Graphite/bezier-rs/lib/src/lib.rs

use glam::DVec2;

mod utils;

/// Representation of the handle point(s) in a bezier segment
#[derive(Copy, Clone)]
pub enum BezierHandles {
	/// Handles for a quadratic segment
	Quadratic {
		/// Point representing the location of the single handle
		handle: DVec2,
	},
	/// Handles for a cubic segment
	Cubic {
		/// Point representing the location of the handle associated to the start point
		handle_start: DVec2,
		/// Point representing the location of the handle associated to the end point
		handle_end: DVec2,
	},
}

/// Representation of a bezier segment with 2D points
#[derive(Copy, Clone)]
pub struct Bezier {
	/// Start point of the bezier segment
	start: DVec2,
	/// Start point of the bezier segment
	end: DVec2,
	/// Handles of the bezier segment
	handles: BezierHandles,
}

impl Bezier {
	// TODO: Consider removing this function
	/// Create a quadratic bezier using the provided coordinates as the start, handle, and end points
	pub fn from_quadratic_coordinates(x1: f64, y1: f64, x2: f64, y2: f64, x3: f64, y3: f64) -> Self {
		Bezier {
			start: DVec2::new(x1, y1),
			handles: BezierHandles::Quadratic { handle: DVec2::new(x2, y2) },
			end: DVec2::new(x3, y3),
		}
	}

	/// Create a quadratc bezier using the provided DVec2s as the start, handle, and end points
	pub fn from_quadratic_dvec2(p1: DVec2, p2: DVec2, p3: DVec2) -> Self {
		Bezier {
			start: p1,
			handles: BezierHandles::Quadratic { handle: p2 },
			end: p3,
		}
	}

	// TODO: Consider removing this function
	/// Create a cubic bezier using the provided coordinates as the start, handles, and end points
	pub fn from_cubic_coordinates(x1: f64, y1: f64, x2: f64, y2: f64, x3: f64, y3: f64, x4: f64, y4: f64) -> Self {
		Bezier {
			start: DVec2::new(x1, y1),
			handles: BezierHandles::Cubic {
				handle_start: DVec2::new(x2, y2),
				handle_end: DVec2::new(x3, y3),
			},
			end: DVec2::new(x4, y4),
		}
	}

	/// Create a cubic bezier using the provided DVec2s as the start, handles, and end points
	pub fn from_cubic_dvec2(p1: DVec2, p2: DVec2, p3: DVec2, p4: DVec2) -> Self {
		Bezier {
			start: p1,
			handles: BezierHandles::Cubic { handle_start: p2, handle_end: p3 },
			end: p4,
		}
	}

	/// Create a quadratic bezier curve that goes through 3 points, where the middle point will be at the corresponding position `t` on the curve.
	/// Note that when `t = 0` or `t = 1`, the expectation is that the `point_on_curve` should be equal to `start` and `end` respectively.
	/// In these cases, if the provided values are not equal, this function will use the `point_on_curve` as the `start`/`end` instead.
	pub fn quadratic_through_points(start: DVec2, point_on_curve: DVec2, end: DVec2, t: f64) -> Self {
		if t == 0. {
			return Bezier::from_quadratic_dvec2(point_on_curve, point_on_curve, end);
		}
		if t == 1. {
			return Bezier::from_quadratic_dvec2(start, point_on_curve, point_on_curve);
		}
		let [a, _, _] = utils::compute_abc_for_quadratic_through_points(start, point_on_curve, end, t);
		Bezier::from_quadratic_dvec2(start, a, end)
	}

	/// Create a cubic bezier curve that goes through 3 points, where the middle point will be at the corresponding position `t` on the curve.
	/// Note that when `t = 0` or `t = 1`, the expectation is that the `point_on_curve` should be equal to `start` and `end` respectively.
	/// In these cases, if the provided values are not equal, this function will use the `point_on_curve` as the `start`/`end` instead.
	/// - `midpoint_separation` is a representation of the how wide the resulting curve will be around `t` on the curve. This parameter designates the distance between the `e1` and `e2` defined in [the projection identity section](https://pomax.github.io/bezierinfo/#abc) of Pomax's bezier curve primer.
	pub fn cubic_through_points(start: DVec2, point_on_curve: DVec2, end: DVec2, t: f64, midpoint_separation: f64) -> Self {
		if t == 0. {
			return Bezier::from_cubic_dvec2(point_on_curve, point_on_curve, end, end);
		}
		if t == 1. {
			return Bezier::from_cubic_dvec2(start, start, point_on_curve, point_on_curve);
		}
		let [a, b, _] = utils::compute_abc_for_cubic_through_points(start, point_on_curve, end, t);
		let distance_between_start_and_end = (end - start) / (start.distance(end));
		let e1 = b - (distance_between_start_and_end * midpoint_separation);
		let e2 = b + (distance_between_start_and_end * midpoint_separation * (1. - t) / t);

		// TODO: these functions can be changed to helpers, but need to come up with an appropriate name first
		let v1 = (e1 - t * a) / (1. - t);
		let v2 = (e2 - (1. - t) * a) / t;
		let handle_start = (v1 - (1. - t) * start) / t;
		let handle_end = (v2 - t * end) / (1. - t);
		Bezier::from_cubic_dvec2(start, handle_start, handle_end, end)
	}

	/// Convert to SVG
	// TODO: Allow modifying the viewport, width and height
	pub fn to_svg(&self) -> String {
		let m_path = format!("M {} {}", self.start.x, self.start.y);
		let handles_path = match self.handles {
			BezierHandles::Quadratic { handle } => {
				format!("Q {} {}", handle.x, handle.y)
			}
			BezierHandles::Cubic { handle_start, handle_end } => {
				format!("C {} {}, {} {}", handle_start.x, handle_start.y, handle_end.x, handle_end.y)
			}
		};
		let curve_path = format!("{}, {} {}", handles_path, self.end.x, self.end.y);
		format!(
			r#"<svg xmlns="http://www.w3.org/2000/svg" viewBox="{} {} {} {}" width="{}px" height="{}px"><path d="{} {} {}" stroke="black" fill="transparent"/></svg>"#,
			0, 0, 100, 100, 100, 100, "\n", m_path, curve_path
		)
	}

	/// Set the coordinates of the start point
	pub fn set_start(&mut self, s: DVec2) {
		self.start = s;
	}

	/// Set the coordinates of the end point
	pub fn set_end(&mut self, e: DVec2) {
		self.end = e;
	}

	/// Set the coordinates of the first handle point. This represents the only handle in a quadratic segment.
	pub fn set_handle_start(&mut self, h1: DVec2) {
		match self.handles {
			BezierHandles::Quadratic { ref mut handle } => {
				*handle = h1;
			}
			BezierHandles::Cubic { ref mut handle_start, .. } => {
				*handle_start = h1;
			}
		};
	}

	/// Set the coordinates of the second handle point. This will convert a quadratic segment into a cubic one.
	pub fn set_handle_end(&mut self, h2: DVec2) {
		match self.handles {
			BezierHandles::Quadratic { handle } => {
				self.handles = BezierHandles::Cubic { handle_start: handle, handle_end: h2 };
			}
			BezierHandles::Cubic { ref mut handle_end, .. } => {
				*handle_end = h2;
			}
		};
	}

	/// Get the coordinates of the bezier segment's start point.
	pub fn start(&self) -> DVec2 {
		self.start
	}

	/// Get the coordinates of the bezier segment's end point.
	pub fn end(&self) -> DVec2 {
		self.end
	}

	/// Get the coordinates of the bezier segment's first handle point. This represents the only handle in a quadratic segment.
	pub fn handle_start(&self) -> DVec2 {
		match self.handles {
			BezierHandles::Quadratic { handle } => handle,
			BezierHandles::Cubic { handle_start, .. } => handle_start,
		}
	}

	/// Get the coordinates of the second handle point. This will return `None` for a quadratic segment.
	pub fn handle_end(&self) -> Option<DVec2> {
		match self.handles {
			BezierHandles::Quadratic { .. } => None,
			BezierHandles::Cubic { handle_end, .. } => Some(handle_end),
		}
	}

	/// Get the coordinates of all points in an array of 4 optional points.
	/// For a quadratic segment, the order of the points will be: `start`, `handle`, `end`. The fourth element will be `None`.
	/// For a cubic segment, the order of the points will be: `start`, `handle_start`, `handle_end`, `end`.
	pub fn get_points(&self) -> [Option<DVec2>; 4] {
		match self.handles {
			BezierHandles::Quadratic { handle } => [Some(self.start), Some(handle), Some(self.end), None],
			BezierHandles::Cubic { handle_start, handle_end } => [Some(self.start), Some(handle_start), Some(handle_end), Some(self.end)],
		}
	}

	///  Calculate the point on the curve based on the `t`-value provided.
	///  Basis code based off of pseudocode found here: <https://pomax.github.io/bezierinfo/#explanation>
	pub fn compute(&self, t: f64) -> DVec2 {
		assert!((0.0..=1.0).contains(&t));

		let t_squared = t * t;
		let one_minus_t = 1.0 - t;
		let squared_one_minus_t = one_minus_t * one_minus_t;

		match self.handles {
			BezierHandles::Quadratic { handle } => squared_one_minus_t * self.start + 2.0 * one_minus_t * t * handle + t_squared * self.end,
			BezierHandles::Cubic { handle_start, handle_end } => {
				let t_cubed = t_squared * t;
				let cubed_one_minus_t = squared_one_minus_t * one_minus_t;
				cubed_one_minus_t * self.start + 3.0 * squared_one_minus_t * t * handle_start + 3.0 * one_minus_t * t_squared * handle_end + t_cubed * self.end
			}
		}
	}

	/// Return a selection of equidistant points on the bezier curve
	/// If no value is provided for `steps`, then the function will default `steps` to be 10
	pub fn compute_lookup_table(&self, steps: Option<i32>) -> Vec<DVec2> {
		let steps_unwrapped = steps.unwrap_or(10);
		let ratio: f64 = 1.0 / (steps_unwrapped as f64);
		let mut steps_array = Vec::with_capacity((steps_unwrapped + 1) as usize);

		for t in 0..steps_unwrapped + 1 {
			steps_array.push(self.compute(f64::from(t) * ratio))
		}

		steps_array
	}

	/// Return an approximation of the length of the bezier curve
	/// code example taken from: <https://gamedev.stackexchange.com/questions/5373/moving-ships-between-two-planets-along-a-bezier-missing-some-equations-for-acce/5427#5427>
	pub fn length(&self) -> f64 {
		// We will use an approximate approach where
		// we split the curve into many subdivisions
		// and calculate the euclidean distance between the two endpoints of the subdivision
		const SUBDIVISIONS: i32 = 1000;

		let lookup_table = self.compute_lookup_table(Some(SUBDIVISIONS));
		let mut approx_curve_length = 0.0;
		let mut prev_point = lookup_table[0];
		// calculate approximate distance between subdivision
		for curr_point in lookup_table.iter().skip(1) {
			// calculate distance of subdivision
			approx_curve_length += (*curr_point - prev_point).length();
			// update the prev point
			prev_point = *curr_point;
		}

		approx_curve_length
	}

	/// Returns a vector representing the derivative at the point designated by `t` on the curve
	pub fn derivative(&self, t: f64) -> DVec2 {
		let one_minus_t = 1. - t;
		match self.handles {
			BezierHandles::Quadratic { handle } => {
				let p1_minus_p0 = handle - self.start;
				let p2_minus_p1 = self.end - handle;
				2. * one_minus_t * p1_minus_p0 + 2. * t * p2_minus_p1
			}
			BezierHandles::Cubic { handle_start, handle_end } => {
				let p1_minus_p0 = handle_start - self.start;
				let p2_minus_p1 = handle_end - handle_start;
				let p3_minus_p2 = self.end - handle_end;
				3. * one_minus_t * one_minus_t * p1_minus_p0 + 6. * t * one_minus_t * p2_minus_p1 + 3. * t * t * p3_minus_p2
			}
		}
	}

	/// Returns a normalized unit vector representing the tangent at the point designated by `t` on the curve
	pub fn tangent(&self, t: f64) -> DVec2 {
		self.derivative(t).normalize()
	}

	/// Returns a normalized unit vector representing the direction of the normal at the point designated by `t` on the curve
	pub fn normal(&self, t: f64) -> DVec2 {
		let derivative = self.derivative(t);
		derivative.normalize().perp()
	}

	/// Returns the pair of Bezier curves that result from splitting the original curve at the point corresponding to `t`
	pub fn split(&self, t: f64) -> [Bezier; 2] {
		let split_point = self.compute(t);

		let t_squared = t * t;
		let t_minus_one = t - 1.;
		let squared_t_minus_one = t_minus_one * t_minus_one;

		match self.handles {
			// TODO: Actually calculate the correct handle locations
			BezierHandles::Quadratic { handle } => [
				Bezier::from_quadratic_dvec2(self.start, t * handle - t_minus_one * self.start, split_point),
				Bezier::from_quadratic_dvec2(split_point, t * self.end - t_minus_one * handle, self.end),
			],
			BezierHandles::Cubic { handle_start, handle_end } => [
				Bezier::from_cubic_dvec2(
					self.start,
					t * handle_start - t_minus_one * self.start,
					t_squared * handle_end - 2. * t * t_minus_one * handle_start + squared_t_minus_one * self.start,
					split_point,
				),
				Bezier::from_cubic_dvec2(
					split_point,
					t_squared * self.end - 2. * t * t_minus_one * handle_end + squared_t_minus_one * handle_start,
					t * self.end - t_minus_one * handle_end,
					self.end,
				),
			],
		}
	}

	/// Returns the Bezier curve representing the sub-curve starting at the point corresponding to `t1` and ending at the point corresponding to `t2`
	pub fn trim(&self, t1: f64, t2: f64) -> Bezier {
		// Depending on the order of `t1` and `t2`, determine which half of the split we need to keep
		let t1_split_side = if t1 <= t2 { 1 } else { 0 };
		let t2_split_side = if t1 <= t2 { 0 } else { 1 };
		let bezier_starting_at_t1 = self.split(t1)[t1_split_side];
		// Adjust the ratio `t2` to its corresponding value on the new curve that was split on `t1`
		let adjusted_t2 = if t1 < t2 || (t1 == t2 && t1 == 0.) {
			// Case where we took the split from t1 to the end
			// Also cover the `t1` == t2 case where there would otherwise be a divide by 0
			(t2 - t1) / (1. - t1)
		} else {
			// Case where we took the split from the beginning to `t1`
			t2 / t1
		};
		bezier_starting_at_t1.split(adjusted_t2)[t2_split_side]
	}

	/// Returns the closest point on the curve to the provided point.
	/// Uses a searching algorithm akin to binary search that can be customized using the following parameters:
	/// - `lut_size` - Size of the lookup table for the initial passthrough
	/// - `convergence_epsilon` - Difference used between floating point numbers to be considered as equal
	/// - `convergence_limit` - Controls the number of iterations needed to consider that minimum distance to have converged
	/// - `iteration_limit` - Controls the maximum total number of iterations to be used
	pub fn project(&self, point: DVec2, lut_size: i32, convergence_epsilon: f64, convergence_limit: i32, iteration_limit: i32) -> DVec2 {
		// First find the closest point from the results of a lookup table
		let lut = self.compute_lookup_table(Some(lut_size));
		let (minimum_position, minimum_distance) = utils::get_closest_point_in_lut(&lut, point);

		// Get the t values to the left and right of the closest result in the lookup table
		let mut left_t = (0.max(minimum_position - 1) as f64) / lut_size as f64;
		let mut right_t = (lut_size.min(minimum_position + 1)) as f64 / lut_size as f64;

		// Perform a finer search by finding closest t from 5 points between [left_t, right_t] inclusive
		// Choose new left_t and right_t for a smaller range around the closest t and repeat the process
		let mut final_t = left_t;
		let mut distance;

		// Increment minimum_distance to ensure that the distance < minimum_distance comparison will be true for at least one iteration
		let mut new_minimum_distance = minimum_distance + 1.;
		// Maintain the previous distance to identify convergence
		let mut previous_distance;
		// Counter to limit the number of iterations
		let mut iteration_count = 0;
		// Counter to identify how many iterations have had a similar result. Used for convergence test
		let mut convergence_count = 0;
		// Store calculated distances to minimize unnecessary recomputations
		const NUM_DISTANCES: usize = 5;
		let mut distances: [f64; NUM_DISTANCES] = [
			point.distance(lut[0.max(minimum_position - 1) as usize]),
			0.,
			0.,
			0.,
			point.distance(lut[lut_size.min(minimum_position + 1) as usize]),
		];

		while left_t <= right_t && convergence_count < convergence_limit && iteration_count < iteration_limit {
			previous_distance = new_minimum_distance;
			let step = (right_t - left_t) / ((NUM_DISTANCES - 1) as f64);
			let mut iterator_t = left_t;
			let mut target_index = 0;
			// Iterate through first 4 points and will handle the right most point later
			for (step_index, table_distance) in distances.iter_mut().enumerate().take(4) {
				// Use previously computed distance for the left most point, and compute new values for the others
				if step_index == 0 {
					distance = *table_distance;
				} else {
					distance = point.distance(self.compute(iterator_t));
					*table_distance = distance;
				}
				if distance < new_minimum_distance {
					new_minimum_distance = distance;
					target_index = step_index;
					final_t = iterator_t
				}
				iterator_t += step;
			}
			// Check right most edge separately since step may not perfectly add up to it (floating point errors)
			if distances[NUM_DISTANCES - 1] < new_minimum_distance {
				new_minimum_distance = distances[NUM_DISTANCES - 1];
				final_t = right_t;
			}

			// Update left_t and right_t to be the t values (final_t +/- step), while handling the edges (i.e. if final_t is 0, left_t will be 0 instead of -step)
			// Ensure that the t values never exceed the [0, 1] range
			left_t = (final_t - step).max(0.);
			right_t = (final_t + step).min(1.);

			// Re-use the corresponding computed distances (target_index is the index corresponding to final_t)
			// Since target_index is a u_size, can't subtract one if it is zero
			distances[0] = distances[if target_index == 0 { 0 } else { target_index - 1 }];
			distances[NUM_DISTANCES - 1] = distances[(target_index + 1).min(NUM_DISTANCES - 1)];

			iteration_count += 1;
			// update count for consecutive iterations of similar minimum distances
			if previous_distance - new_minimum_distance < convergence_epsilon {
				convergence_count += 1;
			} else {
				convergence_count = 0;
			}
		}

		self.compute(final_t)
	}
}

#[cfg(test)]
mod tests {
	use crate::Bezier;
	use glam::DVec2;

	fn compare_points(p1: DVec2, p2: DVec2) -> bool {
		p1.abs_diff_eq(p2, 0.001)
	}

	#[test]
	fn quadratic_from_points() {
		let p1 = DVec2::new(30., 50.);
		let p2 = DVec2::new(140., 30.);
		let p3 = DVec2::new(160., 170.);

		let bezier1 = Bezier::quadratic_through_points(p1, p2, p3, 0.5);
		assert!(compare_points(bezier1.compute(0.5), p2));

		let bezier2 = Bezier::quadratic_through_points(p1, p2, p3, 0.8);
		assert!(compare_points(bezier2.compute(0.8), p2));

		let bezier3 = Bezier::quadratic_through_points(p1, p2, p3, 0.);
		assert!(compare_points(bezier3.compute(0.), p2));
	}

	#[test]
	fn cubic_through_points() {
		let p1 = DVec2::new(30., 30.);
		let p2 = DVec2::new(60., 140.);
		let p3 = DVec2::new(160., 160.);

		let bezier1 = Bezier::cubic_through_points(p1, p2, p3, 0.3, 10.);
		assert!(compare_points(bezier1.compute(0.3), p2));

		let bezier2 = Bezier::cubic_through_points(p1, p2, p3, 0.8, 91.7);
		assert!(compare_points(bezier2.compute(0.8), p2));

		let bezier3 = Bezier::cubic_through_points(p1, p2, p3, 0., 91.7);
		assert!(compare_points(bezier3.compute(0.), p2));
	}

	#[test]
	fn project() {
		let bezier1 = Bezier::from_cubic_coordinates(4., 4., 23., 45., 10., 30., 56., 90.);
		assert!(bezier1.project(DVec2::new(100., 100.), 20, 0.0001, 3, 10) == DVec2::new(56., 90.));
		assert!(bezier1.project(DVec2::new(0., 0.), 20, 0.0001, 3, 10) == DVec2::new(4., 4.));

		let bezier2 = Bezier::from_quadratic_coordinates(0., 0., 0., 100., 100., 100.);
		assert!(bezier2.project(DVec2::new(100., 0.), 20, 0.0001, 3, 10) == DVec2::new(0., 0.));
	}
}