射线与椭球相交

Question

我正在尝试通过 "squishing" space 实现射线与椭球相交，并执行射线与球体：

创建 mat3 S，对角线处为椭圆体半径
通过将开始和方向乘以 S
在局部 space
将 hitPoint 乘以 S 以消除它。

这是射线与球体的对比：

float P = glm::dot(dir, sphereCenter-start);
float L = glm::distance(start, sphereCenter);
float d = sqrt(L*L - P*P);
if (d < radius) {
    float x0 = sqrt(1.f - d*d);
    hitPoint = start + dir*(P - x0);
    hitNormal = glm::normalize(hitPoint - sphereCenter);
}
else if (d == radius) {
    hitPoint = start + dir*P;
    hitNormal = glm::normalize(hitPoint - sphereCenter);
}
else {
    return false;
}
if (glm::distance(start, hitPoint) > dist) return false;
return true;

下面是压缩部分：

glm::vec3 S = start;
    glm::vec3 Dir = dir;

    auto sphereCenter = thisEntity()->transform()->getPosition();
    auto scale = thisEntity()->transform()->getScale();

    glm::mat3 q = glm::mat3(0);
    float x = _radius.x * scale.x;
    float y = _radius.y * scale.y;
    float z = _radius.z * scale.z;
    q[0][0] = x;
    q[1][1] = y;
    q[2][2] = z;
    glm::mat3 qI = glm::inverse(q);

    S = qI * S;
    Dir = qI * Dir;

    //calculate hit point in world space squished
    glm::vec3 hitPoint, hitNormal;
    if (!IntersectionsMath::instance()->segmentVsSphere(sphereCenter, S, Dir, dist, 1.f, hitPoint, hitNormal)) return;

    hitPoint = q * hitPoint;

    hit.pushHit(hitPoint, hitNormal, this);

当前的光线球体代码是针对世界位置的，我正试图让它在原点工作，所以它应该无关紧要。 Ray vs regular sphere 工作正常，椭圆体是问题所在。我在这上面花了很多时间，但有些地方出了问题。

Answer 1

问题：

The center of scaling matters.

解决方案：

Perform the scaling about the center of the ellipsoid.

... 而不是您现在正在做的原点。这是因为，尽管光线的方向相同（它只是一个方向向量），scaled 源和球体中心之间的相对位移将不同：

原点缩放（当前代码）：

来源S' = qI * S，中心C' = qI * C --- S' - C' = qI * (S - C)
关于椭球中心的缩放（正确程序）：

来源S" = qI * (S - C)，中心C" = C --- S" - C" = qI * (S - C) - C

两个位移的不同之处在于原始椭球的位置；因此您当前的射线可能会错过/给出误报。

更正后的代码：

// scale about the ellipsoid's position by subtracting before multiplying
// more appropriate name would be "ellipseCenter" to avoid confusion
S_ = qI * (S - sphereCenter);

// this ::normalize should really be in the intersection function
Dir_ = glm::normalize(qI * Dir); 

// calculate hit point in world space squished
// ... but around the origin in the squashed coordinate system
glm::vec3 hitPoint, hitNormal;
if (!IntersectionsMath::instance()->segmentVsSphere(
          glm::vec3::ZERO, S_, Dir_,
          dist, 1.f,
          hitPoint, hitNormal)) return;

// re-apply the offset
hitPoint = q * hitPoint + sphereCenter

// problem: hitNormal will not be correct for the ellipsoid when scaled
// solution: divide through each component by square of respective semi-axis
// (will provide proof upon request)
hitNormal.x /= (x * x); hitNormal.y /= (y * y); hitNormal.z /= (z * z);

射线与椭球相交

Ray vs ellipsoid intersection

c++

math

intersection