paraplane: The plane at a point and parallel to the plane spanned by three distinct 3D points `a`, `b`, and `c`

Description

An object of class "Planes". Returns the equation and \(z\)-coordinates of the plane passing through point p and parallel to the plane spanned by three distinct 3D points a, b, and c with \(x\)- and \(y\)-coordinates are provided in vectors x and y, respectively.

Usage

paraplane(p, a, b, c, x, y)

Value

A list with the elements

desc: Description of the plane passing through point p and parallel to plane spanned by points a, b and c
points: The input points a, b, c, and p. Plane is parallel to the plane spanned by a, b, and c and passes through point p (stacked row-wise, i.e., row 1 is point a, row 2 is point b, row 3 is point c, and row 4 is point p).
x,y: The input vectors which constitutes the \(x\)- and \(y\)-coordinates of the point(s) of interest on the plane. x and y can be scalars or vectors of scalars.
z: The output vector which constitutes the \(z\)-coordinates of the point(s) of interest on the plane. If x and y are scalars, z will be a scalar and if x and y are vectors of scalars, then z needs to be a matrix of scalars, containing the \(z\)-coordinate for each pair of x and y values.
coeff: Coefficients of the plane (in the \(z = A x+B y+C\) form).
equation: Equation of the plane in long form
equation2: Equation of the plane in short form, to be inserted on the plot

Arguments

p: A 3D point which the plane parallel to the plane spanned by three distinct 3D points a, b, and c crosses.
a, b, c: 3D points that determine the plane to which the plane crossing point p is parallel to.
x, y: Scalars or vectors of scalars representing the \(x\)- and \(y\)-coordinates of the plane parallel to the plane spanned by points a, b, and c and passing through point p.

Author

Elvan Ceyhan

Examples

Run this code

# \donttest{
Q<-c(1,10,3); R<-c(1,1,3); S<-c(3,9,12); P<-c(1,1,0)

pts<-rbind(Q,R,S,P)
paraplane(P,Q,R,S,.1,.2)

xr<-range(pts[,1]); yr<-range(pts[,2])
xf<-(xr[2]-xr[1])*.25
#how far to go at the lower and upper ends in the x-coordinate
yf<-(yr[2]-yr[1])*.25
#how far to go at the lower and upper ends in the y-coordinate
x<-seq(xr[1]-xf,xr[2]+xf,l=5)  #try also l=10, 20, or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=5)  #try also l=10, 20, or 100

plP2QRS<-paraplane(P,Q,R,S,x,y)
plP2QRS
summary(plP2QRS)
plot(plP2QRS,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)

paraplane(P,Q,R,Q+R,.1,.2)

z.grid<-plP2QRS$z

plQRS<-Plane(Q,R,S,x,y)
plQRS
pl.grid<-plQRS$z

zr<-max(z.grid)-min(z.grid)
Pts<-rbind(Q,R,S,P)+rbind(c(0,0,zr*.1),c(0,0,zr*.1),
c(0,0,zr*.1),c(0,0,zr*.1))
Mn.pts<-apply(Pts[1:3,],2,mean)

plot3D::persp3D(z = pl.grid, x = x, y = y, theta =225, phi = 30,
ticktype = "detailed",
main="Plane Crossing Points Q, R, S\n and Plane Passing P Parallel to it")
#plane spanned by points Q, R, S
plot3D::persp3D(z = z.grid, x = x, y = y,add=TRUE)
#plane parallel to the original plane and passing thru point \code{P}

plot3D::persp3D(z = z.grid, x = x, y = y, theta =225, phi = 30,
ticktype = "detailed",
main="Plane Crossing Point P \n and Parallel to the Plane Crossing Q, R, S")
#plane spanned by points Q, R, S
#add the defining points
plot3D::points3D(Pts[,1],Pts[,2],Pts[,3], add=TRUE)
plot3D::text3D(Pts[,1],Pts[,2],Pts[,3], c("Q","R","S","P"),add=TRUE)
plot3D::text3D(Mn.pts[1],Mn.pts[2],Mn.pts[3],plP2QRS$equation,add=TRUE)
plot3D::polygon3D(Pts[1:3,1],Pts[1:3,2],Pts[1:3,3], add=TRUE)
# }