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Billiard scales
To run this, you need Rust installed.
- In any folder you want, create a directory
billiard-scales. - Using the command line, run
cargo initin the new directorybilliard-scales. - Create or copy the following .rs files in
billiard-scales/src. Copy the contents of theCargo.tomlprovided into the newCargo.toml - Run
cargo run --release.
planegeometry.rs
// plane_geometry.rs
// v0.0.0 by inthar
// Geometry of points and lines in the plane, implemented with rational numbers. The implementation is janky, though.
// Can be used to enumerate ternary billiard scales.
use std::ops::Add;
use std::ops::Sub;
use std::ops::Neg;
use std::iter::zip;
use num_traits::sign::Signed;
use std::fmt;
use num_rational::Rational64 as r64;
#[derive(Copy, Clone, Debug, Hash, PartialEq, Eq)]
/// The possible configurations of a point and an oriented line on the plane.
pub enum PointLineConfiguration {
Left, // The point is on the left half-plane.
Right, // The point is on the right half-plane.
OnTheLine, // The point is on the line.
}
fn is_between(a:r64, b:r64, c:r64) -> bool {
if b == c {
a == b
} else if b < c {
b < a && a < c
} else {
c < a && a < b
}
}
/// Gives the greatest common denominator of the two inputs, unless that's 2^63.
/// 2^63 doesn't fit in an `i64`, so it returns -2^63, which does.
pub fn gcd(u: i64, v: i64) -> i64 {
// `wrapping_abs` gives a number's absolute value, unless that's 2^63. 2^63
// won't fit in `i64`, so it gives -2^63 instead.
let mut v = v.wrapping_abs() as u64;
if u == 0 {
return v as i64;
}
let mut u = u.wrapping_abs() as u64;
if v == 0 {
return u as i64;
}
// `|` is bitwise OR. `trailing_zeros` quickly counts a binary number's
// trailing zeros, giving its prime factorization's exponent on two.
let gcd_exponent_on_two = (u | v).trailing_zeros();
// `>>=` divides the left by two to the power of the right, storing that in
// the left variable. `u` divided by its prime factorization's power of two
// turns it odd.
u >>= u.trailing_zeros();
v >>= v.trailing_zeros();
while u != v {
if u < v {
// Swap the variables' values with each other.
core::mem::swap(&mut u, &mut v);
}
u -= v;
u >>= u.trailing_zeros();
}
// `<<` multiplies the left by two to the power of the right.
(u << gcd_exponent_on_two) as i64
}
/// A rational number or unsigned infinity; represents a valid slope for a line in Q^2.
#[derive(Copy, Clone, Debug, Hash)]
pub struct Slope {
numer: i64,
denom: i64,
}
impl PartialEq for Slope {
fn eq(&self, other: &Self) -> bool {
self.numer * other.denom == other.numer * self.denom
}
}
impl fmt::Display for Slope {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write!(f, "{}/{}", self.numer, self.denom)
}
}
impl Slope {
/// Converts a `Slope` to a rational; returns Err if `denom` is 0.
pub fn as_rational(&self) -> Result<r64, String> {
if self.denom != 0 {
Ok(r64::new(self.numer, self.denom))
} else {
Err("Attempted to convert a Slope with `denom` == 0 into a Rational64".to_string())
}
}
/// Converts the projective rational to a rational; panics if the denominator is zero.
pub fn as_rational_raw(&self) -> r64 {
r64::new(self.numer, self.denom)
}
/// Create a new reduced Slope; panics if both numer and denom are 0.
pub fn new(n: i64, d: i64) -> Result<Self, String> {
let (mut numer, mut denom) = (n, d);
if numer == 0 && denom == 0 {
Err("Attempted to create a Slope with both `numer` and `denom` equal to 0".to_string())
} else {
if denom < 0 {
numer = -numer;
denom = -denom;
}
let d = gcd(numer, denom);
if d > 1 {
numer = numer/d;
denom = denom/d;
}
Ok(Self {
numer,
denom,
})
}
}
/// Create a new Slope without checking that `numer` == `denom` == 0, or reducing the rational afterwards.
pub fn raw(numer: i64, denom: i64) -> Self {
Self {
numer,
denom,
}
}
/// Create a new `Slope` from a rational.
pub fn rational(rat: r64) -> Self {
Self {
numer: *rat.numer(),
denom: *rat.denom(),
}
}
/// Create a new `Slope` from an integer.
pub fn integer(n: i64) -> Self {
// use 'raw' since an integer is always reduced
Self::raw(n, 1)
}
/// Shorthand for slope 0.
pub fn zero() -> Self {
Self::raw(0, 1)
}
/// Shorthand for slope 1.
pub fn one() -> Self {
Self::raw(1, 1)
}
/// Shorthand for infinite slope.
pub fn infinity() -> Self {
Self::raw(1, 0)
}
}
/// A finite point; a member of Q^2.
#[derive(Copy, Clone, Debug, Hash, PartialEq)]
pub struct Point {
// The x-coordinate.
x: r64,
// The y-coordinate.
y: r64,
}
impl Add for Point {
type Output = Self;
fn add(self, other: Self) -> Self {
Self {
x: self.x + other.x,
y: self.y + other.y,
}
}
}
impl Sub for Point{
type Output = Self;
fn sub(self, other: Self) -> Self {
Self {
x: self.x - other.x,
y: self.y - other.y,
}
}
}
impl Neg for Point{
type Output = Self;
fn neg(self) -> Self {
Self {
x: -self.x,
y: -self.y,
}
}
}
impl Point {
pub fn new(x: r64, y: r64) -> Self {
Self {
x,
y,
}
}
pub fn zero() -> Self {
Self {
x: r64::new(0,1),
y: r64::new(0,1),
}
}
pub fn scalar_mult(lambda: r64, v: Point) -> Self {
Self {
x: lambda*v.x,
y: lambda*v.y,
}
}
pub fn average(points: &Vec<Self>) -> Self {
let mut sum = Self::zero();
for p in points {
sum = sum + *p;
}
Self::scalar_mult(r64::new(1,1)/r64::new(points.len() as i64,1), sum)
}
pub fn theta(v: Point) -> f64 {
( *v.y.numer() as f64 / *v.y.denom() as f64 ).atan2( *v.x.numer() as f64 / *v.x.denom() as f64 )
}
/// Return a copy of `points` sorted using the direction they make from the centroid, according to theta(v-centroid) in (-pi, pi].
fn ordered_ccw(mut points: Vec<Self>) -> Vec<Self> {
let centroid = Self::average(&points);
points.sort_by(|a,b| Self::theta(*a-centroid).partial_cmp(&Self::theta(*b-centroid)).unwrap() );
points
}
pub fn midpoint(p1: &Self, p2: &Self) -> Self {
Self {
x: (p1.x+p2.x)/r64::new(2,1),
y: (p1.y+p2.y)/r64::new(2,1),
}
}
// Use the slope-intercept formula including the case where the slope is infinite.
pub fn are_collinear(p1: Self, p2: Self, p3: Self) -> bool {
(p1.y - p3.y)*(p1.x - p2.x) == (p1.y - p2.y)*(p1.x - p3.x)
}
}
impl fmt::Display for Point {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write!(f, "({}, {})", self.x, self.y)
}
}
/// An oriented line in Q^2 of rational or infinite slope.
#[derive(Copy, Clone, Debug, Hash)]
pub struct Line {
point1: Point,
point2: Point,
}
impl PartialEq for Line {
fn eq(&self, other: &Self) -> bool {
self.slope() == other.slope() && self.has_point(other.point())
}
}
impl Line {
pub fn slope(self: Line) -> Slope {
Slope::new(
(*(self.point2.y-self.point1.y).numer()) * (*(self.point2.x-self.point1.x).denom()),
(*(self.point2.x-self.point1.x).numer()) * (*(self.point2.y-self.point1.y).denom())).unwrap()
}
pub fn point(self: Line) -> Point {
self.point1
}
pub fn from_slope_and_point(sl: Slope, point: Point) -> Self {
Self {
point1: point,
point2: point + Point::new(r64::new_raw(sl.denom, 1), r64::new_raw(sl.numer, 1)),
}
}
pub fn from_points(point1: Point, point2: Point) -> Result<Self, String> {
if point1 == point2 {
Err("The points {point1} and {point2} are equal".to_string())
} else {
Ok(Line{point1, point2})
}
}
/// Test whether a line is vertical.
fn is_vertical(&self) -> bool {
self.point1.x == self.point2.x
}
/// Test whether a line is horizontal.
fn is_horizontal(&self) -> bool {
self.point1.y == self.point2.y
}
/// Test whether two lines are parallel.
fn is_parallel(&self, other: Self) -> bool {
self.slope() == other.slope()
}
/// Test whether `p` satisfies the equation of the line.
fn has_point(&self, p: Point) -> bool {
(p.y - self.point().y)*r64::new_raw(self.slope().denom,1) == r64::new_raw(self.slope().numer,1)*(p.x - self.point().x)
}
/// Determine whether `p1` and `p2` are on the same side of the line.
fn on_same_side(&self, p1: Point, p2: Point) -> bool {
if self.is_vertical() {
(p1.x < self.point1.x && p2.x < self.point1.x ) || (p1.x > self.point1.x && p2.x > self.point1.x)
} else { // both `p1` and `p2` are above `self`
((p1.y - self.point1.y)*r64::new_raw(self.slope().denom,1) > r64::new_raw(self.slope().numer,1)*(p1.x - self.point1.x) &&
(p2.y - self.point1.y)*r64::new_raw(self.slope().denom,1) > r64::new_raw(self.slope().numer,1)*(p2.x - self.point1.x))
|| // ... or both are below `self`
((p1.y - self.point1.y)*r64::new_raw(self.slope().denom,1) < r64::new_raw(self.slope().numer,1)*(p1.x - self.point1.x) &&
(p2.y - self.point1.y)*r64::new_raw(self.slope().denom,1) < r64::new_raw(self.slope().numer,1)*(p2.x - self.point1.x))
}
}
/// Depending on the orientation of the line, determine whether the point `p` is to the left, to the right, or on the line.
pub fn point_line_config(&self, p: Point) -> PointLineConfiguration {
if self.point2.x != self.point1.x { // For non-vertical lines
if ((p.y - self.point().y)*r64::new_raw(self.slope().denom, 1) - r64::new_raw(self.slope().numer, 1)*(p.x - self.point().x)) * (self.point2.x - self.point1.x).signum() > r64::new_raw(0,1) {
PointLineConfiguration::Left
} else if ((p.y - self.point().y)*r64::new_raw(self.slope().denom, 1) - r64::new_raw(self.slope().numer, 1)*(p.x - self.point().x)) * (self.point2.x - self.point1.x).signum() == r64::new_raw(0,1) {
PointLineConfiguration::OnTheLine
} else {
PointLineConfiguration::Right
}
} else { // For vertical lines
// If point2.y < point1.y, then the line is oriented downwards, so multiply by -1 to compensate.
if (p.x-self.point().x) * (self.point2.y-self.point1.y) < r64::new_raw(0,1) { // upwards vertical line and point to the left of it, or downwards vertical line and point to the right of it.
PointLineConfiguration::Left
} else if (p.x-self.point().x) * (self.point2.y-self.point1.y) == r64::new_raw(0,1) {
PointLineConfiguration::OnTheLine
} else {
PointLineConfiguration::Right
}
}
}
}
impl fmt::Display for Line {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write!(f, "(slope: {}, point: {})", self.slope(), self.point1)
}
}
/// For two non-parallel ones, find the intersection; for two parallel lines, return `None`.
pub fn intersection(l1: Line, l2: Line) -> Option<Point> {
let (x1, y1, x2, y2) = (l1.point().x, l1.point().y, l2.point().x, l2.point().y);
match (l1.slope(), l2.slope()) {
(s1, s2) if s1 == s2 => None, // Same slope
(s1, s2) if s1 == Slope::infinity() && s2 != Slope::infinity() => Some(Point {
x: x1,
y: s2.as_rational_raw() * (x1 - x2) + y2,
}),
(s1, s2) if s1 != Slope::infinity() && s2 == Slope::infinity() => Some(Point {
x: x2,
y: s1.as_rational_raw() * (x2 - x1) + y1,
}),
(s1, s2) => Some(Point {
x: (y1 - s1.as_rational_raw() * x1 + s2.as_rational_raw() * x2 - y2)
/ (s2.as_rational_raw() - s1.as_rational_raw()),
y: (s1.as_rational_raw() * s2.as_rational_raw() * (x2 - x1) + s2.as_rational_raw() * y1 - s1.as_rational_raw() * y2)
/ (s2.as_rational_raw() - s1.as_rational_raw()),
}),
}
}
#[derive(Clone, Debug, Hash)]
pub struct ConvexPolygon {
vertices: Vec<Point>, // Assumes that the vertices are ordered in clockwise or counterclockwise order.
}
impl PartialEq for ConvexPolygon {
fn eq(&self, other: &Self) -> bool {
// If all the points in both polygons are equal then they are equal; we ensure this by enforcing a standard order when a `ConvexPolygon` is created.
if other.vertices.len() != self.vertices.len() {
return false;
}
for (a, b) in zip(self.vertices.iter(), other.vertices.iter()) {
if a != b {
return false;
}
}
true
}
}
impl ConvexPolygon { // Assumes that the vertices form a convex polygon. Does not check if the `Vec` of vertices consists of distinct elements.
pub fn new(vertices: Vec<Point>) -> Option<Self> {
if vertices.len() >= 3 {
Some(Self {
vertices: Point::ordered_ccw(vertices), // This will work for vertices of a convex polygon.
})
} else {
None
}
}
fn raw(vertices: Vec<Point>) -> Self { // Does not check for ordering.
Self {
vertices: Point::ordered_ccw(vertices), // This will work for vertices of a convex polygon.
}
}
pub fn centroid(&self) -> Point {
Point::average(&self.vertices)
}
pub fn has_point_inside(&self, point: Point) -> bool {
// Do `point` and edge_i+1 x edge_i+2 lie on the same side of edge_i?
for i in 0..self.vertices.len()-2 {
if !Line::from_points(self.vertices[i], self.vertices[i+1]).unwrap().on_same_side(point, self.vertices[i+2]) {
return false;
}
}
// Check last two triples.
if !Line::from_points(self.vertices[self.vertices.len()-2], self.vertices[self.vertices.len()-1]).unwrap().on_same_side(point, self.vertices[0]) {
return false;
} else if !Line::from_points(self.vertices[self.vertices.len()-1], self.vertices[0]).unwrap().on_same_side(point, self.vertices[1]) {
return false;
} else {
return true;
}
}
/// A vector storing the edges of the polygon in order.
fn lines_for_edges(&self) -> Vec<Line> {
let mut result: Vec<Line> = (0..self.vertices.len()-1).into_iter().map(|i| Line::from_points(self.vertices[i], self.vertices[i+1]).unwrap()).collect();
result.push(Line::from_points(self.vertices[self.vertices.len()-1], self.vertices[0]).unwrap());
result
}
/// Subdivide a polygon with the given line and return the two pieces.
pub fn subdivide(&self, line: Line) -> (Option<ConvexPolygon>, Option<ConvexPolygon>) {
// Collect vertices that fall to the left, on thhe line, and to the right, respectively. The left and right collections are missing two vertices.
let mut vertices_left: Vec<Point> = self.vertices
.iter()
.filter(|&x| line.point_line_config(*x) == PointLineConfiguration::Left).cloned().collect();
let mut vertices_right: Vec<Point> = self.vertices
.iter()
.filter(|&x| line.point_line_config(*x) == PointLineConfiguration::Right).cloned().collect();
let vertices_on_line: Vec<Point> = self.vertices
.iter()
.filter(|&x| line.point_line_config(*x) == PointLineConfiguration::OnTheLine).cloned().collect();
if !vertices_left.is_empty() || !vertices_right.is_empty() {
// Add any vertices on the line both to the new left polygon and the new right polygon.
for point in &vertices_on_line {
vertices_left.push(*point);
vertices_right.push(*point);
}
// Look for intersection points on edges with the line.
for edge in &self.lines_for_edges() {
if let Some(intsn) = intersection(*edge, line) {
if is_between(intsn.x, edge.point1.x, edge.point2.x) && is_between(intsn.y, edge.point1.y, edge.point2.y) {
vertices_left.push(intsn);
vertices_right.push(intsn);
}
}
}
// At most two vertices are added, so if there was no vertex to one side originally, that side doesn't have a polygon.
}
(ConvexPolygon::new(vertices_left), ConvexPolygon::new(vertices_right))
}
fn triangle_from_lines(l1: Line, l2: Line, l3: Line) -> Option<Self> {
if !l1.is_parallel(l2) && !l2.is_parallel(l3) && !l3.is_parallel(l1) {
let p1 = intersection(l1, l2).unwrap();
let p2 = intersection(l2, l3).unwrap();
let p3 = intersection(l3, l1).unwrap();
Self::new(vec![p1, p2, p3])
} else {
None
}
}
fn parallelogram_from_lines(l1: Line, l2: Line, m1: Line, m2: Line) -> Option<Self> {
if l1.is_parallel(l2) && l1 != l2 && m1.is_parallel(m2) && m1 != m2 {
let p1 = intersection(l1, m1).unwrap();
let p2 = intersection(l1, m2).unwrap();
let p3 = intersection(l2, m1).unwrap();
let p4 = intersection(l2, m2).unwrap();
Self::new(vec![p1, p2, p3, p4])
} else {
None
}
}
}
fn projected_constraint_planes(a: u32, b: u32, c: u32) -> (Vec<Line>, Vec<Line>, Vec<Line>) {
let first = (0..b+c+1).map(|i| Line::from_slope_and_point(Slope::raw(0, 1), Point::new(r64::new(0, 1), r64::new(-(b as i64) + (i as i64), c as i64 ) ) )).collect(); // horizontal lines going up (Left)
let second = (0..a+c+1).map(|j| Line::from_slope_and_point(Slope::raw(1, 0), Point::new(r64::new((c as i64)-(j as i64), c as i64), r64::new(0, 1) ) )).collect(); // vertical lines going left (Left)
let third = (0..a+b+1).map(|k| Line::from_slope_and_point(Slope::raw(b as i64, a as i64), Point::new( r64::new(0, 1), r64::new(-(b as i64)+(k as i64), a as i64))) ).collect(); // lines of pos. slope going up (Left)
(first, second, third)
}
/// Project the unit cube in R^3 via a linear map that has (a,b,c) in its kernel.
pub fn projected_cube(a: u32, b: u32, c: u32) -> Option<ConvexPolygon> {
if a > 0 && b > 0 && c > 0 {
let projected_cube_vertices = vec![Point::new(r64::new(0,1), r64::new(1,1)),
Point::new(r64::new(1,1), r64::new(1,1)),
Point::new(r64::new(1,1), r64::new(0,1)),
Point::new(r64::new(-(a as i64), c as i64), r64::new(-(b as i64), c as i64) + r64::new(1,1)),
Point::new(r64::new(-(a as i64), c as i64), r64::new(-(b as i64), c as i64)),
Point::new(r64::new(-(a as i64), c as i64) + r64::new(1,1), r64::new(-(b as i64), c as i64))];
ConvexPolygon::new(projected_cube_vertices)
} else {
None
}
}
/// Partition the projected unit cube with the projected constraint planes.
pub fn project_and_partition(a: u32, b: u32, c: u32) -> Vec<ConvexPolygon> {
let projected_cube = projected_cube(a, b, c).unwrap();
let lineses = projected_constraint_planes(a, b, c); // it's a vector of vector of `Line`s
let mut first_shapes = Vec::<ConvexPolygon>::new();
let mut second_shapes = Vec::<ConvexPolygon>::new();
let mut third_shapes = Vec::<ConvexPolygon>::new();
let mut i: usize = 0;
let mut remaining: ConvexPolygon = projected_cube;
while let (maybe_remaining, maybe_next) = remaining.subdivide((lineses.0)[i]) { // The remaining portion is to the Left.
match (maybe_remaining, maybe_next) {
(Some(r), None) => {
remaining = r;
},
(Some(r), Some(next)) => {
remaining = r;
first_shapes.push(next);
},
(None, Some(next)) => { // This is the last `next` you need to look at.
first_shapes.push(next);
break;
},
(_, _) => {
break;
},
}
i = i+1;
}
for polygon in first_shapes {
let mut j: usize = 0;
let mut remaining2: ConvexPolygon = polygon;
loop {
let (maybe_remaining, maybe_next) = remaining2.subdivide((lineses.1)[j]); // The remaining portion is to the Left.
match (maybe_remaining, maybe_next) {
(Some(r), None) => {
remaining2 = r;
},
(Some(r), Some(next)) => {
remaining2 = r;
second_shapes.push(next);
},
(None, Some(next)) => {
second_shapes.push(next);
break;
},
(_, _) => {
break;
},
}
j = j+1;
}
}
for polygon in second_shapes {
let mut j: usize = 0;
let mut remaining3: ConvexPolygon = polygon;
while let (maybe_remaining, maybe_next) = remaining3.subdivide((lineses.2)[j]) { // The remaining portion is to the Left.
match (maybe_remaining, maybe_next) {
(Some(r), None) => {
remaining3 = r;
},
(Some(r), Some(next)) => {
remaining3 = r;
third_shapes.push(next);
},
(None, Some(next)) => {
third_shapes.push(next);
break;
},
(_, _) => {
break;
},
}
j = j+1;
}
}
third_shapes
}
/// Advance a projected point, `pt`, by one letter according to what integer coordinate it next hits as it traverses in the given velocity,
/// and return the next point and the corresponding letter, one of "x", "y", and "z".
pub fn advance(a: u32, b: u32, c: u32, pt: Point) -> Result<(Point, String), String> {
let projected_cube = projected_cube(a, b, c).unwrap();
// Consider: _ /
// |
if !projected_cube.has_point_inside(pt) {
Err(format!("Point passed to `advance`, {}, is not inside the projected cube", pt))
} else {
let projected_x_y_equals_1_1 = Line::from_slope_and_point(Slope::raw(b as i64, a as i64), Point::new( r64::new(0, 1), r64::new(-(b as i64)+(a as i64), a as i64))); // the /
let projected_x_z_equals_1_1 = Line::from_slope_and_point(Slope::raw(0, 1), Point::new(r64::new(0, 1), r64::new(-(b as i64)+(c as i64), c as i64))); // the _, oriented right
let projected_y_z_equals_1_1 = Line::from_slope_and_point(Slope::raw(1, 0), Point::new(r64::new((c as i64)-(a as i64), c as i64), r64::new(0, 1))); // the |, oriented up
if projected_x_y_equals_1_1.point_line_config(pt) == PointLineConfiguration::Right
&& projected_y_z_equals_1_1.point_line_config(pt) == PointLineConfiguration::Right { // Below the / and to the right of the | => x has changed, change x coordinate to 0 and project; move x coordinate to the left
Ok((pt - Point::new(r64::new(1,1), r64::new(0,1)), "x".to_string())) // new point is one unit to the left
} else if projected_x_z_equals_1_1.point_line_config(pt) == PointLineConfiguration::Left
&& projected_x_y_equals_1_1.point_line_config(pt) == PointLineConfiguration::Left { // Above the _ and to the left of the / => y has changed, change y coordinate to 0 and project; move y coordinate down
Ok((pt - Point::new(r64::new(0,1), r64::new(1,1)), "y".to_string())) // new point is one unit below
} else if projected_x_z_equals_1_1.point_line_config(pt) == PointLineConfiguration::Right
&& projected_y_z_equals_1_1.point_line_config(pt) == PointLineConfiguration::Left { // Below the _ and to the left of the | => z has changed, change y coordinate to 0 and project
Ok((pt + Point::new(r64::new((a as i64), (c as i64)), r64::new((b as i64), (c as i64) ) ), "z".to_string())) // new point is one unit below in the z-direction, and e3 is projected to (-a/c, -b/c)
} else {
Err(format!("Point passed to `advance`, {}, is inside the projected cube but falls on a projected constraint plane", pt))
}
}
}
lib.rs
// scaletheory.rs
// Inthar's higher-rank scale theory library
// v0.0.0
use std::collections::BTreeSet;
use std::collections::HashSet;
use num_rational::Rational64 as r64;
mod plane_geometry;
use plane_geometry::*;
use std::cmp::{min, max};
type JIScaleProperty = fn(s: [r64]) -> bool;
type CentsScaleProperty = fn(s: [f64]) -> bool;
type AbstractScaleProperty = fn(s: &str) -> bool;
type EquivalenceRelation<T> = fn(&T, &T) -> bool;
pub fn are_rotationally_equivalent(scale1: &String, scale2: &String)-> bool{
if scale1.len() != scale2.len() {
return false;
} else {
let length = scale1.len();
for i in 0..length {
if slice_cyclic_string(scale1, i, length) == *scale2 {
return true;
}
}
false
}
}
// Assumes that equiv : T x T -> bool is an equivalence relation.
pub fn equivalence_class_representatives<T>(set: Vec<T>, equiv: EquivalenceRelation<T>) -> Vec<T>
where T : Clone + PartialEq {
let mut result : Vec<T> = Vec::new();
for i in 0..set.len() {
let mut found_class_equivalent = false;
'a: for j in 0..result.len(){ // All vecs in result are nonempty.
if equiv(&result[j], &set[i]){
found_class_equivalent = true;
continue 'a;
}
}
if found_class_equivalent == false {
result.push(set[i].clone());
}
}
result
}
fn gcd(m: i64, n: i64) -> i64 {
let mut x = m;
let mut y = n;
while x != 0 && y != 0 {
if x > y {
x = x % y;
} else {
y = y % x;
}
}
if y == 0 {x} else {y}
}
fn gcd_u32(m: u32, n: u32) -> u32 {
let mut x = m;
let mut y = n;
while x != 0 && y != 0 {
if x > y {
x = x % y;
} else {
y = y % x;
}
}
if y == 0 {x} else {y}
}
fn lcm(m: i64, n: i64) -> i64 {
m*n/gcd(m,n)
}
// Is the scale word s = s(x, y, z) abstractly mv3?
pub fn maximum_variety(s: &str) -> usize {
let mut result = 0;
let floor_half: usize = s.len()/2;
for l in 1..(floor_half+1) {
let mut sizes: BTreeSet<(usize, usize, usize)> = BTreeSet::new();
for b in 0..(s.len()) {
let mut size: (usize, usize, usize) = (0, 0, 0);
let sl = &slice_cyclic_string(s, b, l);
let chars_in_sl = sl.chars();
for ch in chars_in_sl {
match ch {
'x' => {size.0 += 1;},
'y' => {size.1 += 1;},
'z' => {size.2 += 1;},
_ => {panic!()},
}
}
sizes.insert(size);
}
result = max(result, sizes.len()); // update result
}
result
}
// Is the scale word s = s(x, y, z) pairwise well-formed (pwf)?
pub fn is_pwf(s: &str) -> bool {
let s1 = s.replace("z", "y");
let s2 = s.replace("z", "x");
let s3 = s.replace("y", "x").replace("z", "y");
return is_sv2(&s1)&&is_sv2(&s2)&&is_sv2(&s3);
}
// Is the scale word s = s(a,b,c) pairwise well-formed (pwf)?
pub fn is_pmos(s: &str) -> bool {
let s1 = s.replace("z", "y");
let s2 = s.replace("z", "x");
let s3 = s.replace("y", "x").replace("z", "y");
maximum_variety(&s1) == 2 && maximum_variety(&s2) == 2 && maximum_variety(&s3) == 2
}
// Is the scale word s = s(x, y, z), x > y > z > 0, monotone-mos?
pub fn is_mmos(scale: &str) -> bool {
let s1 = scale.replace("y", "x").replace("z", "y"); // m = L
let s2 = scale.replace("z", "y"); // m = s
let s3 = scale.replace("z", ""); // s = 0
maximum_variety(&s1) == 2 && maximum_variety(&s2) == 2 && maximum_variety(&s3) == 2
}
// Is the scale s = s(x, y) a single period mos (aka well-formed, wf)?
pub fn is_sv2(scale: &str) -> bool {
if scale.len() <= 1{
return false;
}
let floor_half : usize = scale.len()/2;
for slice_length in 1..(floor_half+1) {
let mut sizes: BTreeSet<(usize, usize)> = BTreeSet::new();
for basepoint in 0..(scale.len()) {
let mut size: (usize, usize) = (0, 0);
let sl = &slice_cyclic_string(scale, basepoint, slice_length);
let chars_in_sl = sl.chars();
for c in chars_in_sl {
match c {
'x' => {size.0 += 1;},
'y' => {size.1 += 1;},
_ => {panic!()},
}
}
sizes.insert(size);
}
if sizes.len() != 2 {
return false;
}
}
return true;
}
// Return the balance of `s` = `s`(x, y, z), defined as max { | |w|_{x_i} - |w'|_{x_i} | : x_i is a letter of `s` and k = len(w) = len(w') }.
pub fn balance(scale: &str) -> usize {
if scale.len() <= 1 {
scale.len()
} else {
let mut result = 0;
let floor_half: usize = scale.len()/2;
for slice_length in 1..(floor_half+1) {
let (mut min_x, mut max_x, mut min_y, mut max_y, mut min_z, mut max_z) = (usize::MAX, 0, usize::MAX, 0, usize::MAX, 0); // min and max for each length
let mut sizes: BTreeSet<(usize, usize, usize)> = BTreeSet::new();
for basepoint in 0..(scale.len()) {
let mut size: (usize, usize, usize) = (0, 0, 0);
let sl = &slice_cyclic_string(scale, basepoint, slice_length);
let chars_in_sl = sl.chars();
for ch in chars_in_sl {
match ch {
'x' => {size.0 += 1;},
'y' => {size.1 += 1;},
'z' => {size.2 += 1;},
_ => {panic!()},
}
}
sizes.insert(size);
}
for vector in &sizes {
min_x = min(min_x, vector.0);
max_x = max(max_x, vector.0);
min_y = min(min_y, vector.1);
max_y = max(max_y, vector.1);
min_z = min(min_z, vector.2);
max_z = max(max_z, vector.2);
}
result = max(result, max(max(max_x.abs_diff(min_x), max_y.abs_diff(min_y)), max_z.abs_diff(min_z)));
}
result
}
}
// Function for computing two properties, maximum variety and balance, while executing the subroutine to extract words only once.
pub fn max_variety_and_balance(s: &str) -> (usize, usize) {
if s.len() <= 1 {
(s.len(), 0)
} else {
let mut mv_result = 0;
let mut balance_result = 0;
let (mut min_x, mut max_x, mut min_y, mut max_y, mut min_z, mut max_z) = (usize::MAX, 0, usize::MAX, 0, usize::MAX, 0);
let floor_half: usize = s.len()/2;
for l in 1..(floor_half+1) {
let mut sizes: BTreeSet<(usize, usize, usize)> = BTreeSet::new();
for b in 0..s.len() {
let mut size: (usize, usize, usize) = (0, 0, 0);
let sl = &slice_cyclic_string(s, b, l);
let chars_in_sl = sl.chars();
for ch in chars_in_sl {
match ch {
'x' => {size.0 += 1;},
'y' => {size.1 += 1;},
'z' => {size.2 += 1;},
_ => {panic!()},
}
}
sizes.insert(size);
}
for vector in &sizes {
min_x = min(min_x, vector.0);
max_x = max(max_x, vector.0);
min_y = min(min_y, vector.1);
max_y = max(max_y, vector.1);
min_z = min(min_z, vector.2);
max_z = max(max_z, vector.2);
}
mv_result = max(mv_result, sizes.len()); // update result
balance_result = max(balance_result, max(max(max_x.abs_diff(min_x), max_y.abs_diff(min_y)), max_z.abs_diff(min_z)));
}
(mv_result, balance_result)
}
}
// Return a Christoffel word with `a` x's and `b` y's.
// Algorithm from Bulgakova et al, 2023, "On balanced and abelian properties of circular words over a ternary alphabet".
pub fn christoffel_word(a: u32, b: u32) -> String {
let d = gcd_u32(a, b);
if d == 1 {
let mut result : String = String::from("");
let (mut current_x, mut current_y) = (0u32, 0u32); // Start from the (0,0) vector.
while current_x < a || current_y < b {
if (current_y+1)*a <= b * (current_x) { // if making the (0,1) step doesn't lead to going above the line
current_y += 1; // append the b step and reflect that in the plane vector.
result.push('y');
} else {
current_x += 1;
result.push('x');
}
}
result
} else {
std::iter::repeat(christoffel_word(a/d, b/d)).take(d.try_into().unwrap()).collect::<String>()
}
}
/// Billiard scales with the given signature up to rotation.
pub fn billiard_scales(a: u32, b: u32, c: u32) -> Vec<String> {
let mut result = Vec::<String>::new();
let regions: Vec<ConvexPolygon> = plane_geometry::project_and_partition(a, b, c);
let n = a + b + c;
for region in regions {
let mut word = String::new();
let mut next_point = region.centroid();
let first_point = region.centroid().clone();
for i in 0..n {
let res = plane_geometry::advance(a, b, c, next_point).unwrap();
next_point = res.0;
debug_assert!(i == n-1 || first_point != next_point); // There should not be a subperiod in the orbit of the "billiard ball".
word.push_str(&(res.1));
}
result.push(word);
debug_assert_eq!(first_point, next_point); // The last point should come back to the first point. When i = n - 1, next_point gets set to point number n.
}
result = equivalence_class_representatives::<String>(result, are_rotationally_equivalent);
result
}
main.rs
use scale_theory;
use std::env;
use std::fs::File;
use std::io::Write;
use std::time::Instant;
/// Gives the greatest common denominator of the two inputs, unless that's 2^63.
/// 2^63 doesn't fit in an `i64`, so it returns -2^63, which does.
pub fn gcd(u: i64, v: i64) -> i64 {
// `wrapping_abs` gives a number's absolute value, unless that's 2^63. 2^63
// won't fit in `i64`, so it gives -2^63 instead.
let mut v = v.wrapping_abs() as u64;
if u == 0 {
return v as i64;
}
let mut u = u.wrapping_abs() as u64;
if v == 0 {
return u as i64;
}
// `|` is bitwise OR. `trailing_zeros` quickly counts a binary number's
// trailing zeros, giving its prime factorization's exponent on two.
let gcd_exponent_on_two = (u | v).trailing_zeros();
// `>>=` divides the left by two to the power of the right, storing that in
// the left variable. `u` divided by its prime factorization's power of two
// turns it odd.
u >>= u.trailing_zeros();
v >>= v.trailing_zeros();
while u != v {
if u < v {
// Swap the variables' values with each other.
core::mem::swap(&mut u, &mut v);
}
u -= v;
u >>= u.trailing_zeros();
}
// `<<` multiplies the left by two to the power of the right.
(u << gcd_exponent_on_two) as i64
}
fn main() -> std::io::Result<()> {
env::set_var("RUST_BACKTRACE", "1");
let mut f = File::create("billiard_scales_mediawiki")?;
let start_time = Instant::now();
for a in 1..=29 {
for b in 1..=a {
for c in 1..=b {
if a + b + c <= 31 && gcd(gcd(a as i64, b as i64), c as i64) == 1 {
writeln!(&f, "== {}x{}y{}z ==", a, b, c);
let billiard_scales = scale_theory::billiard_scales(a, b, c);
for scale in &billiard_scales{
writeln!(&f, "# {}\n#* maximum variety {}, balance {}", scale, scale_theory::maximum_variety(scale), scale_theory::balance(scale));
}
}
}
}
}
let elapsed_time = start_time.elapsed();
println!("Done in {:?}", elapsed_time);
Ok(())
}
Cargo.toml
[package]
name = "billiard_scales"
version = "0.0.0"
edition = "2021"
# See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html
[dependencies]
num-rational = "0.4"
num-traits = "0.2.17"