简单定向边界盒OBB冲突检测说明

Question

我可以实现AABB方法来检测碰撞，这很容易而且便宜，但我想实现OBB以获得更高的准确性，所以我创建了带有模型初始化的边界框，它由8个边界顶点和中心组成，每个帧我用转换矩阵转换所有顶点以适应定向边界框，但我无法理解检测两个OBB之间碰撞的方法，我无法理解找到一个简化而清晰的教程，用代码视角而不是数学来解释算法，因为我不是数学家。

如果我有

struct Box {
    glm::vec3 vertices[8];
    Box() {
        for (int i = 0; i < 8; i++) {
            vertices[i] = glm::vec3(0);
        }
    }
    glm::vec3 max;
    glm::vec3 min;
    glm::vec3 origin;

    void reCompute() {
        max = vertices[0];
        min = vertices[0];
        for (int i = 1; i < 8; i++) {
            max.x = max.x > vertices[i].x ? max.x : vertices[i].x;
            max.y = max.y > vertices[i].y ? max.y : vertices[i].y;
            max.z = max.z > vertices[i].z ? max.z : vertices[i].z;

            min.x = min.x < vertices[i].x ? min.x : vertices[i].x;
            min.y = min.y < vertices[i].y ? min.y : vertices[i].y;
            min.z = min.z < vertices[i].z ? min.z : vertices[i].z;
        }
        origin = glm::vec3((max.x + min.x) / 2.0f, (max.y + min.y) / 2.0f, (max.z + min.z) / 2.0f);
    }
//AABB intersection
    bool intersects(const Box &b) const {
        return (min.x < b.max.x) && (max.x > b.min.x) && (min.y < b.max.y) && (max.y > b.min.y) && (min.z < b.max.z) && (max.z > b.min.z) && *this != b;
    }

    bool operator==(const Box& b) const {
        return (max.x == b.max.x && max.y == b.max.y && max.z == b.max.z && min.x == b.min.x && min.y == b.min.y && min.z == b.min.z);
    }
    bool operator!=(const Box& b) const {
        return (max.x != b.max.x) || (max.y != b.max.y) || (max.z != b.max.z) || (min.x != b.min.x) || (min.y != b.min.y) || (min.z != b.min.z);
    }
};

在模型初始化时，我创建了框

    box.vertices[0] = glm::vec3(meshMinX, meshMinY, meshMinZ);
    box.vertices[1] = glm::vec3(meshMaxX, meshMinY, meshMinZ);
    box.vertices[2] = glm::vec3(meshMinX, meshMaxY, meshMinZ);
    box.vertices[3] = glm::vec3(meshMaxX, meshMaxY, meshMinZ);
    box.vertices[4] = glm::vec3(meshMinX, meshMinY, meshMaxZ);
    box.vertices[5] = glm::vec3(meshMaxX, meshMinY, meshMaxZ);
    box.vertices[6] = glm::vec3(meshMinX, meshMaxY, meshMaxZ);
    box.vertices[7] = glm::vec3(meshMaxX, meshMaxY, meshMaxZ);

对于每一帧，我用模型的变换矩阵重新计算盒子

for (int n = 0; n < 8; n++) {
        boxs[j].vertices[n] = glm::vec3(matrix * glm::vec4(box.vertices[n], 1));
    }
boxs[j].reCompute();

Answer 1

要知道两个OBB是否碰撞，你可以使用SAT(分离轴定理):你必须将两个形状的所有点投影到两个形状的每一条法线上。然后你会看到，如果两个形状的投影在每条法线上重叠，就会发生碰撞。如果至少有一条法线没有重叠，那么它们就不会碰撞。仅此而已，要做到这一点，你需要一个方法，将一个向量正交投影到另一个向量上，返回一个标量，还有一个方法，查看两个区间是否重叠。

我有一些Java代码：

U在V上的正交投影：

/**
 * Vec u is projected on Vec v
 * @param u 2d point
 * @param v 2d axe
 * @return the orthogonal projection
 */
public static float orthagonalProjectionOf(Vector2f u, Vector2f v){
    float norme_u = u.lenght();
    float norme_v = v.lenght();
    float dot_u_v = dot(u, v);
    float buffer = (dot_u_v/(norme_u*norme_v))*norme_u;
    if(Float.isNaN(buffer))return 0;//If the vector u is null, then is orthogonal projection is 0, not a NaN
    else return buffer;
}

两个间隔的重叠：

/**
 * Get the overlapping of two interval on an axis.
 * @param minA
 * @param maxA
 * @param minB
 * @param maxB
 * @return true overlapping. false if there is no overlapping
 */
public static boolean isOverlapping(float minA, float maxA, float minB, float maxB) {

    float minOverlap = Float.NaN;
    float maxOverlap = Float.NaN;


    //If B contain in A
    if(minA <= minB && minB <= maxA) {
        if(Float.isNaN(minOverlap) || minB < minOverlap)minOverlap = minB;
    }
    if(minA <= maxB && maxB <= maxA) {
        if(Float.isNaN(maxOverlap) || maxB > minOverlap)maxOverlap = maxB;
    }

    //If A contain in B
    if(minB <= minA && minA <= maxB) {
        if(Float.isNaN(minOverlap) || minA < minOverlap)minOverlap = minA;
    }
    if(minB <= maxA && maxA <= maxB) {
        if(Float.isNaN(maxOverlap) || maxA > minOverlap)maxOverlap = maxA;
    }

    if(Float.isNaN(minOverlap) || Float.isNaN(maxOverlap))return false; //Pas d'intersection
    else return true;//Intersection

}

有了这个，你就可以做一个方法来测试两个OBB之间的碰撞:

public boolean OBBwOBB(RigidBody bodyA, RigidBody bodyB) {
    Shape shapeA = bodyA.getObb().getShape();
    Shape shapeB = bodyB.getObb().getShape();

    short overlapCompt = 0;

    //We test for each normal the projection of the two shape
        //Shape A :
    for(int i = 0; i < shapeA.getNbrOfNormals(); i++) {
        Vector2f normal = shapeA.getNormal(i, bodyA.getAngle());
        boolean overlap = overlapOnThisNormal(bodyA, bodyB, normal);
        if(overlap) {
            overlapCompt++;
        }
    }
        //Shape B :
    for(int i = 0; i < shapeB.getNbrOfNormals(); i++) {
        Vector2f normal = shapeB.getNormal(i, bodyB.getAngle());
        boolean overlap = overlapOnThisNormal(bodyA, bodyB, normal);
        if(overlap){
            overlapCompt++;
        }
    }

    //Now we see if there is a collision
    short howManyNormals = (short) (shapeA.getNbrOfNormals() + shapeB.getNbrOfNormals());
    if(overlapCompt == howManyNormals){//If the number of overlap equal the number of normal in both shape :
        return true;
    }
    else return false;

}

您将需要它来获得投影在向量上的两个形状的最小和最大投影：

/**
 * Test if the orthogonal projection of two shape on a vector overlap.
 * @param bodyA
 * @param bodyB
 * @param normal
 * @return null if no overlap, else Vector2f(minOverlaping, maxOverlaping).
 */
public static boolean overlapOnThisNormal(RigidBody bodyA, RigidBody bodyB, Vector2f normal) {
    Shape shapeA = bodyA.getObb().getShape();
    Shape shapeB = bodyB.getObb().getShape();

    //We test each vertex of A
    float minA = Float.NaN;
    float maxA = Float.NaN;
    for(short j = 0; j < shapeA.getNbrOfPoint(); j++){
        Vector2f vertex = shapeA.getVertex(j, bodyA.getScale().x, bodyA.getScale().y, bodyA.getPosition().x, bodyA.getPosition().y, bodyA.getAngle());
        float bufferA = Vector2f.orthagonalProjectionOf(vertex, normal);
        if(Float.isNaN(minA) || bufferA < minA)minA = bufferA;//Set min interval
        if(Float.isNaN(maxA) || bufferA > maxA)maxA = bufferA;//Set max interval
    }

    //We test each vertex of B
    float minB = Float.NaN;
    float maxB = Float.NaN;
    for(short j = 0; j < shapeB.getNbrOfPoint(); j++){
        Vector2f vertex = shapeB.getVertex(j, bodyB.getScale().x, bodyB.getScale().y, bodyB.getPosition().x, bodyB.getPosition().y, bodyB.getAngle());
        float bufferB = Vector2f.orthagonalProjectionOf(vertex, normal);
        if(Float.isNaN(minB) || bufferB < minB)minB = bufferB;//Set min interval
        if(Float.isNaN(maxB) || bufferB > maxB)maxB = bufferB;//Set max interval
    }

    //We test if there overlap
    boolean overlap = isOverlapping(minA, maxA, minB, maxB);
    return overlap;
}

我希望这对您有所帮助；）

Answer 2

分离轴定理的C代码实现，用于两个3D OBB之间的简单碰撞检测，如下所示：

#include <iostream>

// define the operations to be used in our 3D vertices
struct vec3
{
    float x, y, z;
    vec3 operator- (const vec3 & rhs) const { return{ x - rhs.x, y - rhs.y, z - rhs.z }; }
    float operator* (const vec3 & rhs) const { return{ x * rhs.x + y * rhs.y + z * rhs.z }; } // DOT PRODUCT
    vec3 operator^ (const vec3 & rhs) const { return{ y * rhs.z - z * rhs.y, z * rhs.x - x * rhs.z, x * rhs.y - y * rhs.x }; } // CROSS PRODUCT
    vec3 operator* (const float& rhs)const { return vec3{ x * rhs, y * rhs, z * rhs }; }
};

// set the relevant elements of our oriented bounding box
struct OBB
{
    vec3 Pos, AxisX, AxisY, AxisZ, Half_size;
};

// check if there's a separating plane in between the selected axes
bool getSeparatingPlane(const vec3& RPos, const vec3& Plane, const OBB& box1, const OBB&box2)
{
    return (fabs(RPos*Plane) > 
        (fabs((box1.AxisX*box1.Half_size.x)*Plane) +
        fabs((box1.AxisY*box1.Half_size.y)*Plane) +
        fabs((box1.AxisZ*box1.Half_size.z)*Plane) +
        fabs((box2.AxisX*box2.Half_size.x)*Plane) + 
        fabs((box2.AxisY*box2.Half_size.y)*Plane) +
        fabs((box2.AxisZ*box2.Half_size.z)*Plane)));
}

// test for separating planes in all 15 axes
bool getCollision(const OBB& box1, const OBB&box2)
{
    static vec3 RPos;
    RPos = box2.Pos - box1.Pos;

    return !(getSeparatingPlane(RPos, box1.AxisX, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisY, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisZ, box1, box2) ||
        getSeparatingPlane(RPos, box2.AxisX, box1, box2) ||
        getSeparatingPlane(RPos, box2.AxisY, box1, box2) ||
        getSeparatingPlane(RPos, box2.AxisZ, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisX^box2.AxisX, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisX^box2.AxisY, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisX^box2.AxisZ, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisY^box2.AxisX, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisY^box2.AxisY, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisY^box2.AxisZ, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisZ^box2.AxisX, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisZ^box2.AxisY, box1, box2) ||
        getSeparatingPlane(RPos, box1.AxisZ^box2.AxisZ, box1, box2));
}

// a quick test to see the code working
int _tmain(int argc, _TCHAR* argv[])
{
    // create two obbs
    OBB A, B;

    // set the first obb's properties
    A.Pos = { 0.f, 0.f, 0.f }; // set its center position

    // set the half size
    A.Half_size.x = 10.f; 
    A.Half_size.y = 1.f; 
    A.Half_size.z = 1.f;

    // set the axes orientation
    A.AxisX = { 1.f, 0.f, 0.f };
    A.AxisY = { 0.f, 1.f, 0.f };
    A.AxisZ = { 0.f, 0.f, 1.f };

    // set the second obb's properties
    B.Pos = { 20.f, 0.f, 0.f }; // set its center position

    // set the half size
    B.Half_size.x = 10.f;
    B.Half_size.y = 1.f;
    B.Half_size.z = 1.f;

    // set the axes orientation
    B.AxisX = { 1.f, 0.f, 0.f };
    B.AxisY = { 0.f, 1.f, 0.f };
    B.AxisZ = { 0.f, 0.f, 1.f };

    // run the code and get the result as a message
    if (getCollision(A, B)) std::cout << "Collision!!!" << std::endl;
    else std::cout << "No collision." << std::endl;

    // pause and quit
    std::cout << std::endl;
    system("pause");
    return 0;
}