c++ - 如何获取 3D 世界中对象相对于父对象的绝对位置？

标签 c++ math 3d

我正在构建一个 3D 引擎只是为了学习，并且我有一个对象是另一个对象的父对象。这个父对象具有相对于父对象的位置。所以我想知道如果父对象有旋转和缩放，如何获取该对象位置。

我已经进入Unity来看看效果如何，我看到他们在旋转时使用了正弦和余弦，但我不知道如何。 我想使用父级的前向，但这只有在子级位置为 (0,0,1) 时才有效，对吧？因为如果它的位置不应该是什么？空间中两个位置之间唯一剩下的操作是将它们相乘(矩阵乘法)(因为将它们相加不会得到我们想要的)，但我想我只是放弃了这些选项。

我没有代码，因为我不知道从哪里开始，但主要是复制 Unity 的类和结构名称。 为了对相机进行投影，我使用了维基百科中的 3D 投影页面 ( https://en.wikipedia.org/wiki/3D_projection#Perspective_projection )。

我使用四元数进行旋转(还有讨论欧拉到四元数和四元数到欧拉的维基百科页面:https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles#Source_Code)。

我认为关于代码的信息已经足够了。

最后，我希望 (0,0,1) 中的子对象成为 (0,0,0) 中对象的父对象，该对象旋转到 (0,90,0) 子世界位置是 (1,0,0)。

我使用这种方法使用 3x3 矩阵而不是 4x4 矩阵，这些是函数:

inline Vector3 getPositionAbsoluteFromRelative(const Vector3 &relativeposition, const Vector3 &parentrotation) {
    return Vector3(
        relativeposition.x * (1 - 2 * (parentrotation.y * parentrotation.y + parentrotation.z * parentrotation.z)) + relativeposition.y * 2 * (parentrotation.x * parentrotation.y - parentrotation.w * parentrotation.z) + relativeposition.z * 2 * (parentrotation.x * parentrotation.z + parentrotation.w * parentrotation.y),
        relativeposition.x * 2 * (parentrotation.x * parentrotation.y + parentrotation.w * parentrotation.z) + relativeposition.y * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.z * parentrotation.z)) + relativeposition.z * 2 * (parentrotation.y * parentrotation.z - parentrotation.w * parentrotation.x),
        relativeposition.x * 2 * (parentrotation.x * parentrotation.z - parentrotation.w * parentrotation.y) + relativeposition.y * 2 * (parentrotation.y * parentrotation.z + parentrotation.w * parentrotation.x) + relativeposition.z * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.y * parentrotation.y))
    );
};

inline Vector3 getPositionAbsoluteFromRelative(const Vector3 &relativeposition, const Vector3 &parentrotation, const Vector3 &parentposition) {
    Vector3 absolute(
        relativeposition.x * (1 - 2 * (parentrotation.y * parentrotation.y + parentrotation.z * parentrotation.z)) + relativeposition.y * 2 * (parentrotation.x * parentrotation.y - parentrotation.w * parentrotation.z) + relativeposition.z * 2 * (parentrotation.x * parentrotation.z + parentrotation.w * parentrotation.y),
        relativeposition.x * 2 * (parentrotation.x * parentrotation.y + parentrotation.w * parentrotation.z) + relativeposition.y * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.z * parentrotation.z)) + relativeposition.z * 2 * (parentrotation.y * parentrotation.z - parentrotation.w * parentrotation.x),
        relativeposition.x * 2 * (parentrotation.x * parentrotation.z - parentrotation.w * parentrotation.y) + relativeposition.y * 2 * (parentrotation.y * parentrotation.z + parentrotation.w * parentrotation.x) + relativeposition.z * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.y * parentrotation.y))
    );

    return Vector3(
        absolute.x + parentposition.x,
        absolute.y + parentposition.y,
        absolute.z + parentposition.z
    );
};

inline Vector3 getPositionAbsoluteFromRelative(const Vector3 &relativeposition, const Vector3 &parentrotation, const Vector3 &parentposition, const Vector3 &parentscale) {
    Vector3 absolute(
        relativeposition.x * (1 - 2 * (parentrotation.y * parentrotation.y + parentrotation.z * parentrotation.z)) + relativeposition.y * 2 * (parentrotation.x * parentrotation.y - parentrotation.w * parentrotation.z) + relativeposition.z * 2 * (parentrotation.x * parentrotation.z + parentrotation.w * parentrotation.y),
        relativeposition.x * 2 * (parentrotation.x * parentrotation.y + parentrotation.w * parentrotation.z) + relativeposition.y * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.z * parentrotation.z)) + relativeposition.z * 2 * (parentrotation.y * parentrotation.z - parentrotation.w * parentrotation.x),
        relativeposition.x * 2 * (parentrotation.x * parentrotation.z - parentrotation.w * parentrotation.y) + relativeposition.y * 2 * (parentrotation.y * parentrotation.z + parentrotation.w * parentrotation.x) + relativeposition.z * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.y * parentrotation.y))
    );

    return Vector3(
        absolute.x * parentscale.x + parentposition.x,
        absolute.y * parentscale.y + parentposition.y,
        absolute.z * parentscale.z + parentposition.z
    );
};

inline Vector3 getPositionRelativeFromAbsolute(const Vector3 &absoluteposition, const Vector3 &parentrotation) {
    return Vector3(
        absoluteposition.x * (1 - 2 * (parentrotation.y * parentrotation.y + parentrotation.z * parentrotation.z)) + absoluteposition.y * 2 * (parentrotation.x * parentrotation.y + parentrotation.w * parentrotation.z) + absoluteposition.z * 2 * (parentrotation.x * parentrotation.z - parentrotation.w * parentrotation.y),
        absoluteposition.x * 2 * (parentrotation.x * parentrotation.y - parentrotation.w * parentrotation.z) + absoluteposition.y * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.z * parentrotation.z)) + absoluteposition.z * 2 * (parentrotation.y * parentrotation.z + parentrotation.w * parentrotation.x),
        absoluteposition.x * 2 * (parentrotation.x * parentrotation.z + parentrotation.w * parentrotation.y) + absoluteposition.y * 2 * (parentrotation.y * parentrotation.z - parentrotation.w * parentrotation.x) + absoluteposition.z * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.y * parentrotation.y))
    );
};

inline Vector3 getPositionRelativeFromAbsolute(const Vector3 &absoluteposition, const Vector3 &parentrotation, const Vector3 &parentposition) {
    Vector3 __position = {
        absoluteposition.x - parentposition.x,
        absoluteposition.y - parentposition.y,
        absoluteposition.z - parentposition.z
    };

    return Vector3(
        __position.x * (1 - 2 * (parentrotation.y * parentrotation.y + parentrotation.z * parentrotation.z)) + __position.y * 2 * (parentrotation.x * parentrotation.y + parentrotation.w * parentrotation.z) + __position.z * 2 * (parentrotation.x * parentrotation.z - parentrotation.w * parentrotation.y),
        __position.x * 2 * (parentrotation.x * parentrotation.y - parentrotation.w * parentrotation.z) + __position.y * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.z * parentrotation.z)) + __position.z * 2 * (parentrotation.y * parentrotation.z + parentrotation.w * parentrotation.x),
        __position.x * 2 * (parentrotation.x * parentrotation.z + parentrotation.w * parentrotation.y) + __position.y * 2 * (parentrotation.y * parentrotation.z - parentrotation.w * parentrotation.x) + __position.z * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.y * parentrotation.y))
    );
};

inline Vector3 getPositionRelativeFromAbsolute(const Vector3 &absoluteposition, const Vector3 &parentrotation, const Vector3 &parentposition, const Vector3 &parentscale) {
    Vector3 __position = {
        absoluteposition.x - parentposition.x,
        absoluteposition.y - parentposition.y,
        absoluteposition.z - parentposition.z
    };

    Vector3 relative(
        __position.x * (1 - 2 * (parentrotation.y * parentrotation.y + parentrotation.z * parentrotation.z)) + __position.y * 2 * (parentrotation.x * parentrotation.y + parentrotation.w * parentrotation.z) + __position.z * 2 * (parentrotation.x * parentrotation.z - parentrotation.w * parentrotation.y),
        __position.x * 2 * (parentrotation.x * parentrotation.y - parentrotation.w * parentrotation.z) + __position.y * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.z * parentrotation.z)) + __position.z * 2 * (parentrotation.y * parentrotation.z + parentrotation.w * parentrotation.x),
        __position.x * 2 * (parentrotation.x * parentrotation.z + parentrotation.w * parentrotation.y) + __position.y * 2 * (parentrotation.y * parentrotation.z - parentrotation.w * parentrotation.x) + __position.z * (1 - 2 * (parentrotation.x * parentrotation.x + parentrotation.y * parentrotation.y))
    );

    return Vector3(relative.x / parentscale.x, relative.y / parentscale.y, relative.z / parentscale.z);
};

最佳答案

3D 对象的位置和方向可以组合并存储在 4x4 矩阵中

  • 位置是从世界原点到对象原点的转换
  • 方向是围绕对象原点的旋转。

为什么是 4x4 矩阵？

所有常见的变换(平移、旋转、缩放、剪切、投影)都可以表示为这样的矩阵。

变换的串联相当于相应矩阵的乘法。

虽然将一系列变换函数组合成一个可能很困难(并且不灵活)，但很容易预乘矩阵(表示变换序列)，以便可以一次应用所有变换(通过一个矩阵乘法) )。

这就是 4×4 矩阵在 3D 合成中如此常见的原因。图形。

OP示例:

poschild' = Mparent · poschild

同时

M父级 = T父级 · R父级.

在代码中:

#include <iostream>

#include "linmath.h"

int main()
{
  Vec3f posChild(0.0f, 0.0f, 1.0f);

  Vec3f posParent(0.0f, 0.0f, 0.0f);
  float abcParent[] = { 0.0f, 90.0f, 0.0f };

  // child pos as homogeneous coordinate
  Vec4f posChildH(posChild, 1.0f);

  // compose parent matrix of pos and ori
  Mat4x4f matParent
    = Mat4x4f(InitTrans, posParent)
    * makeEuler(RotZYX,
      degToRad(abcParent[0]),
      degToRad(abcParent[1]),
      degToRad(abcParent[2]));

  // make posChildH global
  Vec4f posChildHW = matParent * posChildH;

  // homogeneous coordinate -> pos in 3d
  Vec3f posChildW(
    posChildHW.x / posChildHW.w,
    posChildHW.y / posChildHW.w,
    posChildHW.z / posChildHW.w);

  // print result
  std::cout << "posChild in WCS: " << std::fixed << posChildW << '\n';
}

输出:

posChild in WCS: ( 1.000000, 0.000000, -0.000000 )

Live Demo on Wandbox

linmath可以在 github: linmath.h 上找到, github: linmath.cc .

相关部分:

3d vector :

template <typename VALUE>
struct Vec3T {
  typedef VALUE Value;
  Value x, y, z;
  Vec3T(Value x, Value y, Value z): x(x), y(y), z(z) { }
};

typedef Vec3T<float> Vec3f;

4d vector (齐次坐标):

template <typename VALUE>
struct Vec4T {
  typedef VALUE Value;
  Value x, y, z, w;
  Vec4T(const Vec3T<Value> &xyz, Value w):
    x(xyz.x), y(xyz.y), z(xyz.z), w(w)
  { }
};

typedef Vec4T<float> Vec4f;

4×4矩阵:

enum ArgInitTrans { InitTrans };
enum ArgInitRot { InitRot };

template <typename VALUE>
struct Mat4x4T {
  union {
    VALUE comp[4 * 4];
    struct {
      VALUE _00, _01, _02, _03;
      VALUE _10, _11, _12, _13;
      VALUE _20, _21, _22, _23;
      VALUE _30, _31, _32, _33;
    };
  };

  // constructor to build a matrix for translation
  Mat4x4T(ArgInitTrans, const Vec3T<VALUE> &t):
    _00((VALUE)1), _01((VALUE)0), _02((VALUE)0), _03((VALUE)t.x),
    _10((VALUE)0), _11((VALUE)1), _12((VALUE)0), _13((VALUE)t.y),
    _20((VALUE)0), _21((VALUE)0), _22((VALUE)1), _23((VALUE)t.z),
    _30((VALUE)0), _31((VALUE)0), _32((VALUE)0), _33((VALUE)1)
  { }
  // constructor to build a matrix for rotation about axis
  Mat4x4T(ArgInitRot, const Vec3T<VALUE> &axis, VALUE angle):
    _03((VALUE)0), _13((VALUE)0), _23((VALUE)0),
    _30((VALUE)0), _31((VALUE)0), _32((VALUE)0), _33((VALUE)1)
  {
    //axis.normalize();
    const VALUE sinAngle = sin(angle), cosAngle = cos(angle);
    const VALUE xx = axis.x * axis.x, xy = axis.x * axis.y;
    const VALUE xz = axis.x * axis.z, yy = axis.y * axis.y;
    const VALUE yz = axis.y * axis.z, zz = axis.z * axis.z;
    _00 = xx + cosAngle * ((VALUE)1 - xx) /* + sinAngle * 0 */;
    _01 = xy - cosAngle * xy - sinAngle * axis.z;
    _02 = xz - cosAngle * xz + sinAngle * axis.y;
    _10 = xy - cosAngle * xy + sinAngle * axis.z;
    _11 = yy + cosAngle * ((VALUE)1 - yy) /* + sinAngle * 0 */;
    _12 = yz - cosAngle * yz - sinAngle * axis.x;
    _20 = xz - cosAngle * xz - sinAngle * axis.y;
    _21 = yz - cosAngle * yz + sinAngle * axis.x;
    _22 = zz + cosAngle * ((VALUE)1 - zz) /* + sinAngle * 0 */;
  }
  // multiply matrix with matrix -> matrix
  Mat4x4T operator * (const Mat4x4T &mat) const
  {
    return Mat4x4T(
      _00 * mat._00 + _01 * mat._10 + _02 * mat._20 + _03 * mat._30,
      _00 * mat._01 + _01 * mat._11 + _02 * mat._21 + _03 * mat._31,
      _00 * mat._02 + _01 * mat._12 + _02 * mat._22 + _03 * mat._32,
      _00 * mat._03 + _01 * mat._13 + _02 * mat._23 + _03 * mat._33,
      _10 * mat._00 + _11 * mat._10 + _12 * mat._20 + _13 * mat._30,
      _10 * mat._01 + _11 * mat._11 + _12 * mat._21 + _13 * mat._31,
      _10 * mat._02 + _11 * mat._12 + _12 * mat._22 + _13 * mat._32,
      _10 * mat._03 + _11 * mat._13 + _12 * mat._23 + _13 * mat._33,
      _20 * mat._00 + _21 * mat._10 + _22 * mat._20 + _23 * mat._30,
      _20 * mat._01 + _21 * mat._11 + _22 * mat._21 + _23 * mat._31,
      _20 * mat._02 + _21 * mat._12 + _22 * mat._22 + _23 * mat._32,
      _20 * mat._03 + _21 * mat._13 + _22 * mat._23 + _23 * mat._33,
      _30 * mat._00 + _31 * mat._10 + _32 * mat._20 + _33 * mat._30,
      _30 * mat._01 + _31 * mat._11 + _32 * mat._21 + _33 * mat._31,
      _30 * mat._02 + _31 * mat._12 + _32 * mat._22 + _33 * mat._32,
      _30 * mat._03 + _31 * mat._13 + _32 * mat._23 + _33 * mat._33);
  }
  // constructor to build a matrix for rotation about axis
  Mat4x4T(ArgInitRot, const Vec3T<VALUE> &axis, VALUE angle):
    _03((VALUE)0), _13((VALUE)0), _23((VALUE)0),
    _30((VALUE)0), _31((VALUE)0), _32((VALUE)0), _33((VALUE)1)
  {
    //axis.normalize();
    const VALUE sinAngle = sin(angle), cosAngle = cos(angle);
    const VALUE xx = axis.x * axis.x, xy = axis.x * axis.y;
    const VALUE xz = axis.x * axis.z, yy = axis.y * axis.y;
    const VALUE yz = axis.y * axis.z, zz = axis.z * axis.z;
    _00 = xx + cosAngle * ((VALUE)1 - xx) /* + sinAngle * 0 */;
    _01 = xy - cosAngle * xy - sinAngle * axis.z;
    _02 = xz - cosAngle * xz + sinAngle * axis.y;
    _10 = xy - cosAngle * xy + sinAngle * axis.z;
    _11 = yy + cosAngle * ((VALUE)1 - yy) /* + sinAngle * 0 */;
    _12 = yz - cosAngle * yz - sinAngle * axis.x;
    _20 = xz - cosAngle * xz - sinAngle * axis.y;
    _21 = yz - cosAngle * yz + sinAngle * axis.x;
    _22 = zz + cosAngle * ((VALUE)1 - zz) /* + sinAngle * 0 */;
}
  // multiply matrix with vector -> vector
  Vec4T<VALUE> operator * (const Vec4T<VALUE> &vec) const
  {
    return Vec4T<VALUE>(
      _00 * vec.x + _01 * vec.y + _02 * vec.z + _03 * vec.w,
      _10 * vec.x + _11 * vec.y + _12 * vec.z + _13 * vec.w,
      _20 * vec.x + _21 * vec.y + _22 * vec.z + _23 * vec.w,
      _30 * vec.x + _31 * vec.y + _32 * vec.z + _33 * vec.w);
  }
};

typedef Mat4x4T<float> Mat4x4f;

角度到弧度:

extern const double Pi;

template <typename VALUE>
inline VALUE degToRad(VALUE angle)
{
  return (VALUE)Pi * angle / (VALUE)180;
}

欧拉(和泰特-布莱恩)角:

// enumeration of rotation axes
enum RotAxis {
  RotX, // rotation about x axis
  RotY, // rotation about y axis
  RotZ // rotation about z axis
};

// enumeration of possible Euler angles
enum EulerAngle {
  RotXYX = RotX + 3 * RotY + 9 * RotX, // 0 + 3 + 0 = 3
  RotXYZ = RotX + 3 * RotY + 9 * RotZ, // 0 + 3 + 18 = 21
  RotXZX = RotX + 3 * RotZ + 9 * RotX, // 0 + 6 + 0 = 6
  RotXZY = RotX + 3 * RotZ + 9 * RotY, // 0 + 6 + 9 = 15
  RotYXY = RotY + 3 * RotX + 9 * RotY, // 1 + 0 + 9 = 10
  RotYXZ = RotY + 3 * RotX + 9 * RotZ, // 1 + 0 + 18 = 19
  RotYZX = RotY + 3 * RotZ + 9 * RotX, // 1 + 6 + 0 = 7
  RotYZY = RotY + 3 * RotZ + 9 * RotY, // 1 + 6 + 9 = 16
  RotZXY = RotZ + 3 * RotX + 9 * RotY, // 2 + 0 + 9 = 11
  RotZXZ = RotZ + 3 * RotX + 9 * RotZ, // 2 + 0 + 18 = 20
  RotZYX = RotZ + 3 * RotY + 9 * RotX, // 2 + 3 + 0 = 5
  RotZYZ = RotZ + 3 * RotY + 9 * RotZ, // 2 + 3 + 18 = 23
  RotHPR = RotZXY, // used in OpenGL Performer
  RotABC = RotZYX // used in German engineering
};

/* decomposes the combined EULER angle type into the corresponding
 * individual EULER angle axis types.
 */
inline void decompose(
  EulerAngle type, RotAxis &axis1, RotAxis &axis2, RotAxis &axis3)
{
  unsigned type_ = (unsigned)type;
  axis1 = (RotAxis)(type_ % 3); type_ /= 3;
  axis2 = (RotAxis)(type_ % 3); type_ /= 3;
  axis3 = (RotAxis)type_;
}

4×4 矩阵的欧拉角:

template <typename VALUE>
Mat4x4T<VALUE> makeEuler(
  EulerAngle mode, VALUE rot1, VALUE rot2, VALUE rot3)
{
  RotAxis axis1, axis2, axis3;
  decompose(mode, axis1, axis2, axis3);
  const static VALUE axes[3][3] = {
    { (VALUE)1, (VALUE)0, (VALUE)0 },
    { (VALUE)0, (VALUE)1, (VALUE)0 },
    { (VALUE)0, (VALUE)0, (VALUE)1 }
  };
  return
      Mat4x4T<VALUE>(InitRot,
        Vec3T<VALUE>(axes[axis1][0], axes[axis1][1], axes[axis1][2]),
        rot1)
    * Mat4x4T<VALUE>(InitRot,
        Vec3T<VALUE>(axes[axis2][0], axes[axis2][1], axes[axis2][2]),
        rot2)
    * Mat4x4T<VALUE>(InitRot,
        Vec3T<VALUE>(axes[axis3][0], axes[axis3][1], axes[axis3][2]),
        rot3);
}

glm 提供了一个类似但更全面的库。 .

关于c++ - 如何获取 3D 世界中对象相对于父对象的绝对位置？，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/56920622/

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