# R/utils-geometry.R In trevorld/piecepack: Board Game Graphics

```# "collision detection" via Separating Axis Theorem
# Arguments of point is vectorized
Point2D <- R6Class("point2d",
public = list(x=NULL, y=NULL,
initialize = function(x=0.5, y=0.5) {
if (is.list(x) || inherits(x, "point2d")) {
xt <- x\$x
y <- x\$y
} else {
xt <- x
y <- y
}
n <- max(lengths(list(xt, y)))
self\$x <- rep(xt, length.out = n)
self\$y <- rep(y, length.out = n)
},
angle_to = function(p) {
p1 = self\$diff(p)
to_t(p1\$x, p1\$y)
},
diff = function(p) {
Vector\$new(p\$x-self\$x, p\$y-self\$y)
},
distance_to = function(p) {
sqrt((p\$x - self\$x)^2 + (p\$y - self\$y)^2)
},
midpoint = function(p) {
x <- mean(c(p\$x, self\$x))
y <- mean(c(p\$y, self\$y))
Point2D\$new(x, y)
},
translate = function(x = 0, y = 0) {
Point2D\$new(self\$x + x, self\$y + y)
},
translate_polar = function(t = 0, r = 1) {
x <- self\$x + to_x(t, r)
y <- self\$y + to_y(t, r)
Point2D\$new(x, y)
},
rotate = function(t = 0) {
Point2D\$new(x, y)
},
dilate = function(width = 1, height = width) {
x <- width * self\$x
y <- height * self\$y
Point2D\$new(x, y)
},
npc_to_in = function(x=0.5, y=0.5, w=1, h=1, t=0) {
self\$translate(-0.5, -0.5)\$dilate(w, h)\$rotate(t)\$translate(x, y)
},
plot = function(gp = gpar()) {
grid.points(x=self\$x, y=self\$y, default.units="in", gp = gp)
}),
active = list(c = function() {
Point2D\$new(mean(self\$x), mean(self\$y))
},
convex_hull = function() {
hull <- grDevices::chull(as.matrix(self))
x <- self\$x[hull]
y <- self\$y[hull]
Point2D\$new(x, y)
})
)
#' @export
`[.point2d` <- function(x, i) Point2D\$new(x\$x[i], x\$y[i])
#' @export
length.point2d <- function(x) length(x\$x)
#' @export
as.matrix.point2d <- function(x, ...) {
m <- cbind(x\$x, x\$y)
colnames(m) <- c("x", "y")
m
}
#' @export
as.list.point2d <- function(x, ...) {
n <- length(x)
ll <- vector("list", n)
for (i in seq_len(n)) {
ll[[i]] <- x[i]
}
ll
}
#' @export
as.data.frame.point2d <- function(x, ...) {
as.data.frame(as.matrix(x))
}
#' @export
Ops.point2d <- function(e1, e2) {
if (missing(e2)) {
switch(.Generic,
"-" = e1\$dilate(-1),
stop(paste0("unary operation '", .Generic, "' not defined for `point2d` objects"))
)
} else {
if (inherits(e1, "point2d") && inherits(e2, "point2d")) {
switch(.Generic,
"+" = e1\$translate(e2),
"-" = e2\$diff(e1),
stop(paste0("binary operation '", .Generic, "' not defined for `point2d` objects"))
)
} else if (is.numeric(e1) && inherits(e2, "point2d")) {
switch(.Generic,
"*" = e2\$dilate(e1),
stop(paste0("binary operation '", .Generic, "' not defined for `point2d` objects"))
)
} else {
switch(.Generic,
stop(paste0("binary operation '", .Generic, "' not defined for `point2d` objects"))
)
}
}
}

Point3D <- R6Class("point3d",
public = list(x=NULL, y=NULL, z=NULL,
initialize = function(x=0, y=0, z=0) {
if (is.list(x) || inherits(x, "point2d") || inherits(x, "point3d")) {
xt <- x\$x
y <- x\$y
z <- x\$z %||% z
} else if (is.matrix(x)) {
xt <- x[, 1]
y <- x[, 2]
z <- x[, 3]
} else {
xt <- x
y <- y
z <- z
}
n <- max(lengths(list(xt, y, z)))
self\$x <- rep(xt, length.out = n)
self\$y <- rep(y, length.out = n)
self\$z <- rep(z, length.out = n)
},
diff = function(p) { ### later create Vector3D class?
Point3D\$new(p\$x-self\$x, p\$y-self\$y, p\$z-self\$z)
},
distance_to = function(p) {
sqrt((p\$x - self\$x)^2 + (p\$y - self\$y)^2 + (p\$z  - self\$z)^2)
},
project_op = function(angle, scale) {
x <- self\$x + scale * self\$z * cos(to_radians(angle))
y <- self\$y + scale * self\$z * sin(to_radians(angle))
Point2D\$new(x, y)
},
dilate = function(width = 1, height = width, depth = width) {
if (is.list(width)) {
x <- width\$width * self\$x
y <- width\$height * self\$y
z <- width\$depth * self\$z
} else {
x <- width * self\$x
y <- height * self\$y
z <- depth * self\$z
}
Point3D\$new(x, y, z)
},
# rotation matrix 'R' is post-multiplied...
rotate = function(t = 0, axis_x = 0, axis_y = 0) {
m <- as.matrix(self)
if (is.matrix(t)) {
R <- t
} else {
R <- AA_to_R(t, axis_x, axis_y)
}
Point3D\$new(m %*% R)
},
translate = function(x = 0, y = 0, z = 0) {
if (is.list(x) || inherits(x, "point2d") || inherits(x, "point3d")) {
xt <- x\$x
y <- x\$y
z <- x\$z %||% z
} else {
xt <- x
y <- y
z <- z
}
Point3D\$new(self\$x + xt, self\$y + y, self\$z + z)
}),
active = list(c = function() {
Point3D\$new(mean(self\$x), mean(self\$y), mean(self\$z))
},
width = function() {
2 * max(self\$distance_to(self\$c))
})
)
#' @export
`[.point3d` <- function(x, i) Point3D\$new(x\$x[i], x\$y[i], x\$z[i])
#' @export
length.point3d <- function(x) length(x\$x)
#' @export
as.matrix.point3d <- function(x, ...) {
m <- cbind(x\$x, x\$y, x\$z)
colnames(m) <- c("x", "y", "z")
m
}
#' @export
as.data.frame.point3d <- function(x, ...) {
as.data.frame(as.matrix(x))
}
#' @export
Ops.point3d <- function(e1, e2) {
if (missing(e2)) {
switch(.Generic,
"-" = e1\$dilate(-1),
stop(paste0("unary operation '", .Generic, "' not defined for `point3d` objects"))
)
} else {
if (inherits(e1, "point3d") && inherits(e2, "point3d")) {
switch(.Generic,
"+" = e1\$translate(e2),
"-" = e2\$diff(e1),
stop(paste0("binary operation '", .Generic, "' not defined for `point3d` objects"))
)
} else if (is.numeric(e1) && inherits(e2, "point3d")) {
switch(.Generic,
"*" = e2\$dilate(e1),
stop(paste0("binary operation '", .Generic, "' not defined for `point3d` objects"))
)
} else {
switch(.Generic,
stop(paste0("binary operation '", .Generic, "' not defined for `point3d` objects"))
)
}
}
}

Vector <- R6Class("geometry_vector", # vector is R builtin class
inherit = Point2D,
public = list(dot = function(v) {
self\$x * v\$x + self\$y * v\$y
}),
active = list(orthogonal = function() {
Vector\$new(self\$y, -self\$x)
}))
`[.geometry_vector` <- function(x, i) Vector\$new(x\$x[i], x\$y[i])

Circle <- R6Class("circle",
public = list(c=NULL, r=NULL,
initialize = function(x=0.5, y=0.5, r=0.5) {
if (is.list(x) || inherits(x, "point2d")) {
self\$c <- Point2D\$new(x)
} else {
self\$c <- Point2D\$new(x, y)
}
self\$r <- r
},
project = function(v) {
center <- v\$dot(self\$c)
c(center - self\$r, center + self\$r)
}))

Polygon <- R6Class("polygon",
public = list(vertices=NULL, edges=NULL, normals=NULL,
initialize = function(x=c(0, 0.5, 1, 0.5), y=c(0.5, 1, 0.5, 0)) {
self\$vertices <- Point2D\$new(x, y)
xy <- cbind(self\$x, self\$y)
n <- length(self\$vertices)
# edges
p <- self\$vertices[c(seq(2, n), 1)]
self\$edges <- LineSegment\$new(self\$vertices, p)
self\$normals <- self\$edges\$orthogonal
},
plot = function(gp = gpar()) {
grid.polygon(x=self\$x, y=self\$y, default.units="in", gp = gp)
},
project = function(v) {
n <- length(self\$x)
projections <- numeric(n)
for (ii in seq(n)) {
projections[ii] <- v\$dot(self\$vertices[ii])
}
range(projections)
},
op_edge_order = function(angle) {
op_ref <- self\$c\$translate_polar(angle + 180, 10 * self\$width)
dists <- sapply(self\$edges\$mid_point, function(x) x\$distance_to(op_ref))
order(dists, decreasing = TRUE)
},
op_edges = function(angle) {
self\$edges[self\$op_edge_order(angle)]
}),
private = list(center = NULL),
active = list(x = function() self\$vertices\$x,
y = function() self\$vertices\$y,
c = function() {
if (is.null(private\$center)) {
x <- mean(self\$vertices\$x)
y <- mean(self\$vertices\$y)
private\$center <- Point2D\$new(x, y)
}
private\$center
},
width = function() {
dx <- diff(range(self\$vertices\$x))
dy <- diff(range(self\$vertices\$y))
max(dx, dy)
})
)

LineSegment <- R6Class("line_segment",
public = list(p1=NULL, p2=NULL,
initialize = function(p1, p2) {
self\$p1 <- p1
self\$p2 <- p2
},
face_matrix = function(z=0, depth=1) {
vs <- list()
vs[[1]] <- Point3D\$new(self\$p1, z = z - 0.5*depth)
vs[[2]] <- Point3D\$new(self\$p1, z = z + 0.5*depth)
vs[[3]] <- Point3D\$new(self\$p2, z = z + 0.5*depth)
vs[[4]] <- Point3D\$new(self\$p2, z = z - 0.5*depth)
n <- max(lengths(c(self\$p1, self\$p2)))
m <- matrix(0, nrow = 4 * n, ncol = 3)
for (i_v in seq_len(n)) {
i_m <- 4 * i_v - 4
for (i in 1:4) {
v <- vs[[i]][i_v]
m[i_m + i, ] <- c(v\$x, v\$y, v\$z)
}
}
colnames(m) <- c("x", "y", "z")
m
}),
active = list(mid_point = function() {
x <- (self\$p1\$x + self\$p2\$x) / 2
y <- (self\$p1\$y + self\$p2\$y) / 2
Point2D\$new(x, y)
},
orthogonal = function() {
self\$p1\$diff(self\$p2)\$orthogonal
})
)
#' @export
`[.line_segment` <- function(x, i) LineSegment\$new(x\$p1[i], x\$p2[i])

Line <- R6Class("line",
public = list(a=NULL, b=NULL, c=NULL,
initialize = function(theta, p) {
# a * x + b * y + c = 0
# cos(theta) * x + sin(theta) * y + c = 0
self\$c = -self\$a * p\$x + -self\$b * p\$y
},
distance_to = function(p) {
abs(self\$a * p\$x + self\$b * p\$y + self\$c) / sqrt(self\$a^2 + self\$b^2)
}))

ConvexPolygon <- R6Class("convex_polygon", inherit = Polygon)
#### ConcavePolygon, add a list of convex polygons that cover it to test SAT
#### Most value adding if adding something like megahexes but could be used for stars as well

do_projections_overlap <- function(r1, r2) {
do_ranges_overlap(r1[1], r1[2], r2[1], r2[2])
}

do_convex_polygons_overlap <- function(s1, s2) {
for (n in c(as.list(s1\$normals), as.list(s2\$normals))) {
if (!do_projections_overlap(s1\$project(n), s2\$project(n))) {
return(FALSE)
}
}
TRUE
}

does_convex_polygon_overlap_circle <- function(p, c) { # nolint
n_vertices <- length(p\$vertices)
c_normals <- p\$vertices\$diff(c\$c)\$orthogonal
for (n in c(as.list(p\$normals), as.list(c_normals))) {
if (!do_projections_overlap(p\$project(n), c\$project(n))) {
return(FALSE)
}
}
TRUE
}

do_shapes_overlap <- function(s1, s2) {
if (inherits(s1, "circle") && inherits(s2, "circle")) {
less_than(s1\$c\$distance_to(s2\$c), s1\$r + s2\$r)
} else if (inherits(s1, "convex_polygon") && inherits(s2, "convex_polygon")) {
do_convex_polygons_overlap(s1, s2)
} else if (inherits(s1, "convex_polygon") && inherits(s2, "circle")) {
does_convex_polygon_overlap_circle(s1, s2)
} else if (inherits(s1, "circle") && inherits(s2, "convex_polygon")) {
does_convex_polygon_overlap_circle(s2, s1)
} else {
TRUE
}
}

# Axis-angle representation to rotation matrix
# https://en.wikipedia.org/wiki/Axis-angle_representation
# Because we do rotation matrix post-multiplication instead of pre-multiplication we usually need to multiply angles
# in following algorithms by -1

#' @rdname geometry_utils
#' @inheritParams save_piece_obj
#' @param axis_z Third coordinate of the axis unit vector (usually inferred).
#' @param angle Angle in degrees (counter-clockwise)
#' @param ... Ignored
#' @export
AA_to_R <- function(angle = 0, axis_x = 0, axis_y = 0, axis_z = NA, ...) {
if (is.na(axis_z)) {
inner <- 1 - axis_x^2 - axis_y^2
if (nigh(inner, 0)) {
axis_z <- 0
} else {
axis_z <- sqrt(inner)
}
} else {
norm <- sqrt(axis_x^2 + axis_y^2 + axis_z^2)
if (!nigh(norm, 1)) {
axis_x <- axis_x / norm
axis_y <- axis_y / norm
axis_z <- axis_z / norm
}
}
e <- c(axis_x, axis_y, axis_z)
I <- diag(3)
K <- cross_matrix(e)
R <- I + s * K + (1 - c) * K %*% K
R
}

# "cross" product matrix
# https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication
cross_matrix <- function(v) {
m <- matrix(0, nrow=3, ncol=3)
m[1, 2] <- -v[3]
m[1, 3] <- v[2]
m[2, 1] <- v[3]
m[2, 3] <- -v[1]
m[3, 1] <- -v[2]
m[3, 2] <- v[1]
m
}

# trace of a (square) matrix
trace <- function(m) sum(diag(m))

# Name 'nigh' to avoid potential conflict with 'dplyr::near()'
nigh <- function(x, y, tolerance = 1e-6) {
if (length(y) < length(x)) y <- rep(y, length.out=length(x))
isTRUE(all.equal(x, y, tolerance = tolerance))
}

# more robust handling of arccosine input
arccos <- function(x) {
if (nigh(x, 1) && x > 1) x <- 1
if (nigh(x, -1) && x < -1) x <- -1
acos(x)
}

# Rotation matrix to Axis-angle representation
# https://en.wikipedia.org/wiki/Axis-angle_representation
#' @param R 3D rotation matrix (post-multiplied)
#' @rdname geometry_utils
#' @export
R_to_AA <- function(R = diag(3)) {
t <- arccos(0.5 * (trace(R) - 1))
if (nigh(R, diag(3))) { # no rotation
t <- 0
e <- c(0, 0, 1)
} else if (nigh(t, pi)) { # 180 degree rotation
t <- pi
B <- 0.5 * (R + diag(3))
e <- sqrt(diag(B))
sB <- sign(B)
if (nigh(sB, ppn)) {
e[3] <- -e[3]
t <- -pi
} else if (nigh(sB, pnp)) {
e[2] <- -e[2]
} else if (nigh(sB, npp)) {
e[1] <- -e[1]
}
} else {
e <- numeric(3)
e[1] <- R[3,2] - R[2,3]
e[2] <- R[1,3] - R[3,1]
e[3] <- R[2,1] - R[1,2]
e <- e / (2 * sin(-t))
}
if (e[3] < 0) { # Force z-axis element positive
e <- -e
t <- -t
}
list(angle = to_degrees(t), axis_x = e[1], axis_y = e[2], axis_z = e[3])
}

# Sign matrices for "B" matrix used to tell signs for axis unit vector (up to sign ambiguity) when angle = 180 degrees
# signs for ppp and nnn don't need to be changed
# ppn and npp
ppn <- matrix(c(1, 1, -1,
1, 1, -1,
-1, -1, 1), ncol = 3, byrow = TRUE)
# pnp and npn
pnp <- matrix(c(1, -1, 1,
-1, 1, -1,
1, -1, 1), ncol = 3, byrow = TRUE)
# pnn and npp
npp <- matrix(c(1, -1, -1,
-1, 1, 1,
-1, 1, 1), ncol = 3, byrow = TRUE)

# Basic 3D rotation matrices
# https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions

#' Helper functions for making geometric calculations.
#'
#' \code{to_x}, \code{to_y}, \code{to_r}, \code{to_t} convert
#' between polar coordinates (in degrees) and Cartesian coordinates.
#' \code{AA_to_R} and \code{R_to_AA} convert back and forth between (post-multiplied) rotation matrix
#' and axis-angle representations of 3D rotations.
#' \code{R_x}, \code{R_y}, and \code{R_z} build (post-multiplied) rotation matrices for simple rotations around
#' the x, y, and z axes.
#'
#' \code{pp_cfg} uses polar coordinates to determine where the "primary" and "directional"
#'      symbols are located on a game piece.
#' They are also useful for drawing certain shapes and for making game diagrams on hex boards.
#'
#' \code{piecepackr} and \code{grid} functions use angles in degrees
#'  but the \code{base} trigonometry functions usually use radians.
#'
#' \code{piecepackr}'s 3D graphics functions \code{save_piece_obj}, \code{piece}, and \code{piece3d}
#' use the axis-angle representation for 3D rotations.
#' The axis-angle representation involves specifying a unit vector
#' indicating the direction of an axis of rotation and an angle describing the (counter-clockwise)
#' rotation around that axis.  Because it is a unit vector one only needs to specify the first two elements,
#' \code{axis_x} and \code{axis_y}, and we are able to infer the 3rd element \code{axis_z}.  The default of
#' \code{axis = 0}, \code{axis_y = 0}, and implied \code{axis_z = 1}
#' corresponds to a rotation around the z-axis which is reverse-compatible
#' with the originally 2D \code{angle} interpretation in \code{grid.piece}.  In order to figure out the appropriate
#' axis-angle representation parameters \code{R_to_AA}, \code{R_x}, \code{R_y}, and \code{R_z} allow one
#' to first come up with an appropriate (post-multiplied) 3D rotation matrix by chaining simple rotations
#' and then convert them to the corresponding axis-angle representation.
#' Pieces are rotated as if their center was at the origin.
#'
#' @examples
#'  to_x(90, 1)
#'  to_y(180, 0.5)
#'  to_t(0, -1)
#'  to_r(0.5, 0)
#'  # default axis-angle axis is equivalent to a rotation about the z-axis
#'  all.equal(AA_to_R(angle=60), R_z(angle=60))
#'  # axis-angle representation of 90 rotation about the x-axis
#'  R_to_AA(R_x(90))
#'  # find Axis-Angle representation of first rotating about x-axis 180 degrees
#'  # and then rotating about z-axis 45 degrees
#'  R_to_AA(R_x(180) %*% R_z(45))
#'
#' @seealso \url{https://en.wikipedia.org/wiki/Axis-angle_representation} for more details
#'   about the Axis-angle representation of 3D rotations.
#'   See \code{\link[base]{Trig}} for R's built-in trigonometric functions.
#' @name geometry_utils
NULL

#' @rdname geometry_utils
#' @export
R_x <- function(angle = 0) {
matrix(c(1, 0, 0,
0, c, -s,
0, s, c),
ncol = 3, byrow = TRUE)
}

#' @rdname geometry_utils
#' @export
R_y <- function(angle = 0) {
matrix(c(c, 0, s,
0, 1, 0,
-s, 0, c),
ncol = 3, byrow = TRUE)
}

#' @rdname geometry_utils
#' @export
R_z <- function(angle = 0) {
matrix(c(c, -s, 0,
s, c, 0,
0, 0, 1),
ncol = 3, byrow = TRUE)
}

#' @rdname geometry_utils
#' @export
to_radians <- function(t) pi * t / 180

#' @rdname geometry_utils
#' @export
to_degrees <- function(t) 180 * t / pi

#' @rdname geometry_utils
#' @param t Angle in degrees (counter-clockwise)
#' @export
to_x <- function(t, r) {
}

#' @rdname geometry_utils
#' @export
to_y <- function(t, r) {
}

#' @rdname geometry_utils
#' @param x Cartesian x coordinate
#' @param y Cartesian y coordinate
#' @export
to_r <- function(x, y) {
sqrt(x^2 + y^2)
}

#' @rdname geometry_utils
#' @export
to_t <- function(x, y) {
to_degrees(atan2(y, x))
}
```
trevorld/piecepack documentation built on May 20, 2024, 4:31 a.m.