找到两个3D多边形的交集

Question

a.k.a。 3D中的多边形裁剪算法

a.k.a。找到2个碰撞多边形之间的碰撞流形

多边形裁剪的大多数算法都针对2D进行了详细描述，并描述为可扩展到3D但没有细节。例如sutherland-hodgman clipping algorithm

我无法在互联网上找到任何3D实现或伪代码，我现在在这里问（并尝试回答我自己的问题）

该算法将采用两种形状，如下所示：

并输出两个形状的交集，如下所示：

请注意，尽管Sutherland-Hodgman算法找到了两个多边形的交集，但它（和大多数其他多边形）在剪切多边形和剪切多边形之间进行了区分。修剪的多边形可以是凹的或凸的，但是修剪的形状必须是凸的。 然而，我在扩展到3D时的实现要求两个形状都是凸，这表明它不是真正的3D sutherland-hodgman算法，而其他答案（使用任何算法）解除这个要求将非常多理解

Answer 1

2D sutherland-hodgman剪切算法通过剪裁形状的每个边缘剪切剪裁形状的每个边缘。然而，3D算法通过裁剪形状的每个面（不是剪裁形状的每个面的每个边缘）剪切剪裁形状的每个面的每个边缘。

我的算法受到这种方法的“启发”，但我必须添加一个额外的步骤才能使所有边缘正确显示出来。基本上我剪了两种方式;通过另一个剪切一个形状然后反向并添加两个;这导致要求两个形状都是凸的。未能包含这种“双向”错过了一些面孔，如下所示

此算法的伪代码如下

    for each clippiING face
        for each clippED face
            for each edge of each clippED face
                clip by clippiING face as per www.jhave.org/learner/misc/sutherlandhodgman/sutherlandhogdmanclipping.shtml
            end
        end

        for each edge of each clippiNG face(this step leads to requirement that both shapes be convex)
            clip clippED face by clippiING face as per www.jhave.org/learner/misc/sutherlandhodgman/sutherlandhogdmanclipping.shtml
        end
    end

这个算法在java中得到了证实，如下所示： （注意JMonkey引擎用于3D可视化，但是被标记为可以很容易地被剥离出来）

import java.util.ArrayList;

//VISUALISATION ONLY IMPORTS ###############
//REQUIRES: JMonkey Engine
import com.jme3.asset.AssetManager;
import com.jme3.math.ColorRGBA;
import com.jme3.scene.Node;

/**
 *
 * @author Richard Tingle
 */

//NOTE: Right handed co-ordinates set; x,y as normal, z towards us

public class Polygon3D {
    //based on sudocode from http://jhave.org/learner/misc/sutherlandhodgman/sutherlandhodgmanclipping.shtml

    ArrayList<Face> faces=new ArrayList<Face>(); 

    public Polygon3D(){} 

    public Polygon3D(ArrayList<Face> faces){
        this.faces=faces;
    }

    public int getNumberOfFaces(){
        return faces.size();
    }

    public void addFace(Face face){
        //for building face by face
        faces.add(face);
    }

    public Face getFace(int faceNumber){
        return faces.get(faceNumber);
    }

    public Polygon3D clip(Polygon3D clippingPolygon){
        Polygon3D workingPolygon=this;

        for(int i=0; i<clippingPolygon.getNumberOfFaces();i++){
            workingPolygon=clip(workingPolygon, clippingPolygon.getFace(i));
        }

        return workingPolygon;
    }

    private Polygon3D clip(Polygon3D inPolygon, Face clippingFace){
        //each edges of each face of the inPolygon is clipped by the clipping face

        Polygon3D outPolygon=new Polygon3D();

        for(int i=0;i<inPolygon.getNumberOfFaces();i++){
            Face clippedFace=inPolygon.getFace(i).clipFace(clippingFace);
            if (clippedFace!=null){
                outPolygon.addFace(clippedFace);
            }
        }

        //additional step, clipping face is also clipped by the inPolygon
        //this step is what causes the requirement for both clipping and clipped 
        //shapes to be convex
        Face workingFace=clippingFace;
        for(int i=0;i<inPolygon.getNumberOfFaces();i++){
            if (workingFace==null){
                //no need for bonus face in this case
                break;
            }
            workingFace=workingFace.clipFace(inPolygon.getFace(i));
        }
        if (workingFace!=null){
            outPolygon.addFace(workingFace);
        }


        return outPolygon;
    }

    //VISUALISATION ONLY ###############
    //REQUIRES: JMonkey Engine

    public void render(AssetManager assetManager, Node node, ColorRGBA color){
        for(int i=0;i<getNumberOfFaces();i++){
            node.attachChild(getFace(i).getGeometry(assetManager,color));
        }
    }

}

  

import java.util.ArrayList;
import javax.vecmath.Vector3d;

//VISUALISATION ONLY IMPORTS ###############
//REQUIRES: JMonkey Engine
import com.jme3.asset.AssetManager;
import com.jme3.material.Material;
import com.jme3.math.ColorRGBA;
import com.jme3.scene.Geometry;
import com.jme3.scene.Mesh;
import com.jme3.scene.VertexBuffer;

/**
 *
 * @author Richard Tingle
 */
public class Face{
    ArrayList<Vector3d> verticies=new ArrayList<Vector3d>(); 
    //NOTE: Face assumes that all its verticies are in the same place, this
    //is not checked and failure to comform to this will lead to errors
    //Face MUST have at least 3 verticies by the time it is used, and the
    //face itself must be convex. Vertex winding must be anticlockwise, but 
    //a function rewind is available to rewind if clockwise winding is used
    //clockwise or anticlockwise winding must be used, randomly putting in 
    //verticies will not end well

    public Face(){};
    public Face(ArrayList<Vector3d> verticies){this.verticies=verticies;}

    public void addVertex(Vector3d vertex){
        if ( Double.isNaN(vertex.x)){
            throw new RuntimeException("NaN Vertex");
        }
        if ( Double.isInfinite(vertex.x)|| Double.isInfinite(vertex.y)|| Double.isInfinite(vertex.z)){
            throw new RuntimeException("infinite Vertex");
        }


        if (verticies.contains(vertex)){
            //degenerate vertex, do not add
        }else{
            verticies.add(vertex);
        }


    }

    public void rewind(Vector3d internalPoint){
        //the winding of the verticies MUST be such that it looks anticlockwise
        //from the "outside", however, this method allows points to be added with
        //either clockwise or anticlockwise winding and then a final point that is 
        //anywhere on the inside of the shape specified in this method and if the
        //wrong winding was used this rewinds it to anticlockwise winding

        if (pointIsInsideFace(internalPoint)==false){
            //winding is incorrect, reverese winding
            ArrayList<Vector3d> verticiesRewound=new ArrayList<Vector3d>(verticies.size());
            for(int i=verticies.size()-1;i>=0;i--){
                verticiesRewound.add(verticies.get(i));
            }
            verticies=verticiesRewound;
        }
    }

    public int getNumberOfEdges(){
        return verticies.size(); //(note because the last vertex connects to the first noOfEdges==noOfVerticies)
    }
    public Vector3d getVertex(int vertex){
        return verticies.get(vertex);
    }

    public Vector3d getStartOfEdge(int edgeNo){
        return verticies.get(edgeNo);
    }
    public Vector3d getEndOfEdge(int edgeNo){
        return verticies.get((edgeNo+1)%verticies.size()); //%verticies.size() allows loop around for last edge
    }

    private double getPointVsFaceDeterminant(Vector3d point){
        //this method is a bit meaningless but its used in 
        //pointIsInsideFace(Vector3d point)
        //and
        //getIntersectionPoint

        //the returned determinant is basically a measure of which side
        //(and how far) a point lies from the plane

        //FOR THIS TO WORK FACE MUST HAVE ANTICLOCKWISE WINDING WHEN LOOKED AT
        //FROM OUTSIDE

        //we define faces as having their verticies in such an order that
        //when looked at from the outside the points are ordered anticlockswise
        //SO this function is equivalent to: pointIsInsideShape

        //see http://math.stackexchange.com/questions/214187/point-on-the-left-or-right-side-of-a-plane-in-3d-space

        //assuming face is convex, we only need the first 3 points to determine
        //the "winding" of the face

        if (verticies.size()<3){
            throw new RuntimeException("Degenerate Face: Face has less than 3 verticies");
        }

        Vector3d a=verticies.get(0);
        Vector3d b=verticies.get(1);
        Vector3d c=verticies.get(2);
        Vector3d x=point;

        Vector3d bDash=new Vector3d();
        bDash.sub(b, x);

        Vector3d cDash=new Vector3d();
        cDash.sub(c, x);

        Vector3d xDash=new Vector3d();
        xDash.sub(x, a);

        //find determinant of the 3 by 3 matrix described in link (see also: http://www.mathsisfun.com/algebra/matrix-determinant.html)
        double determinant=bDash.x*(cDash.y*xDash.z-cDash.z*xDash.y)-bDash.y*(cDash.x*xDash.z-cDash.z*xDash.x)+bDash.z*(cDash.x*xDash.y-cDash.y*xDash.x);

        return determinant;
    }

    public boolean pointIsInsideFace(Vector3d point){
        //FOR THIS TO WORK FACE MUST HAVE ANTICLOCKWISE WINDING WHEN LOOKED AT
        //FROM OUTSIDE

        //we define faces as having their verticies in such an order that
        //when looked at from the outside the points are ordered anticlockswise
        //SO this function is equivalent to: pointIsInsideShape

        //see http://math.stackexchange.com/questions/214187/point-on-the-left-or-right-side-of-a-plane-in-3d-space

        //find determinant of the 3 by 3 matrix described in link (see also: http://www.mathsisfun.com/algebra/matrix-determinant.html)
        double determinant=getPointVsFaceDeterminant(point);
        if (determinant<=0){
            // <= because we define on the face to be "inside the face"
            return true;
        }else{
            return false;
        }


    }

    public Vector3d getIntersectionPoint(Vector3d rayPoint1, Vector3d rayPoint2){
        //NOTE: This method treats the face as if it was an infinite plane
        //this treating as a plane is why convex shapes must be used


        //see http://mathworld.wolfram.com/Plane.html
        //changed from above method as that can get upset with parallel lines

        double determinantPoint1=getPointVsFaceDeterminant(rayPoint1);
        double determinantPoint2=getPointVsFaceDeterminant(rayPoint2);

        if (determinantPoint1==determinantPoint2){
            //paralell line, if we've got into this method then it'll probably
            //be in the plane, the line is in the plane, the middle seems the
            //most reasonable point

            Vector3d average=new Vector3d();
            average.add(rayPoint1, rayPoint2);
            average.scale(0.5);

            return average;
        }else{
            //we want to return the point where the determinant would have been
            //zero

            //linear interpolation
            Vector3d intersect=new Vector3d();
            intersect.sub(rayPoint2, rayPoint1);
            intersect.scale((0-determinantPoint1)/(determinantPoint2-determinantPoint1));
            intersect.add(rayPoint1);

            return intersect;
        }

    }

    public Face clipFace(Face clippingFace){
        //based on sudocode from www.jhave.org/learner/misc/sutherlandhodgman/sutherlandhogdmanclipping.shtml

        //Note, this face may be entirely clipped by the clipping face
        //or clipped to a degenerate edge, in this case null is returned

        Face workingFace=new Face();

        for(int i=0;i<this.getNumberOfEdges();i++){
            //clips all the edges of the working polygon against a plane based upon the clipping face
            //for each edge there are 4 cases, we must determine which it is
            //where we refer to starting and ending verticies they are of workingFace
            //where we refer to "the Face" that is the clipping face
            //and endEdge. The edge of the clipping polygon
            //case 1) both starting verticies are inside face
            //case 2) starting vertex is inside face, ending vertex is inside
            //case 3) Both verticies are outside the face
            //case 4) starting is outside the face, ending is inside

            Vector3d point1=getStartOfEdge(i);
            Vector3d point2=getEndOfEdge(i);

            if (clippingFace.pointIsInsideFace(point1) && clippingFace.pointIsInsideFace(point2)){
                //case 1, the end point is added
                workingFace.addVertex(point2);
            }else if (clippingFace.pointIsInsideFace(point1) && clippingFace.pointIsInsideFace(point2)==false){
                //case 2, only the intersection is added
                Vector3d intersection=clippingFace.getIntersectionPoint(point1, point2);
                workingFace.addVertex(intersection);
            }else if (clippingFace.pointIsInsideFace(point1)==false && clippingFace.pointIsInsideFace(point2)==false){
                //case 3, both verticies are outside the clip shape line, no vertexes added
            }else{
                //case 4 the ending vertex is inside and the starting vertex is outside
                //the line
                //the intercept and the end point are added
                Vector3d intersection=clippingFace.getIntersectionPoint(point1, point2);

                boolean one=clippingFace.pointIsInsideFace(point1);
                boolean two=clippingFace.pointIsInsideFace(point2);

                one=clippingFace.pointIsInsideFace(point1);
                two=clippingFace.pointIsInsideFace(point2);

                intersection=clippingFace.getIntersectionPoint(point1, point2);

                workingFace.addVertex(intersection);
                workingFace.addVertex(point2);
            }

        }

        if (workingFace.getNumberOfEdges()>=3){
            return workingFace;
        }else{
            return null; //degenerate or completely culled face
        }

    }

    private int getNumberOfSegments(){
        return verticies.size()-2;
    }


    //VISUALISATION ONLY ###############
    //REQUIRES: JMonkey Engine

    public Geometry getGeometry(AssetManager assetManager,ColorRGBA color){
        Mesh m = new Mesh();
        m.setMode(Mesh.Mode.LineLoop);

        float[] verticiesForBuffer=new float[3*(getNumberOfEdges())];
        int[] indicies=new int[getNumberOfEdges()];
        for(int i=0;i<getNumberOfEdges();i++){
            Vector3d vertex=getVertex(i);
            verticiesForBuffer[i*3]=(float)vertex.x;
            verticiesForBuffer[i*3+1]=(float)vertex.y;
            verticiesForBuffer[i*3+2]=(float)vertex.z;
            indicies[i]=i;
        }

        m.setBuffer(VertexBuffer.Type.Position, 3, verticiesForBuffer);
        m.setBuffer(VertexBuffer.Type.Index, 1, indicies);

        m.updateBound();
        m.updateCounts();

        Geometry geom = new Geometry("Box", m);

        Material mat = new Material(assetManager, "Common/MatDefs/Misc/Unshaded.j3md");
        mat.setColor("Color", color);
        geom.setMaterial(mat);

        return geom;
    }

}

  

以下主类创建显示的图像并使用JMonkey引擎库

呈现它们

import javax.vecmath.Vector3d;

//VISUALISATION ONLY IMPORTS ###############
//REQUIRES: JMonkey Engine
import com.jme3.app.SimpleApplication;
import com.jme3.math.ColorRGBA;
import com.jme3.renderer.RenderManager;

public class Main extends SimpleApplication {

        public static void main(String[] args) {
        Main app = new Main();
        app.start();

    }

    @Override
    public void simpleInitApp(){
        Polygon3D poly1= getCubePolygon(new Vector3d(-2,-2,-2),new Vector3d(2,2,2),0.5);
        Polygon3D poly2= getCubePolygon(new Vector3d(-1,-5,-1),new Vector3d(1,5,1),-2.5);
        Polygon3D poly3= poly1.clip(poly2);


        //poly1.render(assetManager, rootNode, ColorRGBA.Blue); //comment out to see individual shapes
        //poly2.render(assetManager, rootNode, ColorRGBA.Green);
        poly3.render(assetManager, rootNode,ColorRGBA.Red);
    }


    public Polygon3D getCubePolygon(Vector3d mins, Vector3d maxs, double xSkew){
        //xSkew makes the top and bottom x different (so its not actually a cube)
        Vector3d hhh=new Vector3d(maxs.x+xSkew,maxs.y,maxs.z);
        Vector3d hhl=new Vector3d(maxs.x+xSkew,maxs.y,mins.z);
        Vector3d hlh=new Vector3d(maxs.x-xSkew,mins.y,maxs.z);
        Vector3d hll=new Vector3d(maxs.x-xSkew,mins.y,mins.z);
        Vector3d lhh=new Vector3d(mins.x+xSkew,maxs.y,maxs.z);
        Vector3d lhl=new Vector3d(mins.x+xSkew,maxs.y,mins.z);
        Vector3d llh=new Vector3d(mins.x-xSkew,mins.y,maxs.z);
        Vector3d lll=new Vector3d(mins.x-xSkew,mins.y,mins.z);

        Vector3d centre=new Vector3d(0.5*(mins.x+maxs.x),0.5*(mins.y+maxs.y),0.5*(mins.z+maxs.z)); //just for rewinding

        Face top=new Face();
        Face bottom=new Face();
        Face north=new Face();
        Face south=new Face();
        Face east=new Face();
        Face west=new Face();

        north.addVertex(hll);
        north.addVertex(hhl);
        north.addVertex(hhh);
        north.addVertex(hlh);
        north.rewind(centre);

        south.addVertex(lll);
        south.addVertex(lhl);
        south.addVertex(lhh);
        south.addVertex(llh);
        south.rewind(centre);

        top.addVertex(hhh);
        top.addVertex(hhl);
        top.addVertex(lhl);
        top.addVertex(lhh);
        top.rewind(centre);

        bottom.addVertex(hlh);
        bottom.addVertex(hll);
        bottom.addVertex(lll);
        bottom.addVertex(llh);
        bottom.rewind(centre);

        east.addVertex(hhh);
        east.addVertex(hlh);
        east.addVertex(llh);
        east.addVertex(lhh);
        east.rewind(centre);

        west.addVertex(hhl);
        west.addVertex(hll);
        west.addVertex(lll);
        west.addVertex(lhl);
        west.rewind(centre);

        Polygon3D poly=new Polygon3D();
        poly.addFace(top);
        poly.addFace(bottom);
        poly.addFace(north);
        poly.addFace(south);
        poly.addFace(east);
        poly.addFace(west);


        return poly;
    }

}