用 C 或 C++ 实现从 3D 空间中的 3 个点构建圆

问题描述 投票：0回答：3

我们有 3（三）个 xyz 点在 3D 空间中定义一个圆，这个圆需要转换为折线（用于进一步渲染）。我正在寻找可以完成这项工作的现成 C 或 C++ 函数或免费库。

不明白为什么this被关闭。我什至无法回答我自己的问题。真为你们感到羞耻。但你不会阻止知识传播！

c++ geometry
3个回答
12
投票

有一个更简单的解决方案来查找真实 3D 中的圆参数，只需查看 http://en.wikipedia.org/wiki/Circumscribed_circle 中的“重心坐标”部分即可。 您可以从中提取以下优化代码：

// triangle "edges"
const Vector3d t = p2-p1;
const Vector3d u = p3-p1;
const Vector3d v = p3-p2;

// triangle normal
const Vector3d w = t.crossProduct(u);
const double wsl = w.getSqrLength();
if (wsl<10e-14) return false; // area of the triangle is too small (you may additionally check the points for colinearity if you are paranoid)

// helpers
const double iwsl2 = 1.0 / (2.0*wsl);
const double tt = t*t;
const double uu = u*u;

// result circle
Vector3d circCenter = p1 + (u*tt*(u*v) - t*uu*(t*v)) * iwsl2;
double   circRadius = sqrt(tt * uu * (v*v) * iwsl2*0.5);
Vector3d circAxis   = w / sqrt(wsl);

然后您也可以在真实 3D 中计算圆上的点，例如在 OpenGL 中使用 GL_LINE_STRIP 绘制它们。这应该比使用 2D sin/cos 方法快得多。

// find orthogonal vector to the circle axis
const Vector3d an = circAxis.getNormalized();
const Vector3d ao = Vector3d(4.0+an[0], 4.0+an[0]+an[1], 4.0+an[0]+an[1]+an[2]).crossProduct(an).getNormalized();

// 4x4 rotation matrix around the circle axis
const int steps = 360; // maybe adjust according to circle size on screen
Matrix4d R = makeRotMatrix4d(circCenter, circAxis, 2.0*M_PI/double(steps));

// one point on the circle
Vector3d cp = circCenter + ao*circRadius;

// rotate point on the circle
for (int i=0; i<steps; ++i)
{
   circlePoints.push_back(cp);
   cp = transformPoint(cp, R); // apply the matrix
}

有关变换矩阵（即 makeRotMatrix4d()）的创建，请参见 http://paulbourke.net/geometry/rotate/ 。

请注意，我没有测试上面的代码是否真的可以编译，但它应该给你足够的提示。

6
投票

以下是Mark的answer的C#/Unity移植。它使用来自 Unity 脚本 API 的类型和实用函数。

// triangle "edges"
var t = p2 - p1;
var u = p3 - p1;
var v = p3 - p2;

// triangle normal
var w = Vector3.Cross(t, u);
var wsl = Vector3.Dot(w, w);
// TODO: if (wsl<10e-14) return false; // area of the triangle is too small (you may additionally check the points for colinearity if you are paranoid)

// helpers
var iwsl2 = 1f / (2f * wsl);
var tt = Vector3.Dot(t, t);
var uu = Vector3.Dot(u, u);

// result circle
Vector3 circCenter = p1 + (u * tt * (Vector3.Dot(u, v)) - t * uu * (Vector3.Dot(t, v))) * iwsl2;
var     circRadius = Mathf.Sqrt(tt * uu * (Vector3.Dot(v, v)) * iwsl2 * 0.5f);
Vector3 circAxis   = w / Mathf.Sqrt(wsl);

使用 Unity 的 Gizmos，可以如下绘制圆（在本例中使用 30 条线段来近似）：

// Draw the circle:
Gizmos.color = Color.white;
for (int i = 0; i < 30; ++i) 
{
    Gizmos.DrawLine(
        circCenter + Quaternion.AngleAxis(360f / 30f *  i     , circAxis) * (p1 - circCenter),
        circCenter + Quaternion.AngleAxis(360f / 30f * (i + 1), circAxis) * (p1 - circCenter)
    );
}

顶点位置的结果如下

  • var p1 = new Vector3(0f, 1.44f, 0f);
  • var p2 = new Vector3(0f, 0.73f, 0.65f);
  • var p3 = new Vector3(0f, -1.04f, 0f);
    ：

3
投票

有一篇不错的文章和代码示例介绍了如何在 2D、XY 平面上通过 3 个点构建圆。

http://paulbourke.net/geometry/circlesphere/

http://paulbourke.net/geometry/circlesphere/Circle.cpp

要构建 3D 圆，我们必须：

  • 将我们的 3 个点旋转到 XY 平面
  • 计算圆心
  • 使用文章中的代码在 XY 平面上构建一个圆
  • 将其旋转回原来的平面

对于旋转，最好使用四元数。

为了找到正确的四元数，我查看了 Ogre3d 源代码： void 四元数::FromAngleAxis (const Radian& rfAngle, const Vector3& rkAxis)

还有一个更有用的功能： 四元数 getRotationTo(const Vector3& dest, const Vector3&fallbackAxis = Vector3::ZERO) const 但我没用过。

对于四元数和向量，我使用我们自己的类。这是完成该工作的函数的完整源代码：

bool IsPerpendicular(Point3d *pt1, Point3d *pt2, Point3d *pt3);
double CalcCircleCenter(Point3d *pt1, Point3d *pt2, Point3d *pt3, Point3d *center);

void FindCircleCenter(const Point3d *V1, const Point3d *V2, const Point3d *V3, Point3d *center)
{
    Point3d *pt1=new Point3d(*V1);
    Point3d *pt2=new Point3d(*V2);
    Point3d *pt3=new Point3d(*V3);


    if (!IsPerpendicular(pt1, pt2, pt3) )       CalcCircleCenter(pt1, pt2, pt3, center);
    else if (!IsPerpendicular(pt1, pt3, pt2) )  CalcCircleCenter(pt1, pt3, pt2, center);
    else if (!IsPerpendicular(pt2, pt1, pt3) )  CalcCircleCenter(pt2, pt1, pt3, center);
    else if (!IsPerpendicular(pt2, pt3, pt1) )  CalcCircleCenter(pt2, pt3, pt1, center);
    else if (!IsPerpendicular(pt3, pt2, pt1) )  CalcCircleCenter(pt3, pt2, pt1, center);
    else if (!IsPerpendicular(pt3, pt1, pt2) )  CalcCircleCenter(pt3, pt1, pt2, center);
    else {
        delete pt1;
        delete pt2;
        delete pt3;
        return;
    }
    delete pt1;
    delete pt2;
    delete pt3;

}

bool IsPerpendicular(Point3d *pt1, Point3d *pt2, Point3d *pt3)
// Check the given point are perpendicular to x or y axis
{
    double yDelta_a= pt2->y - pt1->y;
    double xDelta_a= pt2->x - pt1->x;
    double yDelta_b= pt3->y - pt2->y;
    double xDelta_b= pt3->x - pt2->x;

    // checking whether the line of the two pts are vertical
    if (fabs(xDelta_a) <= 0.000000001 && fabs(yDelta_b) <= 0.000000001){
        return false;
    }

    if (fabs(yDelta_a) <= 0.0000001){
        return true;
    }
    else if (fabs(yDelta_b) <= 0.0000001){
        return true;
    }
    else if (fabs(xDelta_a)<= 0.000000001){
        return true;
    }
    else if (fabs(xDelta_b)<= 0.000000001){
        return true;
    }
    else
        return false ;
}

double CalcCircleCenter(Point3d *pt1, Point3d *pt2, Point3d *pt3, Point3d *center)
{
    double yDelta_a = pt2->y - pt1->y;
    double xDelta_a = pt2->x - pt1->x;
    double yDelta_b = pt3->y - pt2->y;
    double xDelta_b = pt3->x - pt2->x;

    if (fabs(xDelta_a) <= 0.000000001 && fabs(yDelta_b) <= 0.000000001){
        center->x= 0.5*(pt2->x + pt3->x);
        center->y= 0.5*(pt1->y + pt2->y);
        center->z= pt1->z;

        return 1;
    }

    // IsPerpendicular() assure that xDelta(s) are not zero
    double aSlope=yDelta_a/xDelta_a; //
    double bSlope=yDelta_b/xDelta_b;
    if (fabs(aSlope-bSlope) <= 0.000000001){    // checking whether the given points are colinear.
        return -1;
    }

    // calc center
    center->x= (aSlope*bSlope*(pt1->y - pt3->y) + bSlope*(pt1->x + pt2 ->x)
                         - aSlope*(pt2->x+pt3->x) )/(2* (bSlope-aSlope) );
    center->y = -1*(center->x - (pt1->x+pt2->x)/2)/aSlope +  (pt1->y+pt2->y)/2;

    return 1;
}

//! Builds a circle in 3D space by 3 points on it and an optional center
void buildCircleBy3Pt(const float *pt1,
                      const float *pt2,
                      const float *pt3,
                      const float *c,       // center, can be NULL
                      std::vector<float> *circle)
{
    /*  Get the normal vector to the triangle formed by 3 points
        Calc a rotation quaternion from that normal to the 0,0,1 axis
        Rotate 3 points using quaternion. Points will be in XY plane 
        Build a circle by 3 points on XY plane 
        Rotate a circle back into original plane using quaternion
     */
    Point3d p1(pt1[0], pt1[1], pt1[2]);
    Point3d p2(pt2[0], pt2[1], pt2[2]);
    Point3d p3(pt3[0], pt3[1], pt3[2]);
    Point3d center;
    if (c)
    {
        center.set(c[0], c[1], c[2]);
    }

    const Vector3d p2top1 = p1 - p2;
    const Vector3d p2top3 = p3 - p2;

    const Vector3d circle_normal = p2top1.crossProduct(p2top3).normalize();
    const Vector3d xy_normal(0, 0, 1);


    Quaternion rot_quat;
    // building rotation quaternion
    {
        // Rotation axis around which we will rotate our circle into XY plane
        Vector3d rot_axis = xy_normal.crossProduct(circle_normal).normalize();
        const double rot_angle = xy_normal.angleTo(circle_normal); // radians

        const double w = cos(rot_angle * 0.5);
        rot_axis *= sin(rot_angle * 0.5);

        rot_quat.set(w, rot_axis.x, rot_axis.y, rot_axis.z);
    }

    Quaternion rot_back_quat;
    // building backward rotation quaternion, same as prev. but -angle
    {
        const double rot_angle = -(xy_normal.angleTo(circle_normal)); // radians
        const double w_back = cos(rot_angle * 0.5);
        Vector3d rot_back_axis = xy_normal.crossProduct(circle_normal).normalize();
        rot_back_axis *= sin(rot_angle * 0.5);
        rot_back_quat.set(w_back, rot_back_axis.x, rot_back_axis.y, rot_back_axis.z);
    }

    rot_quat.rotate(p1);
    rot_quat.rotate(p2);
    rot_quat.rotate(p3);
    rot_quat.rotate(center);

    if (!c)
    {
        // calculate 2D center
        FindCircleCenter(&p1, &p2, &p3, &center);
    }

    // calc radius
    const double radius = center.distanceTo(p1);

    const float DEG2RAD = 3.14159f / 180.0f;
    // build circle
    for (int i = 0; i < 360; ++i)
    {
        float degInRad = i * DEG2RAD;
        Point3d pt(cos(degInRad) * radius + center.x, sin(degInRad) * radius + center.y, 0);

        // rotate the point back into original plane 
        rot_back_quat.rotate(pt);

        circle->push_back(pt.x);
        circle->push_back(pt.y);
        circle->push_back(pt.z);
    }
}
最新问题
© www.soinside.com 2019 - 2024. All rights reserved.