# The line crossing the 3D point `p` and parallel to line joining 3D points `a` and `b` An object of class `"Lines3D"`. Returns the equation, $x$-, $y$-, and $z$-coordinates of the line crossing 3D point `p` and parallel to the line joining 3D points `a` and `b` (i.e., the line is in the direction of `vector` `b`-`a`) with the parameter `t` being provided in `vector` `t`. ```r paraline3D(p, a, b, t) ``` ## Arguments - `p`: A 3D point through which the straight line passes. - `a, b`: 3D points which determine the straight line to which the line passing through point `p` would be parallel (i.e., $b-a$ determines the direction of the straight line passing through `p`). - `t`: A scalar or a `vector` of scalars representing the parameter of the coordinates of the line (for the form: $x=p_0 + A t$, $y=y_0 + B t$, and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and $b-a=(A,B,C)$). ## Returns A `list` with the elements - **desc**: A description of the line - **mtitle**: The `"main"` title for the plot of the line - **points**: The input points that determine the line to which the line crossing point `p` would be parallel. - **pnames**: The names of the input points that determine the line to which the line crossing point `p` would be parallel. - **vecs**: The points `p`, `a`, and `b` stacked row-wise in this order. - **vec.names**: The names of the points `p`, `a`, and `b`. - **x,y,z**: The $x$-, $y$-, and $z$-coordinates of the point(s) of interest on the line parallel to the line determined by points `a` and `b`. - **tsq**: The scalar or the `vector` of the parameter in defining each coordinate of the line for the form: $x=p_0 + A t$, $y=y_0 + B t$, and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and $b-a=(A,B,C)$. - **equation**: Equation of the line passing through point `p` and parallel to the line joining points `a` and `b` (i.e., in the direction of the `vector` `b`-`a`). The line equation is in the form: $x=p_0 + A t$, $y=y_0 + B t$, and $z=z_0 + C t$ where $p=(p_0,y_0,z_0)$ and $b-a=(A,B,C)$. ## Examples ```r P<-c(1,10,4); Q<-c(1,1,3); R<-c(3,9,12) vecs<-rbind(P,R-Q) pts<-rbind(P,Q,R) paraline3D(P,Q,R,.1) tr<-range(pts,vecs); tf<-(tr[2]-tr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate tsq<-seq(-tf*10-tf,tf*10+tf,l=5) #try also l=10, 20, or 100 pln3D<-paraline3D(P,Q,R,tsq) pln3D summary(pln3D) plot(pln3D) x<-pln3D$x y<-pln3D$y z<-pln3D$z zr<-range(z) zf<-(zr[2]-zr[1])*.2 Qv<-(R-Q)*tf*5 Xlim<-range(x,pts[,1]) Ylim<-range(y,pts[,2]) Zlim<-range(z,pts[,3]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] zd<-Zlim[2]-Zlim[1] Dr<-P+min(tsq)*(R-Q) plot3D::lines3D(x, y, z, phi = 0, bty = "g", main="Line Crossing P \n in the direction of R-Q", xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.1,.1)+c(-zf,zf), pch = 20, cex = 2, ticktype = "detailed") plot3D::arrows3D(Dr[1],Dr[2],Dr[3]+zf,Dr[1]+Qv[1], Dr[2]+Qv[2],Dr[3]+zf+Qv[3], add=TRUE) plot3D::points3D(pts[,1],pts[,2],pts[,3],add=TRUE) plot3D::text3D(pts[,1],pts[,2],pts[,3],labels=c("P","Q","R"),add=TRUE) plot3D::arrows3D(P[1],P[2],P[3]-2*zf,P[1],P[2],P[3],lty=2, add=TRUE) plot3D::text3D(P[1],P[2],P[3]-2*zf,labels="initial point",add=TRUE) plot3D::arrows3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2, Dr[3]+3*zf+Qv[3]/2,Dr[1]+Qv[1]/2, Dr[2]+Qv[2]/2,Dr[3]+zf+Qv[3]/2,lty=2, add=TRUE) plot3D::text3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2,Dr[3]+3*zf+Qv[3]/2, labels="direction vector",add=TRUE) plot3D::text3D(Dr[1]+Qv[1]/2,Dr[2]+Qv[2]/2, Dr[3]+zf+Qv[3]/2,labels="R-Q",add=TRUE) ``` ## See Also `Line3D`, `perpline2plane`, and `paraline` ## Author(s) Elvan Ceyhan

paraline3D function