# The plane at a point and parallel to the plane spanned by three distinct 3D points `a`, `b`, and `c` 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. ```r paraplane(p, a, b, c, x, y) ``` ## 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`. ## Returns 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 ## Examples ```r 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) ``` ## See Also `Plane` ## Author(s) Elvan Ceyhan

paraplane function