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``````// Copyright (c) 2018-2020 Thomas Kramer.
//

//! Edge intersection functions for rational coordinates.
//!
//! Some computations on edges can be performed exactly when the coordinates
//! have rational type. For example the intersection of two edges with rational coordinates
//! can be expressed exactly as a point with rational coordinates.
//!
//! This modules contains implementations of such computations that take edges with rational coordinates
//! and produce an output in rational coordinates.
//!

use crate::point::Point;

use num_rational::Ratio;
use num_traits::Zero;

use std::cmp::Ordering;

pub use crate::edge::{Edge, EdgeIntersection, LineIntersection};
use crate::traits::BoundingBox;
use crate::CoordinateType;

impl<T: CoordinateType + num_integer::Integer> Edge<Ratio<T>> {
/// Compute the intersection point of the lines defined by the two edges.
///
/// Degenerate lines don't intersect by definition.
///
/// Returns `LineIntersection::None` iff the two lines don't intersect.
/// Returns `LineIntersection::Collinear` iff both lines are equal.
/// Returns `LineIntersection::Point(p,(a,b,c))` iff the lines intersect in exactly one point `p`.
/// `f` is a value such that `self.start + self.vector()*a/c == p` and
/// `other.start + other.vector()*b/c == p`.
///
/// # Examples
///
/// ```
/// extern crate num_rational;
/// use num_rational::Ratio;
/// use iron_shapes::point::Point;
/// use iron_shapes::edge_rational::*;
///
/// let r = |i| Ratio::from_integer(i);
///
/// let e1 = Edge::new((r(0), r(0)), (r(2), r(2)));
/// let e2 = Edge::new((r(0), r(2)), (r(2), r(0)));
///
/// assert_eq!(e1.line_intersection_rational(e2),
///     LineIntersection::Point(Point::new(r(1), r(1)), (r(4), r(4), r(8))));
///
/// ```
pub fn line_intersection_rational(
&self,
other: Edge<Ratio<T>>,
) -> LineIntersection<Ratio<T>, Ratio<T>> {
if self.is_degenerate() || other.is_degenerate() {
LineIntersection::None
} else {
// TODO: faster implementation if both lines are orthogonal

let ab = self.vector();
let cd = other.vector();

// Assert that the vectors have a non-zero length. This should already be the case
// because the degenerate cases are handled before.
debug_assert!(ab.norm2_squared() > Ratio::zero());
debug_assert!(cd.norm2_squared() > Ratio::zero());

let s = ab.cross_prod(cd);

// TODO: What if approximate zero due to rounding error?
if s.is_zero() {
// Lines are parallel
debug_assert!(self.is_parallel(&other));

// TODO: check more efficiently for collinear lines.
if self.line_contains_point(other.start) {
// If the line defined by `self` contains at least one point of `other` then they are equal.
debug_assert!(self.is_collinear(&other));
LineIntersection::Collinear
} else {
LineIntersection::None
}
} else {
let ac = other.start - self.start;
let ac_cross_cd = ac.cross_prod(cd);
let i = ac_cross_cd / s;

let p: Point<Ratio<T>> = self.start + ab * i;

let ca_cross_ab = ac.cross_prod(ab);

// Check that the intersection point lies on the lines indeed.
// TODO: uncomment this checks
//                debug_assert!(self.cast().line_contains_point_approx(p, 1e-4));
//                debug_assert!(other.cast().line_contains_point_approx(p, 1e-2));

// debug_assert!({
//     let j = ca_cross_ab / s;
//     let p2: Point<Ratio<T>> = other.start + cd * j;
//     (p - p2).norm2_squared() < Ratio::new(T::one(), 1_000_000)
// });

let positions = if s < Ratio::zero() {
(
Ratio::zero() - ac_cross_cd,
Ratio::zero() - ca_cross_ab,
Ratio::zero() - s,
)
} else {
(ac_cross_cd, ca_cross_ab, s)
};

LineIntersection::Point(p, positions)
}
}
}

/// Compute the intersection with another edge.
pub fn edge_intersection_rational(
&self,
other: &Edge<Ratio<T>>,
) -> EdgeIntersection<Ratio<T>, Ratio<T>, Edge<Ratio<T>>> {
//        debug_assert!(tolerance >= Ratio::zero(), "Tolerance cannot be negative.");

// Swap direction of other edge such that both have the same direction.
let other = if (self.start < self.end) != (other.start < other.end) {
other.reversed()
} else {
*other
};

// Check endpoints for coincidence.
// This must be handled separately because equality of the intersection point and endpoints
// will not necessarily be detected due to rounding errors.
let same_start_start = self.start == other.start;
let same_start_end = self.start == other.end;

let same_end_start = self.end == other.start;
let same_end_end = self.end == other.end;

// TODO: optimize for chained edges (start1 == end2 ^ start2 == end1)

// Are the edges equal but not degenerate?
let fully_coincident =
(same_start_start & same_end_end) ^ (same_start_end & same_end_start);

let result = if self.is_degenerate() {
// First degenerate case
if other.contains_point(self.start).inclusive_bounds() {
EdgeIntersection::EndPoint(self.start)
} else {
EdgeIntersection::None
}
} else if other.is_degenerate() {
// Second degenerate case
if self.contains_point(other.start).inclusive_bounds() {
EdgeIntersection::EndPoint(other.start)
} else {
EdgeIntersection::None
}
} else if fully_coincident {
EdgeIntersection::Overlap(*self)
} else if !self.bounding_box().touches(&other.bounding_box()) {
// If bounding boxes do not touch, then intersection is impossible.
EdgeIntersection::None
} else {
// Compute the intersection of the lines defined by the two edges.
let line_intersection = self.line_intersection_rational(other);

// Then check if the intersection point is on both edges
// or find the intersection if the edges overlap.
match line_intersection {
LineIntersection::None => EdgeIntersection::None,

// Intersection in one point:
LineIntersection::Point(p, (pos1, pos2, len)) => {
if pos1 >= Ratio::zero() && pos1 <= len && pos2 >= Ratio::zero() && pos2 <= len
{
if pos1 == Ratio::zero()
|| pos1 == len
|| pos2 == Ratio::zero()
|| pos2 == len
{
EdgeIntersection::EndPoint(p)
} else {
EdgeIntersection::Point(p)
}
} else {
EdgeIntersection::None
}
}
LineIntersection::Collinear => {
debug_assert!(self.is_collinear(&other));

// Project all points of the two edges on the line defined by the first edge
// (scaled by the length of the first edge).
// This allows to calculate the interval of overlap in one dimension.

let (pa, pb) = self.into();
let (pc, pd) = other.into();

let b = pb - pa;
let c = pc - pa;
let d = pd - pa;

let dist_a = Ratio::zero();
let dist_b = b.dot(b);

let dist_c = b.dot(c);
let dist_d = b.dot(d);

let start1 = (dist_a, pa);
let end1 = (dist_b, pb);

// Sort end points of other edge.
let (start2, end2) = if dist_c < dist_d {
((dist_c, pc), (dist_d, pd))
} else {
((dist_d, pd), (dist_c, pc))
};

// Find maximum by distance.
let start = if start1.0 < start2.0 { start2 } else { start1 };

// Find minimum by distance.
let end = if end1.0 < end2.0 { end1 } else { end2 };

// Check if the edges overlap in more than one point, in exactly one point or
// in zero points.
match start.0.cmp(&end.0) {
Ordering::Less => EdgeIntersection::Overlap(Edge::new(start.1, end.1)),
Ordering::Equal => EdgeIntersection::EndPoint(start.1),
Ordering::Greater => EdgeIntersection::None,
}
}
}
};

// Sanity check for the result.
debug_assert!({
match result {
EdgeIntersection::Point(p) => {
self.contains_point(p).is_within_bounds()
&& other.contains_point(p).is_within_bounds()
}
EdgeIntersection::EndPoint(p) => {
self.contains_point(p).on_bounds() || other.contains_point(p).on_bounds()
}
EdgeIntersection::None => self.edges_intersect(&other).is_no(),
EdgeIntersection::Overlap(_) => true,
}
});

// Check that the result is consistent with the edge intersection test.

debug_assert_eq!(
result == EdgeIntersection::None,
self.edges_intersect(&other).is_no()
);

result
}
}

#[cfg(test)]
mod tests {
use super::*;
use crate::traits::Scale;
use num_rational::Rational64;

#[test]
fn test_rational_edge() {
let e = Edge::new(
(Rational64::from(0), Rational64::from(0)),
(Rational64::from(1), Rational64::from(1)),
);
let v = e.vector();
assert!(v.norm2_squared() == Rational64::from(2));
assert!(!e.is_degenerate());
}

#[test]
fn test_line_intersection_rational() {
// Helper constructors.
let rp = |a: i64, b: i64| Point::new(Rational64::from(a), Rational64::from(b));
let r = Rational64::new;
let i = Rational64::from;

let e1 = Edge::new(rp(0, 0), rp(2, 2));
let e2 = Edge::new(rp(1, 0), rp(0, 1));
let e3 = Edge::new(rp(1, 0), rp(3, 2));

assert_eq!(
e1.line_intersection_rational(e2),
LineIntersection::Point(Point::new(r(1, 2), r(1, 2)), (i(1), i(2), i(4)))
);

// Parallel lines should not intersect
assert_eq!(e1.line_intersection_rational(e3), LineIntersection::None);

let e4 = Edge::new(rp(-320, 2394), rp(94, -4482));
let e5 = Edge::new(rp(71, 133), rp(-1373, 13847));

if let LineIntersection::Point(intersection, _) = e4.line_intersection_rational(e5) {
let a = e4.vector();
let b = intersection - e4.start;
let area = b.cross_prod(a);
assert!(area.is_zero());

let a = e5.vector();
let b = intersection - e5.start;
let area = b.cross_prod(a);
assert!(area.is_zero());
} else {
assert!(false);
}

// Collinear lines.
let scale = Rational64::new(1, 3);
let e1 = Edge::new(rp(0, 0), rp(2, 2)).scale(scale);
let e2 = Edge::new(rp(4, 4), rp(8, 8)).scale(scale);
assert!(!e1.is_coincident(&e2));
assert!(e1.is_parallel(&e2));
assert_eq!(
e1.line_intersection_rational(e2),
LineIntersection::Collinear
);
}

#[test]
fn test_edge_intersection_rational() {}
}
``````