1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139

use crate::{Line, Point, Scalar, Vector};

/// A plane
#[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Hash, Ord, PartialOrd)]
#[repr(C)]
pub struct Plane {
    origin: Point<3>,
    u: Vector<3>,
    v: Vector<3>,
}

impl Plane {
    /// Create a `Plane` from a parametric description
    pub fn from_parametric(
        origin: impl Into<Point<3>>,
        u: impl Into<Vector<3>>,
        v: impl Into<Vector<3>>,
    ) -> Self {
        let origin = origin.into();
        let u = u.into();
        let v = v.into();

        Self { origin, u, v }
    }

    /// Access the origin of the plane
    pub fn origin(&self) -> Point<3> {
        self.origin
    }

    /// Access the u-vector of the plane
    pub fn u(&self) -> Vector<3> {
        self.u
    }

    /// Access the v-vector of the plane
    pub fn v(&self) -> Vector<3> {
        self.v
    }

    /// Compute the normal of the plane
    pub fn normal(&self) -> Vector<3> {
        self.u().cross(&self.v()).normalize()
    }

    /// Convert the plane to three-point form
    pub fn three_point_form(&self) -> [Point<3>; 3] {
        let a = self.origin();
        let b = self.origin() + self.u();
        let c = self.origin() + self.v();

        [a, b, c]
    }

    /// Convert the plane to constant-normal form
    pub fn constant_normal_form(&self) -> (Scalar, Vector<3>) {
        let normal = self.normal();
        let distance = normal.dot(&self.origin().coords);

        (distance, normal)
    }

    /// Determine whether the plane is parallel to the given vector
    pub fn is_parallel_to_vector(&self, vector: &Vector<3>) -> bool {
        self.normal().dot(vector) == Scalar::ZERO
    }

    /// Project a point into the plane
    pub fn project_point(&self, point: impl Into<Point<3>>) -> Point<2> {
        let origin_to_point = point.into() - self.origin();
        let coords = self.project_vector(origin_to_point);
        Point { coords }
    }

    /// Project a vector into the plane
    pub fn project_vector(&self, vector: impl Into<Vector<3>>) -> Vector<2> {
        // The vector we want to project can be expressed as a linear
        // combination of `self.u()`, `self.v()`, and `self.normal()`:
        // `v = a*u + b*v + c*n`
        //
        // All we need to do is to solve this equation. `a` and `b` are the
        // components of the projected vector. `c` is the distance of the points
        // that the original and projected vectors point to.
        //
        // To solve the equation, let's change it into the standard `Mx = b`
        // form, then we can let nalgebra do the actual solving.
        let m =
            nalgebra::Matrix::<_, _, nalgebra::Const<3>, _>::from_columns(&[
                self.u().to_na(),
                self.v().to_na(),
                self.normal().to_na(),
            ]);
        let b = vector.into();
        let x = m
            .lu()
            .solve(&b.to_na())
            .expect("Expected matrix to be invertible");

        Vector::from([x.x, x.y])
    }

    /// Project a line into the plane
    pub fn project_line(&self, line: &Line<3>) -> Line<2> {
        let line_origin_in_plane = self.project_point(line.origin());
        let line_direction_in_plane = self.project_vector(line.direction());

        Line::from_origin_and_direction(
            line_origin_in_plane,
            line_direction_in_plane,
        )
    }
}

#[cfg(test)]
mod tests {
    use crate::{Plane, Point, Vector};

    #[test]
    fn project_point() {
        let plane =
            Plane::from_parametric([1., 1., 1.], [1., 0., 0.], [0., 1., 0.]);

        assert_eq!(plane.project_point([2., 1., 2.]), Point::from([1., 0.]));
        assert_eq!(plane.project_point([1., 2., 2.]), Point::from([0., 1.]));
    }

    #[test]
    fn project_vector() {
        let plane =
            Plane::from_parametric([1., 1., 1.], [1., 0., 0.], [0., 1., 0.]);

        assert_eq!(plane.project_vector([1., 0., 1.]), Vector::from([1., 0.]));
        assert_eq!(plane.project_vector([0., 1., 1.]), Vector::from([0., 1.]));

        let plane =
            Plane::from_parametric([1., 1., 1.], [1., 0., 0.], [1., 1., 0.]);
        assert_eq!(plane.project_vector([0., 1., 0.]), Vector::from([-1., 1.]));
    }
}