# The line crossing the 3D point `p` and perpendicular to the plane spanned by 3D points `a`, `b`, and `c` An object of class `"Lines3D"`. Returns the equation, $x$-, $y$-, and $z$-coordinates of the line crossing 3D point `p` and perpendicular to the plane spanned by 3D points `a`, `b`, and `c` (i.e., the line is in the direction of normal `vector` of this plane) with the parameter `t` being provided in `vector` `t`. ```r perpline2plane(p, a, b, c, t) ``` ## Arguments - `p`: A 3D point through which the straight line passes. - `a, b, c`: 3D points which determine the plane to which the line passing through point `p` would be perpendicular (i.e., the normal `vector` of this plane 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 normal vector$=(A,B,C)$). ## 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 and plane, line crosses point `p` and plane is determined by 3D points `a`, `b`, and `c`. - **pnames**: The names of the input points that determine the line and plane; line would be perpendicular to the plane. - **vecs**: The point `p` and normal vector. - **vec.names**: The names of the point `p` and the second entry is "normal vector". - **x,y,z**: The $x$-, $y$-, and $z$-coordinates of the point(s) of interest on the line perpendicular to the plane determined by points `a`, `b`, and `c`. - **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 normal vector$=(A,B,C)$. - **equation**: Equation of the line passing through point `p` and perpendicular to the plane determined by points `a`, `b`, and `c` (i.e., line is in the direction of the normal `vector` N of the plane). 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 normal vector$=(A,B,C)$. ## Examples ```r P<-c(1,1,1); Q<-c(1,10,4); R<-c(1,1,3); S<-c(3,9,12) cf<-as.numeric(Plane(Q,R,S,1,1)$coeff) a<-cf[1]; b<-cf[2]; c<- -1; vecs<-rbind(Q,c(a,b,c)) pts<-rbind(P,Q,R,S) perpline2plane(P,Q,R,S,.1) tr<-range(pts,vecs); tf<-(tr[2]-tr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate tsq<-seq(-tf*10-tf,tf*10+tf,l=5) #try also l=10, 20, or 100 pln2pl<-perpline2plane(P,Q,R,S,tsq) pln2pl summary(pln2pl) plot(pln2pl,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE) xc<-pln2pl$x yc<-pln2pl$y zc<-pln2pl$z zr<-range(zc) zf<-(zr[2]-zr[1])*.2 Rv<- -c(a,b,c)*zf*5 Dr<-(Q+R+S)/3 pts2<-rbind(Q,R,S) xr<-range(pts2[,1],xc); yr<-range(pts2[,2],yc) xf<-(xr[2]-xr[1])*.1 #how far to go at the lower and upper ends in the x-coordinate yf<-(yr[2]-yr[1])*.1 #how far to go at the lower and upper ends in the y-coordinate xs<-seq(xr[1]-xf,xr[2]+xf,l=5) #try also l=10, 20, or 100 ys<-seq(yr[1]-yf,yr[2]+yf,l=5) #try also l=10, 20, or 100 plQRS<-Plane(Q,R,S,xs,ys) z.grid<-plQRS$z Xlim<-range(xc,xs,pts[,1]) Ylim<-range(yc,ys,pts[,2]) Zlim<-range(zc,z.grid,pts[,3]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] zd<-Zlim[2]-Zlim[1] plot3D::persp3D(z = z.grid, x = xs, y = ys, theta =225, phi = 30, main="Line Crossing P and \n Perpendicular to the Plane Defined by Q, R, S", col="lightblue", ticktype = "detailed", xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05), zlim=Zlim+zd*c(-.05,.05)) #plane spanned by points Q, R, S plot3D::lines3D(xc, yc, zc, bty = "g",pch = 20, cex = 2,col="red", ticktype = "detailed",add=TRUE) plot3D::arrows3D(Dr[1],Dr[2],Dr[3],Dr[1]+Rv[1],Dr[2]+Rv[2], Dr[3]+Rv[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","S"),add=TRUE) plot3D::arrows3D(P[1],P[2],P[3]-zf,P[1],P[2],P[3],lty=2, add=TRUE) plot3D::text3D(P[1],P[2],P[3]-zf,labels="initial point",add=TRUE) plot3D::text3D(P[1],P[2],P[3]+zf/2,labels="P",add=TRUE) plot3D::arrows3D(Dr[1],Dr[2],Dr[3],Dr[1]+Rv[1]/2,Dr[2]+Rv[2]/2, Dr[3]+Rv[3]/2,lty=2, add=TRUE) plot3D::text3D(Dr[1]+Rv[1]/2,Dr[2]+Rv[2]/2,Dr[3]+Rv[3]/2, labels="normal vector",add=TRUE) ``` ## See Also `Line3D`, `paraline3D`, and `perpline` ## Author(s) Elvan Ceyhan

perpline2plane function