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[package]
name = "bezier-rs"
version = "0.1.0"
rust-version = "1.62.0"
edition = "2021"
authors = ["Graphite Authors <contact@graphite.rs>"]
description = "A wide assortment of useful math functions for Bezier segments and shapes."
license = "MIT OR Apache-2.0"
readme = "./README.md"
homepage = "https://graphite.rs/libraries/bezier-rs"
repository = "https://github.com/GraphiteEditor/Graphite/libraries/bezier-rs"
[dependencies]
glam = { version = "0.17", features = ["serde"] }

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SOFTWARE.

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# Bezier-rs
A wide assortment of useful math functions for Bezier segments and shapes.

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// Implementation constants
/// Constant used to determine if `f64`s are equivalent.
pub const MAX_ABSOLUTE_DIFFERENCE: f64 = 1e-3;
/// A stricter constant used to determine if `f64`s are equivalent.
pub const STRICT_MAX_ABSOLUTE_DIFFERENCE: f64 = 1e-6;
/// Number of distances used in search algorithm for `project`.
pub const NUM_DISTANCES: usize = 5;
/// Maximum allowed angle that the normal of the `start` or `end` point can make with the normal of the corresponding handle for a curve to be considered scalable/simple.
pub const SCALABLE_CURVE_MAX_ENDPOINT_NORMAL_ANGLE: f64 = std::f64::consts::PI / 3.;
// Method argument defaults
/// Default `t` value used for the `curve_through_points` functions.
pub const DEFAULT_T_VALUE: f64 = 0.5;
/// Default LUT step size in `compute_lookup_table` function.
pub const DEFAULT_LUT_STEP_SIZE: usize = 10;
/// Default number of subdivisions used in `length` calculation.
pub const DEFAULT_LENGTH_SUBDIVISIONS: usize = 1000;
/// Default step size for `reduce` function.
pub const DEFAULT_REDUCE_STEP_SIZE: f64 = 0.01;
// SVG constants
pub const SVG_ARG_CUBIC: &str = "C";
pub const SVG_ARG_LINEAR: &str = "L";
pub const SVG_ARG_MOVE: &str = "M";
pub const SVG_ARG_QUADRATIC: &str = "Q";
pub const SVG_ARG_CLOSED: &str = "Z";

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use glam::DVec2;
use std::fmt::{Debug, Formatter, Result};
/// Struct to represent optional parameters that can be passed to the `project` function.
#[derive(Copy, Clone)]
pub struct ProjectionOptions {
/// Size of the lookup table for the initial passthrough. The default value is `20`.
pub lut_size: usize,
/// Difference used between floating point numbers to be considered as equal. The default value is `0.0001`
pub convergence_epsilon: f64,
/// Controls the number of iterations needed to consider that minimum distance to have converged. The default value is `3`.
pub convergence_limit: usize,
/// Controls the maximum total number of iterations to be used. The default value is `10`.
pub iteration_limit: usize,
}
impl Default for ProjectionOptions {
fn default() -> Self {
ProjectionOptions {
lut_size: 20,
convergence_epsilon: 1e-4,
convergence_limit: 3,
iteration_limit: 10,
}
}
}
/// Struct used to represent the different strategies for generating arc approximations.
#[derive(Copy, Clone)]
pub enum ArcStrategy {
/// Start with the greedy strategy of maximizing arc approximations and automatically switch to the divide-and-conquer when the greedy approximations no longer fall within the error bound.
Automatic,
/// Use the greedy strategy to maximize approximated arcs, despite potentially erroneous arcs.
FavorLargerArcs,
/// Use the divide-and-conquer strategy that prioritizes correctness over maximal arcs.
FavorCorrectness,
}
/// Struct to represent optional parameters that can be passed to the `arcs` function.
#[derive(Copy, Clone)]
pub struct ArcsOptions {
/// Determines how the approximated arcs are computed.
/// When maximizing the arcs, the algorithm may return incorrect arcs when the curve contains any small loops or segments that look like a very thin "U".
/// The enum options behave as follows:
/// - `Automatic`: Maximize arcs until an erroneous approximation is found. Compute the arcs of the rest of the curve by first splitting on extremas to ensure no more erroneous cases are encountered.
/// - `FavorLargerArcs`: Maximize arcs using the original algorithm from the [Approximating a Bezier curve with circular arcs](https://pomax.github.io/bezierinfo/#arcapproximation) section of Pomax's bezier curve primer. Erroneous arcs are possible.
/// - `FavorCorrectness`: Prioritize correctness by first spliting the curve by its extremas and determine the arc approximation of each segment instead.
///
/// The default value is `Automatic`.
pub strategy: ArcStrategy,
/// The error used for approximating the arc's fit. The default is `0.5`.
pub error: f64,
/// The maximum number of segment iterations used as attempts for arc approximations. The default is `100`.
pub max_iterations: usize,
}
impl Default for ArcsOptions {
fn default() -> Self {
ArcsOptions {
strategy: ArcStrategy::Automatic,
error: 0.5,
max_iterations: 100,
}
}
}
/// Struct to represent the circular arc approximation used in the `arcs` bezier function.
#[derive(Copy, Clone, PartialEq)]
pub struct CircleArc {
/// The center point of the circle.
pub center: DVec2,
/// The radius of the circle.
pub radius: f64,
/// The start angle of the circle sector in rad.
pub start_angle: f64,
/// The end angle of the circle sector in rad.
pub end_angle: f64,
}
impl Debug for CircleArc {
fn fmt(&self, f: &mut Formatter<'_>) -> Result {
write!(f, "Center: {}, radius: {}, start to end angles: {} to {}", self.center, self.radius, self.start_angle, self.end_angle)
}
}
impl Default for CircleArc {
fn default() -> Self {
CircleArc {
center: DVec2::ZERO,
radius: 0.,
start_angle: 0.,
end_angle: 0.,
}
}
}

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use super::*;
use crate::consts::*;
/// Functionality relating to core `Subpath` operations, such as constructors and `iter`.
impl Subpath {
/// Create a new `Subpath` using a list of [ManipulatorGroup]s.
/// A `Subpath` with less than 2 [ManipulatorGroup]s may not be closed.
pub fn new(manipulator_groups: Vec<ManipulatorGroup>, closed: bool) -> Self {
assert!(!closed || manipulator_groups.len() > 1, "A closed Subpath must contain more than 1 ManipulatorGroup.");
Self { manipulator_groups, closed }
}
/// Create a `Subpath` consisting of 2 manipulator groups from a `Bezier`.
pub fn from_bezier(bezier: Bezier) -> Self {
Subpath::new(
vec![
ManipulatorGroup {
anchor: bezier.start(),
in_handle: None,
out_handle: bezier.handle_start(),
},
ManipulatorGroup {
anchor: bezier.end(),
in_handle: bezier.handle_end(),
out_handle: None,
},
],
false,
)
}
/// Returns true if the `Subpath` contains no [ManipulatorGroup].
pub fn is_empty(&self) -> bool {
self.manipulator_groups.is_empty()
}
/// Returns the number of [ManipulatorGroup]s contained within the `Subpath`.
pub fn len(&self) -> usize {
self.manipulator_groups.len()
}
/// Returns an iterator of the [Bezier]s along the `Subpath`.
pub fn iter(&self) -> SubpathIter {
SubpathIter { sub_path: self, index: 0 }
}
/// Returns an SVG representation of the `Subpath`.
pub fn to_svg(&self, options: ToSVGOptions) -> String {
if self.is_empty() {
return String::new();
}
let curve_start_argument = format!("{SVG_ARG_MOVE}{} {}", self[0].anchor.x, self[0].anchor.y);
let mut curve_arguments: Vec<String> = self.iter().map(|bezier| bezier.svg_curve_argument()).collect();
if self.closed {
curve_arguments.push(String::from(SVG_ARG_CLOSED));
}
let anchor_arguments = options.formatted_anchor_arguments();
let anchor_circles = self
.manipulator_groups
.iter()
.map(|point| format!(r#"<circle cx="{}" cy="{}" {}/>"#, point.anchor.x, point.anchor.y, anchor_arguments))
.collect::<Vec<String>>();
let handle_point_arguments = options.formatted_handle_point_arguments();
let handle_circles: Vec<String> = self
.manipulator_groups
.iter()
.flat_map(|group| [group.in_handle, group.out_handle])
.flatten()
.map(|handle| format!(r#"<circle cx="{}" cy="{}" {}/>"#, handle.x, handle.y, handle_point_arguments))
.collect();
let handle_pieces: Vec<String> = self.iter().filter_map(|bezier| bezier.svg_handle_line_argument()).collect();
format!(
r#"<path d="{} {}" {}/><path d="{}" {}/>{}{}"#,
curve_start_argument,
curve_arguments.join(" "),
options.formatted_curve_arguments(),
handle_pieces.join(" "),
options.formatted_handle_line_arguments(),
handle_circles.join(""),
anchor_circles.join(""),
)
}
}

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use super::*;
/// Functionality relating to looking up properties of the `Subpath` or points along the `Subpath`.
impl Subpath {
/// Return the sum of the approximation of the length of each `Bezier` curve along the `Subpath`.
/// - `num_subdivisions` - Number of subdivisions used to approximate the curve. The default value is `1000`.
pub fn length(&self, num_subdivisions: Option<usize>) -> f64 {
self.iter().fold(0., |accumulator, bezier| accumulator + bezier.length(num_subdivisions))
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::Bezier;
use glam::DVec2;
#[test]
fn length_quadratic() {
let start = DVec2::new(20., 30.);
let middle = DVec2::new(80., 90.);
let end = DVec2::new(60., 45.);
let handle1 = DVec2::new(75., 85.);
let handle2 = DVec2::new(40., 30.);
let handle3 = DVec2::new(10., 10.);
let bezier1 = Bezier::from_quadratic_dvec2(start, handle1, middle);
let bezier2 = Bezier::from_quadratic_dvec2(middle, handle2, end);
let bezier3 = Bezier::from_quadratic_dvec2(end, handle3, start);
let mut subpath = Subpath::new(
vec![
ManipulatorGroup {
anchor: start,
in_handle: None,
out_handle: Some(handle1),
},
ManipulatorGroup {
anchor: middle,
in_handle: None,
out_handle: Some(handle2),
},
ManipulatorGroup {
anchor: end,
in_handle: None,
out_handle: Some(handle3),
},
],
false,
);
assert_eq!(subpath.length(None), bezier1.length(None) + bezier2.length(None));
subpath.closed = true;
assert_eq!(subpath.length(None), bezier1.length(None) + bezier2.length(None) + bezier3.length(None));
}
#[test]
fn length_mixed() {
let start = DVec2::new(20., 30.);
let middle = DVec2::new(70., 70.);
let end = DVec2::new(60., 45.);
let handle1 = DVec2::new(75., 85.);
let handle2 = DVec2::new(40., 30.);
let handle3 = DVec2::new(10., 10.);
let linear_bezier = Bezier::from_linear_dvec2(start, middle);
let quadratic_bezier = Bezier::from_quadratic_dvec2(middle, handle1, end);
let cubic_bezier = Bezier::from_cubic_dvec2(end, handle2, handle3, start);
let mut subpath = Subpath::new(
vec![
ManipulatorGroup {
anchor: start,
in_handle: Some(handle3),
out_handle: None,
},
ManipulatorGroup {
anchor: middle,
in_handle: None,
out_handle: Some(handle1),
},
ManipulatorGroup {
anchor: end,
in_handle: None,
out_handle: Some(handle2),
},
],
false,
);
assert_eq!(subpath.length(None), linear_bezier.length(None) + quadratic_bezier.length(None));
subpath.closed = true;
assert_eq!(subpath.length(None), linear_bezier.length(None) + quadratic_bezier.length(None) + cubic_bezier.length(None));
}
}

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mod core;
mod lookup;
mod structs;
pub use structs::*;
use crate::Bezier;
use std::ops::{Index, IndexMut};
/// Structure used to represent a path composed of [Bezier] curves.
pub struct Subpath {
manipulator_groups: Vec<ManipulatorGroup>,
closed: bool,
}
/// Iteration structure for iterating across each curve of a `Subpath`, using an intermediate `Bezier` representation.
pub struct SubpathIter<'a> {
index: usize,
sub_path: &'a Subpath,
}
impl Index<usize> for Subpath {
type Output = ManipulatorGroup;
fn index(&self, index: usize) -> &Self::Output {
assert!(index < self.len(), "Index out of bounds in trait Index of SubPath.");
&self.manipulator_groups[index]
}
}
impl IndexMut<usize> for Subpath {
fn index_mut(&mut self, index: usize) -> &mut Self::Output {
assert!(index < self.len(), "Index out of bounds in trait IndexMut of SubPath.");
&mut self.manipulator_groups[index]
}
}
impl Iterator for SubpathIter<'_> {
type Item = Bezier;
// Returns the Bezier representation of each `Subpath` segment, defined between a pair of adjacent manipulator points.
fn next(&mut self) -> Option<Self::Item> {
let len = self.sub_path.len() - 1
+ match self.sub_path.closed {
true => 1,
false => 0,
};
if self.index >= len {
return None;
}
let start_index = self.index;
let end_index = (self.index + 1) % self.sub_path.len();
self.index += 1;
let start = self.sub_path[start_index].anchor;
let end = self.sub_path[end_index].anchor;
let out_handle = self.sub_path[start_index].out_handle;
let in_handle = self.sub_path[end_index].in_handle;
if let (Some(handle1), Some(handle2)) = (out_handle, in_handle) {
Some(Bezier::from_cubic_dvec2(start, handle1, handle2, end))
} else if let Some(handle) = out_handle.or(in_handle) {
Some(Bezier::from_quadratic_dvec2(start, handle, end))
} else {
Some(Bezier::from_linear_dvec2(start, end))
}
}
}

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use glam::DVec2;
/// Structure used to represent a single anchor with up to two optional associated handles along a `Subpath`
pub struct ManipulatorGroup {
pub anchor: DVec2,
pub in_handle: Option<DVec2>,
pub out_handle: Option<DVec2>,
}
/// Structure to represent optional parameters that can be passed to the `into_svg` function.
pub struct ToSVGOptions {
/// Color of the line segments along the `Subpath`. Defaulted to `black`.
pub curve_stroke_color: String,
/// Width of the line segments along the `Subpath`. Defaulted to `2.`.
pub curve_stroke_width: f64,
/// Stroke color outlining circles marking anchors on the `Subpath`. Defaulted to `black`.
pub anchor_stroke_color: String,
/// Stroke width outlining circles marking anchors on the `Subpath`. Defaulted to `2.`.
pub anchor_stroke_width: f64,
/// Radius of the circles marking anchors on the `Subpath`. Defaulted to `4.`.
pub anchor_radius: f64,
/// Fill color of the circles marking anchors on the `Subpath`. Defaulted to `white`.
pub anchor_fill: String,
/// Color of the line segments connecting anchors to handle points. Defaulted to `gray`.
pub handle_line_stroke_color: String,
/// Width of the line segments connecting anchors to handle points. Defaulted to `1.`.
pub handle_line_stroke_width: f64,
/// Stroke color outlining circles marking the handles of `Subpath`. Defaulted to `gray`.
pub handle_point_stroke_color: String,
/// Stroke color outlining circles marking the handles of `Subpath`. Defaulted to `1.5`.
pub handle_point_stroke_width: f64,
/// Radius of the circles marking the handles of `Subpath`. Defaulted to `3.`.
pub handle_point_radius: f64,
/// Fill color of the circles marking the handles of `Subpath`. Defaulted to `white`.
pub handle_point_fill: String,
}
impl ToSVGOptions {
/// Combine and format curve styling options for an SVG path.
pub(crate) fn formatted_curve_arguments(&self) -> String {
format!(r#"stroke="{}" stroke-width="{}" fill="none""#, self.curve_stroke_color, self.curve_stroke_width)
}
/// Combine and format anchor styling options an SVG circle.
pub(crate) fn formatted_anchor_arguments(&self) -> String {
format!(
r#"r="{}", stroke="{}" stroke-width="{}" fill="{}""#,
self.anchor_radius, self.anchor_stroke_color, self.anchor_stroke_width, self.anchor_fill
)
}
/// Combine and format handle point styling options for an SVG circle.
pub(crate) fn formatted_handle_point_arguments(&self) -> String {
format!(
r#"r="{}", stroke="{}" stroke-width="{}" fill="{}""#,
self.handle_point_radius, self.handle_point_stroke_color, self.handle_point_stroke_width, self.handle_point_fill
)
}
/// Combine and format handle line styling options an SVG path.
pub(crate) fn formatted_handle_line_arguments(&self) -> String {
format!(r#"stroke="{}" stroke-width="{}" fill="none""#, self.handle_line_stroke_color, self.handle_line_stroke_width)
}
}
impl Default for ToSVGOptions {
fn default() -> Self {
ToSVGOptions {
curve_stroke_color: String::from("black"),
curve_stroke_width: 2.,
anchor_stroke_color: String::from("black"),
anchor_stroke_width: 2.,
anchor_radius: 4.,
anchor_fill: String::from("white"),
handle_line_stroke_color: String::from("gray"),
handle_line_stroke_width: 1.,
handle_point_stroke_color: String::from("gray"),
handle_point_stroke_width: 1.5,
handle_point_radius: 3.,
handle_point_fill: String::from("white"),
}
}
}

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use crate::consts::{MAX_ABSOLUTE_DIFFERENCE, STRICT_MAX_ABSOLUTE_DIFFERENCE};
use glam::{BVec2, DMat2, DVec2};
use std::f64::consts::PI;
/// Helper to perform the computation of a and c, where b is the provided point on the curve.
/// Given the correct power of `t` and `(1-t)`, the computation is the same for quadratic and cubic cases.
/// Relevant derivation and the definitions of a, b, and c can be found in [the projection identity section](https://pomax.github.io/bezierinfo/#abc) of Pomax's bezier curve primer.
fn compute_abc_through_points(start_point: DVec2, point_on_curve: DVec2, end_point: DVec2, t_to_nth_power: f64, nth_power_of_one_minus_t: f64) -> [DVec2; 3] {
let point_c_ratio = nth_power_of_one_minus_t / (t_to_nth_power + nth_power_of_one_minus_t);
let c = point_c_ratio * start_point + (1. - point_c_ratio) * end_point;
let ab_bc_ratio = (t_to_nth_power + nth_power_of_one_minus_t - 1.).abs() / (t_to_nth_power + nth_power_of_one_minus_t);
let a = point_on_curve + (point_on_curve - c) / ab_bc_ratio;
[a, point_on_curve, c]
}
/// Compute `a`, `b`, and `c` for a quadratic curve that fits the start, end and point on curve at `t`.
/// The definition for the `a`, `b`, `c` points are defined in [the projection identity section](https://pomax.github.io/bezierinfo/#abc) of Pomax's bezier curve primer.
pub fn compute_abc_for_quadratic_through_points(start_point: DVec2, point_on_curve: DVec2, end_point: DVec2, t: f64) -> [DVec2; 3] {
let t_squared = t * t;
let one_minus_t = 1. - t;
let squared_one_minus_t = one_minus_t * one_minus_t;
compute_abc_through_points(start_point, point_on_curve, end_point, t_squared, squared_one_minus_t)
}
/// Compute `a`, `b`, and `c` for a cubic curve that fits the start, end and point on curve at `t`.
/// The definition for the `a`, `b`, `c` points are defined in [the projection identity section](https://pomax.github.io/bezierinfo/#abc) of Pomax's bezier curve primer.
pub fn compute_abc_for_cubic_through_points(start_point: DVec2, point_on_curve: DVec2, end_point: DVec2, t: f64) -> [DVec2; 3] {
let t_cubed = t * t * t;
let one_minus_t = 1. - t;
let cubed_one_minus_t = one_minus_t * one_minus_t * one_minus_t;
compute_abc_through_points(start_point, point_on_curve, end_point, t_cubed, cubed_one_minus_t)
}
/// Return the index and the value of the closest point in the LUT compared to the provided point.
pub fn get_closest_point_in_lut(lut: &[DVec2], point: DVec2) -> (usize, f64) {
lut.iter()
.enumerate()
.map(|(i, p)| (i, point.distance_squared(*p)))
.min_by(|x, y| (&(x.1)).partial_cmp(&(y.1)).unwrap())
.unwrap()
}
// TODO: Use an `Option` return type instead of a `Vec`
/// Find the roots of the linear equation `ax + b`.
pub fn solve_linear(a: f64, b: f64) -> Vec<f64> {
let mut roots = Vec::new();
// There exist roots when `a` is not 0
if a.abs() > MAX_ABSOLUTE_DIFFERENCE {
roots.push(-b / a);
}
roots
}
// TODO: Use an `impl Iterator` return type instead of a `Vec`
/// Find the roots of the linear equation `ax^2 + bx + c`.
/// Precompute the `discriminant` (`b^2 - 4ac`) and `two_times_a` arguments prior to calling this function for efficiency purposes.
pub fn solve_quadratic(discriminant: f64, two_times_a: f64, b: f64, c: f64) -> Vec<f64> {
let mut roots = Vec::new();
if two_times_a != 0. {
if discriminant > 0. {
let root_discriminant = discriminant.sqrt();
roots.push((-b + root_discriminant) / (two_times_a));
roots.push((-b - root_discriminant) / (two_times_a));
} else if discriminant == 0. {
roots.push(-b / (two_times_a));
}
} else {
roots = solve_linear(b, c);
}
roots
}
/// Compute the cube root of a number.
fn cube_root(f: f64) -> f64 {
if f < 0. {
-(-f).powf(1. / 3.)
} else {
f.powf(1. / 3.)
}
}
// TODO: Use an `impl Iterator` return type instead of a `Vec`
/// Solve a cubic of the form `x^3 + px + q`, derivation from: <https://trans4mind.com/personal_development/mathematics/polynomials/cubicAlgebra.htm>.
pub fn solve_reformatted_cubic(discriminant: f64, a: f64, p: f64, q: f64) -> Vec<f64> {
let mut roots = Vec::new();
if p.abs() <= STRICT_MAX_ABSOLUTE_DIFFERENCE {
// Handle when p is approximately 0
roots.push(cube_root(-q));
} else if q.abs() <= STRICT_MAX_ABSOLUTE_DIFFERENCE {
// Handle when q is approximately 0
if p < 0. {
roots.push((-p).powf(1. / 2.));
}
} else if discriminant.abs() <= STRICT_MAX_ABSOLUTE_DIFFERENCE {
// When discriminant is 0 (check for approximation because of floating point errors), all roots are real, and 2 are repeated
let q_divided_by_2 = q / 2.;
let a_divided_by_3 = a / 3.;
roots.push(2. * cube_root(-q_divided_by_2) - a_divided_by_3);
roots.push(cube_root(q_divided_by_2) - a_divided_by_3);
} else if discriminant > 0. {
// When discriminant > 0, there is one real and two imaginary roots
let q_divided_by_2 = q / 2.;
let square_root_discriminant = discriminant.powf(1. / 2.);
roots.push(cube_root(-q_divided_by_2 + square_root_discriminant) - cube_root(q_divided_by_2 + square_root_discriminant) - a / 3.);
} else {
// Otherwise, discriminant < 0 and there are three real roots
let p_divided_by_3 = p / 3.;
let a_divided_by_3 = a / 3.;
let cube_root_r = (-p_divided_by_3).powf(1. / 2.);
let phi = (-q / (2. * cube_root_r.powi(3))).acos();
let two_times_cube_root_r = 2. * cube_root_r;
roots.push(two_times_cube_root_r * (phi / 3.).cos() - a_divided_by_3);
roots.push(two_times_cube_root_r * ((phi + 2. * PI) / 3.).cos() - a_divided_by_3);
roots.push(two_times_cube_root_r * ((phi + 4. * PI) / 3.).cos() - a_divided_by_3);
}
roots
}
// TODO: Use an `impl Iterator` return type instead of a `Vec`
/// Solve a cubic of the form `ax^3 + bx^2 + ct + d`.
pub fn solve_cubic(a: f64, b: f64, c: f64, d: f64) -> Vec<f64> {
if a.abs() <= STRICT_MAX_ABSOLUTE_DIFFERENCE {
if b.abs() <= STRICT_MAX_ABSOLUTE_DIFFERENCE {
// If both a and b are approximately 0, treat as a linear problem
solve_linear(c, d)
} else {
// If a is approximately 0, treat as a quadratic problem
let discriminant = c * c - 4. * b * d;
solve_quadratic(discriminant, 2. * b, c, d)
}
} else {
let new_a = b / a;
let new_b = c / a;
let new_c = d / a;
// Refactor cubic to be of the form: a(t^3 + pt + q), derivation from: https://trans4mind.com/personal_development/mathematics/polynomials/cubicAlgebra.htm
let p = (3. * new_b - new_a * new_a) / 3.;
let q = (2. * new_a.powi(3) - 9. * new_a * new_b + 27. * new_c) / 27.;
let discriminant = (p / 3.).powi(3) + (q / 2.).powi(2);
solve_reformatted_cubic(discriminant, new_a, p, q)
}
}
/// Determine if two rectangles have any overlap. The rectangles are represented by a pair of coordinates that designate the top left and bottom right corners (in a graphical coordinate system).
pub fn do_rectangles_overlap(rectangle1: [DVec2; 2], rectangle2: [DVec2; 2]) -> bool {
let [bottom_left1, top_right1] = rectangle1;
let [bottom_left2, top_right2] = rectangle2;
top_right1.x >= bottom_left2.x && top_right2.x >= bottom_left1.x && top_right2.y >= bottom_left1.y && top_right1.y >= bottom_left2.y
}
/// Returns the intersection of two lines. The lines are given by a point on the line and its slope (represented by a vector).
pub fn line_intersection(point1: DVec2, point1_slope_vector: DVec2, point2: DVec2, point2_slope_vector: DVec2) -> DVec2 {
assert!(point1_slope_vector.normalize() != point2_slope_vector.normalize());
// Find the intersection when the first line is vertical
if f64_compare(point1_slope_vector.x, 0., MAX_ABSOLUTE_DIFFERENCE) {
let m2 = point2_slope_vector.y / point2_slope_vector.x;
let b2 = point2.y - m2 * point2.x;
DVec2::new(point1.x, point1.x * m2 + b2)
}
// Find the intersection when the second line is vertical
else if f64_compare(point2_slope_vector.x, 0., MAX_ABSOLUTE_DIFFERENCE) {
let m1 = point1_slope_vector.y / point1_slope_vector.x;
let b1 = point1.y - m1 * point1.x;
DVec2::new(point2.x, point2.x * m1 + b1)
}
// Find the intersection where neither line is vertical
else {
let m1 = point1_slope_vector.y / point1_slope_vector.x;
let b1 = point1.y - m1 * point1.x;
let m2 = point2_slope_vector.y / point2_slope_vector.x;
let b2 = point2.y - m2 * point2.x;
let intersection_x = (b2 - b1) / (m1 - m2);
DVec2::new(intersection_x, intersection_x * m1 + b1)
}
}
/// Check if 3 points are collinear.
pub fn are_points_collinear(p1: DVec2, p2: DVec2, p3: DVec2) -> bool {
let matrix = DMat2::from_cols(p1 - p2, p2 - p3);
f64_compare(matrix.determinant() / 2., 0., MAX_ABSOLUTE_DIFFERENCE)
}
/// Compute the center of the circle that passes through all three provided points. The provided points cannot be collinear.
pub fn compute_circle_center_from_points(p1: DVec2, p2: DVec2, p3: DVec2) -> Option<DVec2> {
if are_points_collinear(p1, p2, p3) {
return None;
}
let midpoint_a = p1.lerp(p2, 0.5);
let midpoint_b = p2.lerp(p3, 0.5);
let midpoint_c = p3.lerp(p1, 0.5);
let tangent_a = (p1 - p2).perp();
let tangent_b = (p2 - p3).perp();
let tangent_c = (p3 - p1).perp();
let intersect_a_b = line_intersection(midpoint_a, tangent_a, midpoint_b, tangent_b);
let intersect_b_c = line_intersection(midpoint_b, tangent_b, midpoint_c, tangent_c);
let intersect_c_a = line_intersection(midpoint_c, tangent_c, midpoint_a, tangent_a);
Some((intersect_a_b + intersect_b_c + intersect_c_a) / 3.)
}
/// Compare two `f64` numbers with a provided max absolute value difference.
pub fn f64_compare(f1: f64, f2: f64, max_abs_diff: f64) -> bool {
(f1 - f2).abs() < max_abs_diff
}
/// Determine if an `f64` number is within a given range by using a max absolute value difference comparison.
pub fn f64_approximately_in_range(value: f64, min: f64, max: f64, max_abs_diff: f64) -> bool {
(min..=max).contains(&value) || f64_compare(value, min, max_abs_diff) || f64_compare(value, max, max_abs_diff)
}
/// Compare the two values in a `DVec2` independently with a provided max absolute value difference.
pub fn dvec2_compare(dv1: DVec2, dv2: DVec2, max_abs_diff: f64) -> BVec2 {
BVec2::new((dv1.x - dv2.x).abs() < max_abs_diff, (dv1.y - dv2.y).abs() < max_abs_diff)
}
/// Determine if the values in a `DVec2` are within a given range independently by using a max absolute value difference comparison.
pub fn dvec2_approximately_in_range(point: DVec2, min: DVec2, max: DVec2, max_abs_diff: f64) -> BVec2 {
(point.cmpge(min) & point.cmple(max)) | dvec2_compare(point, min, max_abs_diff) | dvec2_compare(point, max, max_abs_diff)
}
#[cfg(test)]
mod tests {
use super::*;
use crate::consts::MAX_ABSOLUTE_DIFFERENCE;
/// Compare vectors of `f64`s with a provided max absolute value difference.
fn f64_compare_vector(vec1: Vec<f64>, vec2: Vec<f64>, max_abs_diff: f64) -> bool {
vec1.len() == vec2.len() && vec1.into_iter().zip(vec2.into_iter()).all(|(a, b)| f64_compare(a, b, max_abs_diff))
}
#[test]
fn test_solve_linear() {
// Line that is on the x-axis
assert!(solve_linear(0., 0.).is_empty());
// Line that is parallel to but not on the x-axis
assert!(solve_linear(0., 1.).is_empty());
// Line with a non-zero slope
assert!(solve_linear(2., -8.) == vec![4.]);
}
#[test]
fn test_solve_cubic() {
// discriminant == 0
let roots1 = solve_cubic(1., 0., 0., 0.);
assert!(roots1 == vec![0.]);
let roots2 = solve_cubic(1., 3., 0., -4.);
assert!(roots2 == vec![1., -2.]);
// p == 0
let roots3 = solve_cubic(1., 0., 0., -1.);
assert!(roots3 == vec![1.]);
// discriminant > 0
let roots4 = solve_cubic(1., 3., 0., 2.);
assert!(f64_compare_vector(roots4, vec![-3.196], MAX_ABSOLUTE_DIFFERENCE));
// discriminant < 0
let roots5 = solve_cubic(1., 3., 0., -1.);
assert!(f64_compare_vector(roots5, vec![0.532, -2.879, -0.653], MAX_ABSOLUTE_DIFFERENCE));
// quadratic
let roots6 = solve_cubic(0., 3., 0., -3.);
assert!(roots6 == vec![1., -1.]);
// linear
let roots7 = solve_cubic(0., 0., 1., -1.);
assert!(roots7 == vec![1.]);
}
#[test]
fn test_do_rectangles_overlap() {
// Rectangles overlap
assert!(do_rectangles_overlap([DVec2::new(0., 0.), DVec2::new(20., 20.)], [DVec2::new(10., 10.), DVec2::new(30., 20.)]));
// Rectangles share a side
assert!(do_rectangles_overlap([DVec2::new(0., 0.), DVec2::new(10., 10.)], [DVec2::new(10., 10.), DVec2::new(30., 30.)]));
// Rectangle inside the other
assert!(do_rectangles_overlap([DVec2::new(0., 0.), DVec2::new(10., 10.)], [DVec2::new(2., 2.), DVec2::new(6., 4.)]));
// No overlap, rectangles are beside each other
assert!(!do_rectangles_overlap([DVec2::new(0., 0.), DVec2::new(10., 10.)], [DVec2::new(20., 0.), DVec2::new(30., 10.)]));
// No overlap, rectangles are above and below each other
assert!(!do_rectangles_overlap([DVec2::new(0., 0.), DVec2::new(10., 10.)], [DVec2::new(0., 20.), DVec2::new(20., 30.)]));
}
#[test]
fn test_find_intersection() {
// y = 2x + 10
// y = 5x + 4
// intersect at (2, 14)
let start1 = DVec2::new(0., 10.);
let end1 = DVec2::new(0., 4.);
let start_direction1 = DVec2::new(1., 2.);
let end_direction1 = DVec2::new(1., 5.);
assert!(line_intersection(start1, start_direction1, end1, end_direction1) == DVec2::new(2., 14.));
// y = x
// y = -x + 8
// intersect at (4, 4)
let start2 = DVec2::new(0., 0.);
let end2 = DVec2::new(8., 0.);
let start_direction2 = DVec2::new(1., 1.);
let end_direction2 = DVec2::new(1., -1.);
assert!(line_intersection(start2, start_direction2, end2, end_direction2) == DVec2::new(4., 4.));
}
#[test]
fn test_are_points_collinear() {
assert!(are_points_collinear(DVec2::new(2., 4.), DVec2::new(6., 8.), DVec2::new(4., 6.)));
assert!(!are_points_collinear(DVec2::new(1., 4.), DVec2::new(6., 8.), DVec2::new(4., 6.)));
}
#[test]
fn test_compute_circle_center_from_points() {
// 3/4 of unit circle
let center1 = compute_circle_center_from_points(DVec2::new(0., 1.), DVec2::new(-1., 0.), DVec2::new(1., 0.));
assert_eq!(center1.unwrap(), DVec2::new(0., 0.));
// 1/4 of unit circle
let center2 = compute_circle_center_from_points(DVec2::new(-1., 0.), DVec2::new(0., 1.), DVec2::new(1., 0.));
assert_eq!(center2.unwrap(), DVec2::new(0., 0.));
}
}