如何正确旋转一个矢量以匹配另一个矢量?(OPENGL）

一段时间以来，我一直在努力解决以下问题：我有一个程序，允许用户绘制各种长度的贝塞尔曲线(第一个4点，其他所有3点，简单地将曲线一个接一个地连接起来)。我需要沿着曲线放置小的矩形火车轨道，让用户在他们制作的轨道上骑行。我已经定义了顶点，并制定了一种方法将它们正确地放置在直线上，但旋转被证明是很棘手的。对于平滑曲线，我目前的实现效果很好，但锐角会导致轨迹不再对齐，并且经过该角的所有轨迹都具有完全相同的旋转，从而完全打断它。所有相关代码如下：

用点填充曲线std:：矢量的代码：

p0 = glm::vec2(pointVertexData.at(0), pointVertexData.at(1));
p1 = glm::vec2(pointVertexData.at(3), pointVertexData.at(4));
p2 = glm::vec2(pointVertexData.at(6), pointVertexData.at(7));
p3 = glm::vec2(pointVertexData.at(9), pointVertexData.at(10));
curveVertexData = Subdivide(0.0f, 1.0f, 0.05f, curveVertexData);
for (int i = 0; i < timesToLoop; i++)
{
p0 = p3;
p1 = glm::vec2(pointVertexData.at(n), pointVertexData.at(n+1));
p2 = glm::vec2(pointVertexData.at(n+3), pointVertexData.at(n+4));
p3 = glm::vec2(pointVertexData.at(n+6), pointVertexData.at(n+7));
std::vector<GLfloat> tempVec = Subdivide(0.0f, 1.0f, 0.05f, tempVec);
curveVertexData.insert(curveVertexData.end(), tempVec.begin()+3, tempVec.end());
}

细分代码：

std::vector<GLfloat> Subdivide(GLfloat u0, GLfloat u1, GLfloat maxLineLength, std::vector<GLfloat> recurVertices)
{
GLfloat umid = (u0 + u1) / 2.0;
glm::vec2 x0 = Interpolate(p0, p1, p2, p3, u0, pFinal);
glm::vec2 x1 = Interpolate(p0, p1, p2, p3, u1, pFinal);
GLfloat length = sqrt(pow((x1.x - x0.x), 2) + pow((x1.y - x0.y), 2));
if (length > maxLineLength)
{
std::vector<GLfloat> firstVertices = Subdivide(u0, umid, maxLineLength, firstVertices);
std::vector<GLfloat> secondVertices = Subdivide(umid, u1, maxLineLength, secondVertices);
secondVertices.insert(secondVertices.begin(), firstVertices.begin(), firstVertices.end()-3);
recurVertices = secondVertices;
return recurVertices;
}
else
{
recurVertices.push_back(x0.x);
recurVertices.push_back(x0.y);
recurVertices.push_back(0.1f);
recurVertices.push_back(x1.x);
recurVertices.push_back(x1.y);
recurVertices.push_back(0.1f);
numberOfVertices += 6;
return recurVertices;
}
}

用于设置带有轨迹顶点的std:：矢量的代码：

std::vector<GLfloat> tempVertices;
numberOfTrackVertices = 0;
for (int i = 0; i < curveVertexData.size() - 2; i+=3)
{
std::cout << "Now calculating point # " << i << " : ";
if(i != 0 && i < curveVertexData.size() - 5)
shiftVertices(trainVertices, curveVertexData[i], curveVertexData[i + 1], curveVertexData[i + 2], curveVertexData[i + 3], curveVertexData[i + 4], curveVertexData[i + 5], curveVertexData[i - 3], curveVertexData[i - 2], curveVertexData[i - 1], &tempVertices);
else if (i == 0)
shiftVertices(trainVertices, curveVertexData[i], curveVertexData[i + 1], curveVertexData[i + 2], curveVertexData[i + 3], curveVertexData[i + 4], curveVertexData[i + 5], curveVertexData[i], curveVertexData[i + 1], curveVertexData[i + 2], &tempVertices);
else
shiftVertices(trainVertices, curveVertexData[i], curveVertexData[i + 1], curveVertexData[i + 2], curveVertexData[i], curveVertexData[i + 1], curveVertexData[i + 2], curveVertexData[i - 3], curveVertexData[i - 2], curveVertexData[i - 1], &tempVertices);
}

最后，我认为最有可能是罪魁祸首的代码，用于改变轨道旋转的代码。我目前的算法如下：(请注意，将"currentOrientation"设置为等于顶点的前两个元素彼此相减的原因是，它们表示矩形的后下角，当它们彼此相减时，会给出一个表示长方体方向的向量)

void shiftVertices(GLfloat inVertices[], GLfloat x, GLfloat y, GLfloat z, GLfloat rx, GLfloat ry, GLfloat rz, GLfloat qx, GLfloat qy, GLfloat qz, std::vector<GLfloat> *container)
{
glm::vec3 tempVectors[36];
glm::vec3 moveVector = glm::vec3(x, y, z);
glm::vec3 rotateVector = glm::normalize(glm::vec3(rx - qx, ry - qy, rz - qz));
rotateVector = glm::normalize(glm::cross(rotateVector, UP));
bool unFilled = true;
int i = 0;
int n = 0;
while(unFilled)
{
tempVectors[n].x = inVertices[i];
i++;
tempVectors[n].y = inVertices[i];
i++;
tempVectors[n].z = inVertices[i];
i++;
n++;
if (n >= 36)
unFilled = false;
}
glm::vec3 currentOrientation = glm::normalize(tempVectors[0] - tempVectors[1]);
GLfloat angleToRotate = glm::acos(glm::dot(currentOrientation, rotateVector));
angleToRotate = (180.0f * angleToRotate) / PI;
std::cout << angleToRotate << "n";
glm::mat4 rotationMatrix;
rotationMatrix = glm::rotate(rotationMatrix, angleToRotate, UP);
for (int u = 0; u < 36; u++)
{
tempVectors[u] = glm::vec3(rotationMatrix * glm::vec4(tempVectors[u], 1.0));
tempVectors[u] = tempVectors[u] + moveVector;
}
i = 0;
n = 0;
unFilled = true;
while (unFilled)
{
container->push_back(tempVectors[n].x);
container->push_back(tempVectors[n].y);
container->push_back(tempVectors[n].z);
numberOfTrackVertices++;
n++;
if (n >= 36)
unFilled = false;
}
}

此实施产生以下结果：https://i.stack.imgur.com/h7gbr.jpg(除了最后一张图片。对不起，它不允许我嵌入图片)

我为此查阅了许多资源，但收效甚微。一个实现是Jur van den Berg对这个问题的回答：https://math.stackexchange.com/questions/180418/calculate-rotation-matrix-to-align-vector-a-to-vector-b-in-3d请注意，我在图像中称之为"斜对称方法"。我的算法实现如下，如前所述，无法正常工作：(注意，此代码替换了上一个示例中代码的中间部分，前后循环保持不变)

glm::vec3 crossVector = glm::cross(currentOrientation, rotateVector);
GLfloat sineAngle = crossVector.length();
GLfloat cosAngle = glm::dot(currentOrientation, rotateVector);
glm::mat3 experimentalRMatrix;
glm::mat3 skewSymmetric = { 0, (-1.0f * crossVector.z), crossVector.y,
crossVector.z, 0, (-1.0f *crossVector.x),
(-1.0f * crossVector.y), crossVector.x, 0 };
glm::mat3 skewSecond = skewSymmetric * skewSymmetric;
skewSecond = skewSecond * ((1.0f - cosAngle) / (sineAngle * sineAngle));
experimentalRMatrix = glm::mat3() + skewSymmetric + skewSecond;
testVector = experimentalRMatrix * currentOrientation;
rotationMatrix = glm::mat4(experimentalRMatrix);

所有这些都摆在桌面上，我希望能分析我解决问题的尝试失败的原因，和/或找到一个正确旋转顶点的解决方案。

谢谢。