perp.ln2pl: The line crossing the 3D point `P` and perpendicular to the plane spanned by 3D points `A`, `B`, and `C`

Description

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.

Usage

perp.ln2pl(P, A, B, C, t)

Value

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: \(x=x_0 + a t\), \(y=y_0 + b t\), and \(z=z_0 + c t\) where \(P=(x_0,y_0,z_0)\) 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: \(x=x_0 + a t\), \(y=y_0 + b t\), and \(z=z_0 + c t\) where \(P=(x_0,y_0,z_0)\) and normal vector\(=(a,b,c)\).

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: \(x=x_0 + a t\), \(y=y_0 + b t\), and \(z=z_0 + c t\) where \(P=(x_0,y_0,z_0)\) and normal vector\(=(a,b,c)\)).

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
P<-c(1,1,1); A<-c(1,10,4); B<-c(1,1,3); C<-c(3,9,12)

cf<-as.numeric(Plane(A,B,C,1,1)$coeff)
a<-cf[1]; b<-cf[2]; c<- -1;

vecs<-rbind(A,c(a,b,c))
pts<-rbind(P,A,B,C)
perp.ln2pl(P,A,B,C,.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=20)  #try also l=100

pln2pl<-perp.ln2pl(P,A,B,C,tsq)
pln2pl
summary(pln2pl)
plot(pln2pl)

perp.ln2pl(P,A,B,C,c(.1,.2))

xc<-pln2pl$x
yc<-pln2pl$y
zc<-pln2pl$z

zr<-range(zc)
zf<-(zr[2]-zr[1])*.2
Bv<- -c(a,b,c)*zf*5

Dr<-(A+B+C)/3

pts2<-rbind(A,B,C)
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=20)  #try also l=100
ys<-seq(yr[1]-yf,yr[2]+yf,l=20)  #try also l=100

plABC<-Plane(A,B,C,xs,ys)
z.grid<-plABC$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,
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 A, B, C
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]+Bv[1],Dr[2]+Bv[2],Dr[3]+Bv[3], add=TRUE)
plot3D::points3D(pts[,1],pts[,2],pts[,3],add=TRUE)
plot3D::text3D(pts[,1],pts[,2],pts[,3],labels=c("P","A","B","C"),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]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+Bv[3]/2,lty=2, add=TRUE)
plot3D::text3D(Dr[1]+Bv[1]/2,Dr[2]+Bv[2]/2,Dr[3]+Bv[3]/2,labels="normal vector",add=TRUE)
# }