Z base
        glVertex3fv(p[3]);
        glVertex3fv(p[4]);
        glVertex3fv(q[4]);
        glVertex3fv(q[3]);
        glVertex3fv(p[4]);
        glVertex3fv(p[5]);
        glVertex3fv(q[5]);
        glVertex3fv(q[4]);
        glNormal3f(-c,-s,0.0);      // shaft circumference side
        glVertex3fv(p[5]);
        glVertex3fv(p[0]);
        glVertex3fv(q[0]);
        glVertex3fv(q[5]);
        glNormal3f(c,s,0.0);        // outter circumference side
        glVertex3fv(p[2]);
        glVertex3fv(p[3]);
        glVertex3fv(q[3]);
        glVertex3fv(q[2]);
        glNormal3f(-s,c,0.0);
        glVertex3fv(p[4]);
        glVertex3fv(p[3]);
        glVertex3fv(p[2]);
        glVertex3fv(p[1]);
        glNormal3f(s,-c,0.0);
        glVertex3fv(q[1]);
        glVertex3fv(q[2]);
        glVertex3fv(q[3]);
        glVertex3fv(q[4]);

        // points on circles at angle a
        c=cos(a); s=sin(a);;
        x=r0*c; y=r0*s; q[0][0]=x; q[5][0]=x;
                        q[0][1]=y; q[5][1]=y;
        x=r1*c; y=r1*s; q[1][0]=x; q[4][0]=x;
                        q[1][1]=y; q[4][1]=y;
        c=cos(a+da); s=sin(a+da); a+=da;
        x+=dr*c;y+=dr*s;q[2][0]=x; q[3][0]=x;
                        q[2][1]=y; q[3][1]=y;
        c=cos(a); s=sin(a); a+=da;
        // render gap
        glNormal3f(0.0,0.0,-1.0);   // -Z base
        glVertex3fv(q[0]);
        glVertex3fv(q[1]);
        glVertex3fv(p[1]);
        glVertex3fv(p[0]);
        glNormal3f(0.0,0.0,+1.0);   // +Z base
        glVertex3fv(q[4]);
        glVertex3fv(q[5]);
        glVertex3fv(p[5]);
        glVertex3fv(p[4]);
        glNormal3f(-c,-s,0.0);      // shaft circumference side
        glVertex3fv(q[5]);
        glVertex3fv(q[0]);
        glVertex3fv(p[0]);
        glVertex3fv(p[5]);
        glNormal3f(c,s,0.0);        // outter circumference side
        glVertex3fv(q[1]);
        glVertex3fv(q[4]);
        glVertex3fv(p[4]);
        glVertex3fv(p[1]);
        }
    glEnd();
    }

所以只是点 p[2],q[2],p[3],q[3] 略有变化，基础侧必须用更多的 QUADS 来补偿 ...

这里是 glCog(0.2,0.5,0.6,0.2,10); 的预览：

在 OpenGL 中绘制绕 Y 轴旋转的 Cog

Drawing a Cog which rotates around the Y axis in OpenGL

java

opengl

opengl-compat