# Check whether a point is inside a tetrahedron Checks if the point `p` lies in the tetrahedron, `th`, using the barycentric coordinates, generally denoted as $(\alpha,\beta,\gamma)$. If all (normalized or non-normalized) barycentric coordinates are positive then the point `p` is inside the tetrahedron, if all are nonnegative with one or more are zero, then `p` falls on the boundary. If some of the barycentric coordinates are negative, then `p` falls outside the tetrahedron. `boundary` is a logical argument (default=`FALSE`) to include boundary or not, so if it is `TRUE`, the function checks if the point, `p`, lies in the closure of the tetrahedron (i.e., interior and boundary combined) else it checks if `p` lies in the interior of the tetrahedron. ```r in.tetrahedron(p, th, boundary = TRUE) ``` ## Arguments - `p`: A 3D point to be checked whether it is inside the tetrahedron or not. - `th`: A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron. - `boundary`: A logical parameter (default=`TRUE`) to include boundary or not, so if it is `TRUE`, the function checks if the point, `p`, lies in the closure of the tetrahedron (i.e., interior and boundary combined); else, it checks if `p` lies in the interior of the tetrahedron. ## Returns A `list` with two elements - **in.tetra**: A logical output, if the point, `p`, is inside the tetrahedron, `th`, it is `TRUE`, else it is `FALSE`. - **barycentric**: The barycentric coordinates of the point `p` with respect to the tetrahedron, `th`. ## Examples ```r A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3); P<-c(.1,.1,.1) tetra<-rbind(A,B,C,D) in.tetrahedron(P,tetra,boundary = FALSE) in.tetrahedron(C,tetra) in.tetrahedron(C,tetra,boundary = FALSE) n1<-5; n2<-5; n<-n1+n2 Xp<-rbind(cbind(runif(n1),runif(n1,0,sqrt(3)/2),runif(n1,0,sqrt(6)/3)), runif.tetra(n2,tetra)$g) in.tetra<-vector() for (i in 1:n) {in.tetra<-c(in.tetra,in.tetrahedron(Xp[i,],tetra,boundary = TRUE)$in.tetra) } in.tetra dat.tet<-Xp[in.tetra,] if (is.vector(dat.tet)) {dat.tet<-matrix(dat.tet,nrow=1)} Xlim<-range(tetra[,1],Xp[,1]) Ylim<-range(tetra[,2],Xp[,2]) Zlim<-range(tetra[,3],Xp[,3]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] zd<-Zlim[2]-Zlim[1] plot3D::scatter3D(Xp[,1],Xp[,2],Xp[,3], phi=40,theta=40, bty = "g", pch = 20, cex = 1, ticktype="detailed",xlim=Xlim+xd*c(-.05,.05), ylim=Ylim+yd*c(-.05,.05),zlim=Zlim+zd*c(-.05,.05)) #add the vertices of the tetrahedron plot3D::points3D(tetra[,1],tetra[,2],tetra[,3], add=TRUE) plot3D::points3D(dat.tet[,1],dat.tet[,2],dat.tet[,3],pch=4, add=TRUE) L<-rbind(A,A,A,B,B,C); R<-rbind(B,C,D,C,D,D) plot3D::segments3D(L[,1], L[,2], L[,3], R[,1], R[,2],R[,3], add=TRUE,lwd=2) plot3D::text3D(tetra[,1],tetra[,2],tetra[,3], labels=c("A","B","C","D"), add=TRUE) in.tetrahedron(P,tetra) #this works fine ``` ## See Also `in.triangle` ## Author(s) Elvan Ceyhan

in.tetrahedron function