Seamlessly connecting centers

Question

I'm trying to make a path through the inside of the tube, along the center. The data I have to work with is the center points of the circles that describe the pipe at the beginning and end of each pipe rotation.

Drawing a path through straight pipe sections is trivial, but I'm not sure how to approach the bends. Any rotation between two circles must have a constant radius. Therefore, I have access to two points on this circle and the direction of the circle line at this point.

Does anyone know how I will calculate the rest of the circle from this?

Edit:

Attached photorealistic sketch of a pipe.

So, pretend that the curve does not squeamish, like hell, the blue lines indicate the circle, red - the center, and green - the path through the center.

Answer 1

• Lightening

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

  

• Decision

  Use cube interpolation like this

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

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

  Now some general 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 had not made a stupid mistake, then the coefficients:

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

  So, now just code the generalized equations into some function with the input parameter t and output p(t) and n(t) and / or adjust the segment of the circle or tube and call it for the cycle loop, for example, as follows:

   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’s best to look when the normals have an M_PI (3.1415...) size M_PI (3.1415...) , this is the code above:

the code for my use of my vector library, so you just need to do the following functions:

 vector_ld(a,x,y,z); //a[]={ x,y,z } vector_mul(a,b,c); //a[]=b[] xc[] 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 didn’t forget to copy something ...) ...