---
title: "VS5 - Functions for Euclidean Geometry"
author: "Elvan Ceyhan"
date: '`r Sys.Date()` '
output:
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
bibliography: References.bib
vignette: >
  %\VignetteIndexEntry{VS5 - Functions for Euclidean Geometry}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

<style>
.math {
  font-size: small;
}
</style>

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE,comment = "#>",fig.width=6, fig.height=4, fig.align = "center")
#knitr::write_bib("knitr", file = "References.bib")
```

First we load the `pcds` and `plot2D` packages:
```{r setup, message=FALSE, results='hide'}
library(pcds)
library(plot3D)
```

# Auxiliary Functions for Two and Three Dimensional Euclidean Geometry
We provide the equations for the $d-1$ dimensional hyper-planes and related geometric
structures in $d$-dimensional Euclidean space for $d=2,3$. 
That is, we provide the equations of the lines,
planes, and intersection points, etc. in $\mathbb R^2$ and $\mathbb R^3$.
Some of these functions are used in the construction of proximity regions and the related PCDs
hence are auxiliary functions in the construction of PCDs
(see @ceyhan:Phd-thesis and @ceyhan:comp-geo-2010 for more details).

## Geometric Structures in 2D Euclidean Space
The equation of the line and other related quantities like slope and intercept and 
the plot of the line are provided by the function `Line`.
This function is different from the `line` function in the standard `stats` package
in `R` in the sense that `Line(a,b,x)` fits the line passing
through points `a` and `b` and returns various quantities related to this line (see below) and `x` is
the vector of $x$-coordinates of the points on the line `Line(a,b,x)`
(i.e., want the compute the $y$ coordinates of the points on the line `Line(a,b,x)`
whose $x$-coordinates are given in the vector `x`
while `line(a,b)` fits a straight line robustly
whose $x$-coordinates are in vector `a` and $y$-coordinates are in vector `b`.

### Equation of the Line Crossing two Distinct 2D Points

The function `Line` is an object of class `Lines` and has arguments `a,b,x` where

- `a,b`, 2D points that determine the straight line (i.e., through which the straight line passes) and
- `x`, a scalar or a vector of scalars representing the $x$-coordinates of the line.

Its `call` (with `lnAB` in the below script) just returns 
`Coefficients of the line (for the form: y = slope * x + intercept)`.
Its `summary` returns the defining points,
selected $x$ points (first row) and estimated $y$ points (second row) that fall on the line (first 6 or fewer),
the equation of the line,
and the coefficients of the line (i.e., the `slope` and `intercept`).
The `plot` function (or `plot.Lines`) returns the plot of the line together with the defining points.
```{r lAB, fig.cap="The line joining the points $A$ and $B$ in 2D space."}
A<-c(-1.22,-2.33); B<-c(2.55,3.75)

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

lnAB<-Line(A,B,x)
lnAB
summary(lnAB)
plot(lnAB)
```

The difference of the function `Line` from the `line` function in the standard `stats` package in `R` is illustrated below.
```{r EGch6, eval=F}
line(A,B)  #this takes vector A as the x points and vector B as the y points and fits the line
#> 
#> Call:
#> line(A, B)
#> 
#> Coefficients:
#> [1]   1.231  -1.081

line(x,lnAB$y) #gives the same line as Line(A,B,x) above
#> 
#> Call:
#> line(x, lnAB$y)
#> 
#> Coefficients:
#> [1]  -0.3625   1.6127
c(lnAB$intercept,lnAB$slope)
#>  intercept      slope 
#> -0.3624668  1.6127321
```

The slope of the line joining two distinct 2D points `a` and `b` can be found by the `slope` function
with arguments `a,b` which are as in the function `Line`.
```{r EGch7, eval=F}
slope(A,B)
#> [1] 1.612732
```

### Equation of the Line Crossing a Point and Parallel to the Line Crossing two Distinct 2D Points

The equation of the line crossing a point `p` parallel to the line segment joining two distinct 2D points `a` and `b` is provided by the function `paraline`. 
Other related quantities like slope and intercept and the plot of the line are also provided.
The function `paraline` is an object of class `Lines` which takes arguments `p,a,b,x` where

- `a,b,x` are as in the function `Line` and
- `p` is a 2D point at which the parallel line to line segment joining `a` and `b` crosses.

Its `call`, `summary`, and `plot` are also as in `Line`.
```{r lPpl2AB, eval=F, fig.cap="The line crossingn the point $P$ and parallel to the line segment $[AB]$ in 2D space."}
A<-c(1.1,1.2); B<-c(2.3,3.4); P<-c(.51,2.5)

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

plnAB<-paraline(P,A,B,x)
plnAB
#> Call:
#> paraline(p = P, a = A, b = B, x = x)
#> 
#> Coefficients of the line (in the form: y = slope * x + intercept) 
#>     slope intercept 
#>  1.833333  1.565000
summary(plnAB)
#> Call:
#> paraline(p = P, a = A, b = B, x = x)
#> 
#> Defining Points
#>   [,1] [,2]
#> A 1.10  1.2
#> B 2.30  3.4
#> P 0.51  2.5
#> 
#>  Selected x points (first row) and estimated y points (second row) that fall on the Line
#>       (first 6 or fewer are printed at each row) 
#> [1] 0.06250 0.73375 1.40500 2.07625 2.74750
#> [1] 1.679583 2.910208 4.140833 5.371458 6.602083
#> 
#> Equation of the Line Crossing Point P and Parallel to Line Segment [AB] 
#> [1] "y=1.83333333333333x+1.565"
#> 
#> Coefficients of the line (when the line is in the form: y = slope * x + intercept) 
#> intercept     slope 
#>  1.565000  1.833333
plot(plnAB)
```

### Equation of the Line Crossing a Point and Perpendicular to the Line Crossing two Distinct 2D Points

The equation of the line crossing a point `p` perpendicular to the line segment joining two distinct 2D points `a` and `b` is 
provided by the function `perpline`. 
Other related quantities like slope and intercept and the plot of the line are also provided.
The function `perpline` is an object of class `Lines` and has the same arguments as `paraline`. 
Its `call`, `summary`, and `plot` are as in `Line`.
In the plot of the object, we use `asp=1` so that the lines are depicted correctly, i.e., as perpendicular to each other.
```{r lPprp2AB, eval=F, fig.cap="The line crossing point $P$ and perpendicular to the line joining $A$ and $B$."}
A<-c(1.1,1.2); B<-c(2.3,3.4); P<-c(.51,2.5)

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

plnAB<-perpline(P,A,B,x)
plnAB
#> Call:
#> perpline(p = P, a = A, b = B, x = x)
#> 
#> Coefficients of the line (in the form: y = slope * x + intercept) 
#>      slope  intercept 
#> -0.5454545  2.7781818
summary(plnAB)
#> Call:
#> perpline(p = P, a = A, b = B, x = x)
#> 
#> Defining Points
#>   [,1] [,2]
#> A 1.10  1.2
#> B 2.30  3.4
#> P 0.51  2.5
#> 
#>  Selected x points (first row) and estimated y points (second row) that fall on the Line
#>       (first 6 or fewer are printed at each row) 
#> [1] 0.06250 0.73375 1.40500 2.07625 2.74750
#> [1] 2.744091 2.377955 2.011818 1.645682 1.279545
#> 
#> Equation of the Line Crossing Point P Perpendicular to Line Segment [AB] 
#> [1] "y=-0.545454545454545x+2.77818181818182"
#> 
#> Coefficients of the line (when the line is in the form: y = slope * x + intercept) 
#>  intercept      slope 
#>  2.7781818 -0.5454545
plot(plnAB,asp=1)
```

The point of intersection of two lines, where each line is defined by a pair of points, 
is provided by the function `int.2lines`.
The arguments of the functions are the four points `p1,q1,p2,q2`,
where the first line is defined with the points `p1,q1`, and the second line is defined with the points `p2,q2`.
```{r EGch1, eval=F}
A<-c(-1.22,-2.33); B<-c(2.55,3.75); C<-c(0,6); D<-c(3,-2)
ip<-int.2lines(A,B,C,D)
ip
#> [1] 1.486767 2.035289
```

The angles to draw arcs between two line segments can be computed with the function `angle.str2end`
which takes arguments `a,b,c,radian` where

- `a,b,c` three 2D points which represent the intersecting line segments $[ba]$ and $[bc]$.
- `radian`, a logical argument (default=`TRUE`). If `TRUE`, the smaller angle or counter-clockwise angle
between the line segments $[ba]$ and $[bc]$ is provided in radians, else it is provided in degrees.

This function returns the output

- `small.arc.angles`, angles of $[ba]$ and $[bc]$ with the $x$-axis so that difference between them
is the smaller angle between $[ba]$ and $[bc]$, and
- `ccw.arc.angles`, angles of $[ba]$ and $[bc]$ with the $x$-axis so that difference between them
is the counter-clockwise angle between $[ba]$ and $[bc]$

The two pairs of angles in radians or degrees in the output of this function is used 
to draw arcs between two vectors or line segments
for the `plotrix::draw.arc` function in the `plotrix` package.
More specifically, these angles are used to draw AS proximity regions 
(for the circular parts, i.e., arc-slices inside the triangle),
The angles are provided with respect to the $x$-axis in the coordinate system.
The line segments are $[ba]$ and $[bc]$ for the arguments `a,b,c` of the function.

Below are the angles between line segments $BA$ and $BC$ first in radians and then in degrees.
```{r EGch2, eval=F}
A<-c(.3,.2); B<-c(.6,.3); C<-c(1,1)

angle.str2end(A,B,C)
#> $small.arc.angles
#> [1] 1.051650 3.463343
#> 
#> $ccw.arc.angles
#> [1] -2.819842  1.051650
angle.str2end(A,B,C,radian=FALSE)
#> $small.arc.angles
#> [1]  60.25512 198.43495
#> 
#> $ccw.arc.angles
#> [1] -161.56505   60.25512
```

We plot the line segments $BA$ and $BC$ and annotate the angles between them and 
also the angles between the line segments and the $x$-axis
using the below code, 
type also `? angle.str2end` for the code to generate this figure.
```{r angBA-BC, eval=F, fig.cap="The line segments $BA$ and $BC$ and the angles (in degrees) between them and between the line segments and the $x$-axis."}
pts<-rbind(A,B,C)

Xp<-c(B[1]+max(abs(C[1]-B[1]),abs(A[1]-B[1])),0)

#plot of the line segments
ang.rad<-angle.str2end(A,B,C,radian=TRUE); ang.rad
ang.deg<-angle.str2end(A,B,C,radian=FALSE); ang.deg
ang.deg1<-ang.deg$s; ang.deg1
ang.deg2<-ang.deg$c; ang.deg2

rad<-min(Dist(A,B),Dist(B,C))

Xlim<-range(pts[,1],Xp[1],B+Xp,B[1]+c(+rad,-rad))
Ylim<-range(pts[,2],B[2]+c(+rad,-rad))
xd<-Xlim[2]-Xlim[1]
yd<-Ylim[2]-Ylim[1]

#plot for the counter-clockwise arc
plot(pts,pch=1,asp=1,xlab="x",ylab="y",xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05))
L<-rbind(B,B,B); R<-rbind(A,C,B+Xp)
segments(L[,1], L[,2], R[,1], R[,2], lty=2)
plotrix::draw.arc(B[1],B[2],radius=.3*rad,angle1=ang.rad$c[1],angle2=ang.rad$c[2])
plotrix::draw.arc(B[1],B[2],radius=.6*rad,angle1=0, angle2=ang.rad$s[1],lty=2,col=2)
plotrix::draw.arc(B[1],B[2],radius=.9*rad,angle1=0,angle2=ang.rad$s[2],col=3)
txt<-pts
text(txt+cbind(rep(xd*.02,nrow(txt)),rep(-xd*.02,nrow(txt))),c("A","B","C"))

text(rbind(B)+.5*rad*c(cos(mean(ang.rad$c)),sin(mean(ang.rad$c))),
     paste(abs(round(ang.deg2[2]-ang.deg2[1],2))," degrees",sep=""))
text(rbind(B)+.6*rad*c(cos(ang.rad$s[1]/2),sin(ang.rad$s[1]/2)),paste(round(ang.deg1[1],2)),col=2)
text(rbind(B)+.9*rad*c(cos(ang.rad$s[2]/2),sin(ang.rad$s[2]/2)),paste(round(ang.deg1[2],2)),col=3)
```

The area of a polygon `h` in $\mathbb R^2$ is provided by the function `area.polygon`
with the sole argument `h` which is a vector of $n$ 2D points, 
stacked row-wise, each row representing a vertex of the polygon,
where $n$ is the number of vertices of the polygon..
Here the vertices of the polygon `h` must be provided in clockwise or counter-clockwise order, 
otherwise the function does not yield the area. 
Also, the polygon could be convex or non-convex. 
See (@weisstein-area-poly).
```{r EGch3, eval=F}
A<-c(0,0); B<-c(1,0); C<-c(0.5,.8);
Tr<-rbind(A,B,C);
area.polygon(Tr)
#> [1] 0.4

A<-c(0,0); B<-c(1,0); C<-c(.7,.6); D<-c(0.3,.8);
h1<-rbind(A,B,C,D);  #try also h1<-rbind(A,B,D,C) or h1<-rbind(A,C,B,D) or h1<-rbind(A,D,C,B);
area.polygon(h1)
#> [1] 0.49
```

## Various Distances and Related Functions

The proximity regions are based on *distances* or *dissimilarity measures* between points or
between points and lines (or edges). 
So, we provide various functions that measure the distances between such geometric structures.

The distance from a point to a line defined by two points is provided by the function `dist.point2line`
with arguments `p,a,b` where 

- `p` is a 2D point, distance from `p` to the line passing through points `a` and `b` are to be computed.
- `a,b` are 2D points that determine the straight line (i.e., through which the straight line passes).

This function also returns the closest point on the line to the point `p`.
```{r EGch4, eval=F}
dpl<-dist.point2line(P,A,B);
dpl
#> $distance
#> [1] 2.5
#> 
#> $cl2p
#> [1] 0.51 0.00
```

The distance from a point `p` to a set `Yp` of finite cardinality 
where the set `Yp` is stored as a matrix is returned by the function `dist.point2set`.
The function takes arguments `p,Yp` where 

- `p`, a vector (i.e., a point in $\mathbb R^d$) and
- `Yp`, a set of $d$-dimensional points.

It returns the index and coordinates of the point in the set `Yp` closest to the point `p`.
```{r EGch5, eval=F}
X1<-cbind(runif(10),runif(10))
dist.point2set(c(1,2),X1)
#> $distance
#> [1] 1.086349
#> 
#> $ind.cl.point
#> [1] 3
#> 
#> $closest.point
#> [1] 0.7933641 0.9334844
```

## Geometric Structures in 3D Euclidean Space
There are more possibilities for the geometric structures in $\mathbb R^3$ compared to $\mathbb R^2$.
In $\mathbb R^2$ only lines and their relationships exist, but
in $\mathbb R^3$, in addition to the lines and their relationships,
there are also planes,
and relationships between lines and planes, and planes and planes.

### Equation of the Line Crossing a Point Parallel to (i.e., in the Direction of) a Vector

The equation of the line crossing a point in the direction of a vector 
in the 3D space are provided by the function `Line3D`.
If the option `dir.vec=TRUE` (which is the default) in `Line(p,v,t,dir.vec=T)`, 
the line crosses the point `p` in the direction of the vector `v` 
(i.e., parallel to the line joining the origin and `v`), otherwise,
in `Line(p,v,t,dir.vec=FALSE)`,the line crosses `p` and 
is in the direction of the vector joining `p`  and `v` (i.e.,  in direction of $v-r_0$).

The function `Line3D` is an object of class `Lines3D` and takes arguments `p,v,t,dir.vec`
where 

- `p`, a 3D point through which the straight line passes.
- `v`, a 3D vector which determines the direction of the straight line (i.e., the straight line would be
parallel to this vector) if the `TRUE`, otherwise it is 3D point and $v-r_0$ determines the direction of the
the straight line.
- `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 $r_0=(x_0,y_0,z_0)$
and $v=(A,B,C)$ if `TRUE`, else $v-r_0=(A,B,C)$).
- `dir.vec`, a logical argument about `v`, if `TRUE` `v` is treated as a vector,
else `v` is treated as a point and so the direction vector is taken to be $v-r_0$.

Its `call` (with `lnPQ3D` in the below script with the default option `dir.vec=TRUE`) just returns 
`Coefficients of the parameterized line passing through initial point P =(x0,y0,z0) in the direction of OQ =(A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t) `.
Its `summary` returns the defining vectors, namely, 
`initial point` as viewed as a vector joining the origin and the point and `direction vector`,
estimated $x$ points (first row), $y$ points (second row), and $z$ points (third row)
(first 6 or fewer are printed at each row) ,
the equation of the line passing through point `p` in the direction of $OQ$  with $O$ representing the origin $(0,0,0)$ 
(i.e., parallel to $OQ$ ),
and the coefficients of the parameterized line passing through initial point $P =(x_0,y_0,z_0)$ 
in the direction of  $OQ =(A,B,C)$ (for the form: $x=x_0 + A\,t$, $y=y_0 + B\,t$, and $z=z_0 + C\,t$).
The `plot` function (or `plot.Lines3D`) returns the plot of the line together with the defining points and vectors.
```{r l3dPQ, eval=F, fig.cap="The line crossing the point $P$ parallel to the vector $OQ$."}
P<-c(1,10,3); Q<-c(1,1,3);

vecs<-rbind(P,Q)

Line3D(P,Q,.1)
#> Call:
#> Line3D(p = P, v = Q, t = 0.1)
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of OQ = (A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t) 
#>                  [,1] [,2] [,3]
#> initial point       1   10    3
#> direction vector    1    1    3
Line3D(P,Q,.1,dir.vec=FALSE)
#> Call:
#> Line3D(p = P, v = Q, t = 0.1, dir.vec = FALSE)
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of PQ = (A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t) 
#>                  [,1] [,2] [,3]
#> initial point       1   10    3
#> direction vector    0   -9    0

tr<-range(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

lnPQ3D<-Line3D(P,Q,tsq)
lnPQ3D
#> Call:
#> Line3D(p = P, v = Q, t = tsq)
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of OQ = (A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t) 
#>                  [,1] [,2] [,3]
#> initial point       1   10    3
#> direction vector    1    1    3
summary(lnPQ3D)
#> Call:
#> Line3D(p = P, v = Q, t = tsq)
#> 
#> Defining Vectors
#>                  [,1] [,2] [,3]
#> initial point       1   10    3
#> direction vector    1    1    3
#> 
#>  Estimated x points (first row), y points (second row), and z points (third row)
#>    that fall on the Line
#>       (first 6 or fewer are printed at each row) 
#> [1] -8.90 -3.95  1.00  5.95 10.90
#> [1]  0.10  5.05 10.00 14.95 19.90
#> [1] -26.70 -11.85   3.00  17.85  32.70
#> 
#> Equation of the line passing through point P in the direction of OQ  with O representing the origin (0,0,0)
#>                    (i.e., parallel to OQ ) 
#>      [,1]        
#> [1,] "x = 1 + t" 
#> [2,] "y = 10 + t"
#> [3,] "z = 3 + 3t"
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of OQ = (A,B,C) (in the form: x = x0 + A*t, y = y0 + B*t, and z = z0 + C*t) 
#>    [,1] [,2] [,3]
#> P     1   10    3
#> OQ    1    1    3

plot(lnPQ3D)
```

### Equation of the Line Crossing a Point and Parallel to the Line Crossing two Distinct 3D Points

The equation of the line in 3D space crossing a point `p` and 
parallel to line joining 3D points `a` and `b` is returned by the function `paraline3D`. 
Other related quantities and the plot of the line are also provided.
The function `paraline3D` is an object of class `Lines3D` and takes arguments `p,a,b,t` where 

- `p`, a 3D point through which the straight line passes.
- `a,b`, 3D points which determine the straight line to which the line passing through point `p` would be
parallel (i.e., $b-a$ 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 $b-a=(A,B,C)$).

Its `call`, `summary`, and `plot` are as in `Line3D`.

```{r l3dPpl2QR, eval=F, fig.cap="The line crossing the point $P$ and parallel to the line segment $QR$ in 3D space."}
P<-c(1,10,4); Q<-c(1,1,3); R<-c(3,9,12)

vecs<-rbind(P,R-Q)
pts<-rbind(P,Q,R)
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

pln3D<-paraline3D(P,Q,R,tsq)
pln3D
#> Call:
#> paraline3D(p = P, a = Q, b = R, t = tsq)
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of R - Q = (A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t) 
#>                  [,1] [,2] [,3]
#> initial point       1   10    4
#> direction vector    2    8    9
summary(pln3D)
#> Call:
#> paraline3D(p = P, a = Q, b = R, t = tsq)
#> 
#> Defining Vectors
#>                  [,1] [,2] [,3]
#> initial point       1   10    4
#> direction vector    2    8    9
#> 
#>  Estimated x points (first row), y points (second row), and z points (third row)
#>    that fall on the Line
#>       (first 6 or fewer are printed at each row) 
#> [1] -23.2 -11.1   1.0  13.1  25.2
#> [1] -86.8 -38.4  10.0  58.4 106.8
#> [1] -104.90  -50.45    4.00   58.45  112.90
#> 
#> Equation of the line passing through point P parallel to the line joining points Q and R 
#>      [,1]         
#> [1,] "x = 1 + 2t" 
#> [2,] "y = 10 + 8t"
#> [3,] "z = 4 + 9t" 
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of R - Q = (A,B,C) (in the form: x = x0 + A*t, y = y0 + B*t, and z = z0 + C*t) 
#>       [,1] [,2] [,3]
#> P        1   10    4
#> R - Q    2    8    9

plot(pln3D)
```

### Equation of the Plane Crossing three Distinct 3D Points

The equation of the plane crossing three distinct points in $\mathbb R^3$ is provided by the function `Plane`. 
Other related quantities and the plot of the plane are also provided.
The function `Plane` is an object of class `Planes` and takes arguments `a,b,c,x,y` where

- `a,b,c` are 3D points that determine the plane (i.e., through which the plane is passing) and
- `x,y` are scalars or vectors of scalars representing the $x$- and $y$-coordinates of the plane.

Its `call` (with `plP123` in the below script) just returns 
`Coefficients of the Plane (in the form: z = A*x + B*y + C)`.
Its `summary` returns the defining points,
selected $x$ and $y$ points and estimated $z$ points --- presented row-wise, respectively --- that fall on the plane
(first 6 or fewer are printed on each row),
the equation of the plane passing through points `a`, `b`, and `c`,
and the coefficients of the plane (in the form $z = A\,x + B\,y + C$).
The `plot` function (or `plot.Planes`) returns the plot of the plane together with the defining points.
```{r plP123, eval=F, fig.cap="The plane joining the points $P_1$, $P_2$, and $P_3$ in 3D space."} 
P1<-c(1,10,3); P2<-c(1,1,3); P3<-c(3,9,12) #also try P2=c(2,2,3)

pts<-rbind(P1,P2,P3)
Plane(P1,P2,P3,.1,.2)
#> Call:
#> Plane(a = P1, b = P2, c = P3, x = 0.1, y = 0.2)
#> 
#> Coefficients of the Plane (in the form: z = A*x + B*y + C):
#>    A    B    C 
#>  4.5  0.0 -1.5

xr<-range(pts[,1]); yr<-range(pts[,2])
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
x<-seq(xr[1]-xf,xr[2]+xf,l=3)  #try also l=10, 20, or 100
y<-seq(yr[1]-yf,yr[2]+yf,l=3)  #try also l=10, 20, or 100

plP123<-Plane(P1,P2,P3,x,y)
plP123
#> Call:
#> Plane(a = P1, b = P2, c = P3, x = x, y = y)
#> 
#> Coefficients of the Plane (in the form: z = A*x + B*y + C):
#>    A    B    C 
#>  4.5  0.0 -1.5
summary(plP123)
#> Call:
#> Plane(a = P1, b = P2, c = P3, x = x, y = y)
#> 
#> Defining Points
#>    [,1] [,2] [,3]
#> P1    1   10    3
#> P2    1    1    3
#> P3    3    9   12
#> 
#>  Selected x and y points and estimated z points --- presented row-wise, respectively --- that fall on the Plane
#>       (first 6 or fewer are printed on each row) 
#> [1] 0.8 2.0 3.2
#> [1]  0.1  5.5 10.9
#> [1]  2.1  7.5 12.9
#> 
#> Equation of the Plane Passing through Points P1, P2, and P3 
#> [1] "z = 4.5x -1.5"
#> 
#> Coefficients of the Plane (in the form z = A*x + B*y + C):
#>    A    B    C 
#>  4.5  0.0 -1.5
plot(plP123,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)
```

### Equation of the Plane Crossing a Point and Parallel to the Plane Crossing three Distinct 3D Points

The plane at a point `p` and parallel to the plane spanned by three distinct 3D points `a`, `b`, and `c`
is provided by the function `paraplane`.
Other related quantities and the plot of the plane are also provided.
The function `paraplane` is an object of class `Planes` and takes arguments `p,a,b,c,x,y` where

- `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`.

Its `call`, `summary`, and `plot` are as in `Plane`.
```{r plPpl2QRS, eval=F, fig.cap="The plane crossing the point $P$ and parallel to the plane spanned by the points $Q$, $R$, and $S$ in 3D space."}
Q<-c(1,10,3); R<-c(2,2,3); S<-c(3,9,12); P<-c(1,2,4)

pts<-rbind(Q,R,S,P)
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
#> Call:
#> paraplane(p = P, a = Q, b = R, c = S, x = x, y = y)
#> 
#> Coefficients of the Plane (in the form: z = A*x + B*y + C):
#>    A    B    C 
#>  4.8  0.6 -2.0
summary(plP2QRS)
#> Call:
#> paraplane(p = P, a = Q, b = R, c = S, x = x, y = y)
#> 
#> Defining Points
#>   [,1] [,2] [,3]
#> Q    1   10    3
#> R    2    2    3
#> S    3    9   12
#> P    1    2    4
#> 
#>  Selected x and y points and estimated z points --- presented row-wise, respectively --- that fall on the Plane
#>       (first 6 or fewer are printed on each row) 
#> [1] 0.50 1.25 2.00 2.75 3.50
#> [1]  0  3  6  9 12
#> [1]  0.4  4.0  7.6 11.2 14.8
#> 
#> Equation of the Plane Passing through Point P Parallel to the Plane 
#>  Passing through Points Q, R and S 
#> [1] "z = 4.8x + 0.6y -2"
#> 
#> Coefficients of the Plane (in the form z = A*x + B*y + C):
#>    A    B    C 
#>  4.8  0.6 -2.0

plot(plP2QRS,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)
```

### Equation of the Line Crossing a Point and Perpendicular to the Plane Crossing three Distinct 3D Points

The line crossing the 3D point `p` and perpendicular to the plane spanned by the three points 
`a`, `b`, and `c` in $\mathbb R^3$ is returned by the function `perpline2plane`.
Other related quantities and the plot of the plane are also provided.

The function `perpline2plane` is an object of class `Lines3D` and takes arguments `p,a,b,c,t`
where `p,a,b,c` are as in `paraplane` and `t` is as in `paraline3D`. 
Its `call` (with `pln2pl` in the below script) just returns 
`Coefficients of the parameterized line passing through initial point P =(x0,y0,z0) in the direction of normal vector =(A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t)`.
Its `summary` returns the defining vectors (i.e., the `initial point` viewed as a vector from the origin,
and the `normal vector`),
estimated $x$ points (first row), $y$ points (second row), and $z$ points (third row)
that fall on the Line
(first 6 or fewer are printed at each row),
the equation of the line crossing point $p$ perpendicular to the plane spanned by points `a`, `b`, and `c`,
and the coefficients of the parameterized line passing through initial point $p =(x_0,y_0,z_0)$ in the 
direction of normal vector $(A,B,C)$ (in the form: $x=x_0 + A\,t$, $y=y_0 + B\,t$, and $z=z_0 + C\,t$). 
The `plot` function (or `plot.Lines3D`) returns the plot of the line together with the plane its perpendicular to.
```{r lprp2plQRS, eval=F, fig.cap="The line crossing the point $P$ and perpendicular to the plane spanned by $Q$, $R$, and $S$ in 3D space."}
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)
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
#> Call:
#> perpline2plane(p = P, a = Q, b = R, c = S, t = tsq)
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of normal vector = (A,B,C) (for the form: x=x0 + A*t, y=y0 + B*t, and z=z0 + C*t) 
#>                   [,1]      [,2] [,3]
#> initial point 1.000000 1.0000000    1
#> normal vector 4.055556 0.1111111   -1
summary(pln2pl)
#> Call:
#> perpline2plane(p = P, a = Q, b = R, c = S, t = tsq)
#> 
#> Defining Vectors
#>                   [,1]      [,2] [,3]
#> initial point 1.000000 1.0000000    1
#> normal vector 4.055556 0.1111111   -1
#> 
#>  Estimated x points (first row), y points (second row), and z points (third row)
#>    that fall on the Line
#>       (first 6 or fewer are printed at each row) 
#> [1] -56.99444 -27.99722   1.00000  29.99722  58.99444
#> [1] -0.5888889  0.2055556  1.0000000  1.7944444  2.5888889
#> [1]  15.30   8.15   1.00  -6.15 -13.30
#> 
#> Equation of the line crossing point P perpendicular to the plane spanned by points Q, Rand S 
#>      [,1]                        
#> [1,] "x = 1 + 4.05555555555555t" 
#> [2,] "y = 1 + 0.111111111111111t"
#> [3,] "z = 1-t"                   
#> 
#> Coefficients of the parameterized line passing through initial point P = (x0,y0,z0) in the direction of normal vector = (A,B,C) (in the form: x = x0 + A*t, y = y0 + B*t, and z = z0 + C*t) 
#>                   [,1]      [,2] [,3]
#> P             1.000000 1.0000000    1
#> normal vector 4.055556 0.1111111   -1

plot(pln2pl,theta = 225, phi = 30, expand = 0.7, facets = FALSE, scale = TRUE)
```

The point of intersection of a line and a plane is returned by the function `int.line.plane`
with arguments `p1,p2,p3,p4,p5`,
where line is defined by the points `p1` and `p2`, while the plane is defined by the points `p3`, `p4`, and `p5`.
```{r EGch8, eval=F}
L1<-c(2,4,6); L2<-c(1,3,5);
A<-c(1,10,3); B<-c(1,1,3); C<-c(3,9,12)

Pint<-int.line.plane(L1,L2,A,B,C)
Pint
#> [1] 1.571429 3.571429 5.571429
```

The distance from a point `p` to a plane spanned by three points `a`, `b`, and `c` in $\mathbb R^3$ 
is returned by the function  `dist.point2plane` with arguments `p,a,b,c`
which are same as in the function `paraplane`.
The function also returns the point of orthogonal projection from point `p` to the plane.
```{r EGch9,eval=F}
P<-c(5,2,40)
P1<-c(1,2,3); P2<-c(3,9,12); P3<-c(1,1,3);

dis<-dist.point2plane(P,P1,P2,P3);
dis
#> $distance
#> [1] 4.121679
#> 
#> $proj.pt2plane
#> [1]  9.023529  2.000000 39.105882
```

**References**
