从在C或C 3D空间执行3点建立圈++(Build Circle from 3 Points in

我们有定义3D空间中的圆圈3（三）XYZ点，这个圈子需要转换成折线（进一步渲染）。我在寻找一个现成的C或C ++函数或免费的图书馆，可以做的工作。

不明白为什么这是封闭的。我甚至不能回答我的问题在那里。可耻的是你们。但你不会停止传播知识！

Answer 1:

还有一个更简单的解决方案，以找到真正的3D圆的参数，只是来看看，在“重心坐标”部分http://en.wikipedia.org/wiki/Circumscribed_circle 。您可以提取，下面的优化代码：

// 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，您可以估算出真正的3D圆点过，并如在OpenGL使用GL_LINE_STRIP吸引他们。这应该是比使用2D罪快得多/ COS接近。

// 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
}

为了创建的变换矩阵（即makeRotMatrix4d（））参见http://paulbourke.net/geometry/rotate/例如。

请注意，我没有测试，如果上面的代码编译真的，但它应该给你足够的提示。

Answer 2:

有一个很好的文章，并就如何在2D，XY平面3点建立一个圆代码示例。

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

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

要建立一个3D圈，我们就必须：

旋转我们的3个点为XY平面
计算圆心
建立XY平面圆在文章中使用的代码
旋转它放回它的原平面

对于旋转最好是使用四元数。

为了找到一个正确的四元数我看着OGRE3D源代码：无效四元数:: FromAngleAxis（const的弧度及rfAngle，常量的Vector3＆rkAxis）

还有一个有用的功能有：四元getRotationTo（常量的Vector3＆DEST，常量的Vector3＆fallbackAxis = :: Vector3类型ZERO）const的，但我没有使用它。

对于quaterions和载体，我用我们自己的类。下面是做这项工作的功能的完整源代码：

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);
    }
}