Smoothly connecting circle centers

1. Clarification

   tube has the same circular diameter everywhere so no distortion due to bending is present !!! input is 2 endpoints points (centers of tube) P0,P1 and 2 vectors (normal/direction of tube) N0,N1

   

2. Solution

   Use Interpolation cubic for example this one

   p(t)=a0+a1*t+a2*t*t+a3*t*t*t
   t=<0,1.0>
   

   so write equations for the known data, solve a0,a1,a2,a3 coefficients for each axis you need (2D: x,y) and then you can get the center point and its normal in any point along the bend side which is what you need.

   Now some generic equations:

   p(t)=a0+a1*t+     a2*t*t+     a3*t*t*t // circle center position
   n(t)=   a1   +2.0*a2*t   +3.0*a3*t*t   // circle orientation
   

   • p,n,a0,a1,a2,a3 are vectors !!!
   • t is scalar

   Now add the known data

   I. t=0 -> p(0)=P0
   P0=a0
   a0=P0
   
   II. t=0 -> n(0)=N0
   N0=a1
   a1=N0
   
   III. t=1 -> p(1)=P1
   P1=a0+a1+a2+a3
   P1=P0+N0+a2+a3
   a2=P1-P0-N0-a3
   
   IV. t=1 -> n(1)=N1
   N1=a1+2.0*a2+3.0*a3
   N1=N0+2.0*(P1-P0-N0-a3)+3.0*a3
   a3=N1+N0-2.0*(P1-P0)
   
   III.
   a2=P1-P0-N0-(N1+N0-2.0*(P1-P0))
   a2=P1-P0-N0-N1-N0+2.0*(P1-P0)
   a2=P1-P0-N1+2.0*(P1-P0-N0)
   a2=3.0*(P1-P0)-N1-2.0*N0
   

   So if I did not make any silly mistake then coefficients are:

   a0=P0
   a1=N0
   a2=3.0*(P1-P0)-N1-2.0*N0
   a3=N1+N0-2.0*(P1-P0)
   

   So now just encode generic equations into some function with input parameter t and output p(t) and n(t) and/or render circle or tube segment and the call this in for loop for example like this:

   for (t=0.0;t<=1.0;t+=0.1) f(t);
   

[edit1] C++ implementation

//---------------------------------------------------------------------------
void glCircle3D(double *pos,double *nor,double r,bool _fill)
    {
    int i,n=36;
    double a,da=divide(pi2,n),p[3],dp[3],x[3],y[3];
         if (fabs(nor[0]-nor[1])>1e-6) vector_ld(x,nor[1],nor[0],nor[2]);
    else if (fabs(nor[0]-nor[2])>1e-6) vector_ld(x,nor[2],nor[1],nor[0]);
    else if (fabs(nor[1]-nor[2])>1e-6) vector_ld(x,nor[0],nor[2],nor[1]);
    else                       vector_ld(x,1.0,0.0,0.0);
    vector_mul(x,x,nor);
    vector_mul(y,x,nor);
    vector_len(x,x,r);
    vector_len(y,y,r);
    if (_fill)
        {
        glBegin(GL_TRIANGLE_FAN);
        glVertex3dv(pos);
        }
    else glBegin(GL_LINE_STRIP);
    for (a=0.0,i=0;i<=n;i++,a+=da)
        {
        vector_mul(dp,x,cos(a)); vector_add(p,pos,dp);
        vector_mul(dp,y,sin(a)); vector_add(p,p  ,dp);
        glVertex3dv(p);
        }
    glEnd();
    }
//---------------------------------------------------------------------------
void tube(double *P0,double *N0,double *P1,double *N1,double R)
    {
    int i;
    double a0[3],a1[3],a2[3],a3[3],p[3],n[3],t,tt,ttt;
    // compute coefficients
    for (i=0;i<3;i++)
        {
        a0[i]=P0[i];
        a1[i]=N0[i];
        a2[i]=(3.0*(P1[i]-P0[i]))-N1[i]-(2.0*N0[i]);
        a3[i]=N1[i]+N0[i]-2.0*(P1[i]-P0[i]);
        }
    // step through curve from t=0 to t=1
    for (t=0.0;t<=1.0;t+=0.02)
        {
        tt=t*t;
        ttt=tt*t;
        // compute circle position and orientation
        for (i=0;i<3;i++)
            {
            p[i]=a0[i]+(a1[i]*t)+(a2[i]*tt)+(a3[i]*ttt);
            n[i]=a1[i]+(2.0*a2[i]*t)+(3.0*a3[i]*tt);
            }
        // render it
        glCircle3D(p,n,R,false);
        }
    }
//---------------------------------------------------------------------------
void test()
    {
    // tube parameters
    double P0[3]={-1.0, 0.0, 0.0},N0[3]={+1.0,-1.0, 0.0},p[3];
    double P1[3]={+1.0,+1.0, 0.0},N1[3]={ 0.0,+1.0, 0.0};
    // just normalize normals to size 3.1415...
    vector_len(N0,N0,M_PI);
    vector_len(N1,N1,M_PI);
    // draw normals to visula confirmation of tube direction
    glBegin(GL_LINES);
    glColor3f(0.0,0.0,1.0); vector_add(p,P0,N0); glVertex3dv(P0); glVertex3dv(p);
    glColor3f(0.0,0.0,1.0); vector_add(p,P1,N1); glVertex3dv(P1); glVertex3dv(p);
    glEnd();
    // render tube
    glColor3f(1.0,1.0,1.0); tube(P0,N0,P1,N1,0.2);
    }
//---------------------------------------------------------------------------

Visually it best looks when normals are of size M_PI (3.1415...) this is how it looks for the code above:

the code of mine use mine vector library so you just need to code functions like:

vector_ld(a,x,y,z); //a[]={ x,y,z }
vector_mul(a,b,c);  //a[]=b[] x c[]
vector_mul(a,b,c);  //a[]=b[] * c
vector_add(a,b,c);  //a[]=b[] + c[]
vector_sub(a,b,c);  //a[]=b[] - c[]
vector_len(a,b,c);  //a[]=b[]*  c / |b[]|

which is easy (hope I did not forget to copy something …)…