tilezz 0.1.4

Utilities to work with perfect-precision polygonal tiles built on top of cyclotomic integer rings.
Documentation 
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

//! Core linear algebra utils
use super::traits::IsRing;

/// Sign of the wedge product of `p1` and `p2`.
///
/// By the identity `wedge(p1, p2) = Im(conj(p1) * p2)`, this is the same as
/// the sign of the imaginary component of `conj(p1) * p2`. Returns -1, 0, or 1.
///
/// Stays inside `ZZ` for the multiplication and only extracts a sign at the end.
pub fn wedge_sign<ZZ: IsRing>(p1: &ZZ, p2: &ZZ) -> i8 {
    (p1.conj() * *p2).im_sign()
}

/// Sign of the dot product of `p1` and `p2`.
///
/// By the identity `dot(p1, p2) = Re(conj(p1) * p2)`, this is the same as
/// the sign of the real component of `conj(p1) * p2`. Returns -1, 0, or 1.
///
/// Stays inside `ZZ` for the multiplication and only extracts a sign at the end.
pub fn dot_sign<ZZ: IsRing>(p1: &ZZ, p2: &ZZ) -> i8 {
    (p1.conj() * *p2).re_sign()
}

/// Return true if angle (wrt. positive real line) is in closed interval `[a,b]`,
/// assuming that a and b are in counterclockwise order and their ccw angle
/// is less than a half turn.
pub fn angle_between<ZZ: IsRing>(p: &ZZ, (a, b): (&ZZ, &ZZ)) -> bool {
    wedge_sign(a, p) >= 0 && wedge_sign(p, b) >= 0
}

/// Return whether this point is strictly between the other two.
///
/// NOTE: we already assume all three involved points are colinear.
pub fn is_between<ZZ: IsRing>(p: &ZZ, (a, b): (&ZZ, &ZZ)) -> bool {
    // wedge(a - p, p - b) == 0 AND dot(a - p, p - b) > 0
    let v = *a - *p;
    let w = *p - *b;
    wedge_sign(&v, &w) == 0 && dot_sign(&v, &w) > 0
}

/// Return whether the segments pa and pb (in that order) have a ccw angle,
/// i.e. b is ccw of a with respect to rotation around p.
pub fn is_ccw<ZZ: IsRing>(p: &ZZ, (a, b): (&ZZ, &ZZ)) -> bool {
    // wedge(a - p, b - p) > 0
    wedge_sign(&(*a - *p), &(*b - *p)) > 0
    // NOTE: wedge(*a - *p, *p - *b).is_zero() is subexp. of is_between... interesting symmetry
}

#[cfg(test)]
mod tests {
    use num_traits::{One, Zero};

    use super::super::rings::ZZ12;
    use super::super::traits::{SymNum, Units};
    use super::*;

    type ZZi = ZZ12;

    #[test]
    fn test_is_between() {
        let a: ZZi = ZZi::zero();
        let b: ZZi = ZZi::one();
        let c: ZZi = ZZi::from(2);
        let e: ZZi = <ZZi as Units>::unit(ZZi::hturn() / 2);
        let f: ZZi = b + e;
        let g: ZZi = <ZZi as Units>::unit(1) + <ZZi as Units>::unit(-1) - ZZi::one();

        // is actually in between
        assert!(is_between(&b, (&a, &c)));
        assert!(is_between(&g, (&a, &b)));
        // is an endpoint
        assert!(!is_between(&b, (&a, &b)));
        assert!(!is_between(&a, (&a, &b)));
        // colinear, but not between
        assert!(!is_between(&c, (&a, &b)));
        // not colinear
        assert!(!is_between(&f, (&a, &b)));
    }

    #[test]
    fn test_is_ccw() {
        type ZZ = ZZ12;
        assert!(is_ccw(
            &ZZ::zero(),
            (&<ZZ as Units>::unit(0), &<ZZ as Units>::unit(1))
        ));
        assert!(is_ccw(
            &ZZ::zero(),
            (&<ZZ as Units>::unit(2), &<ZZ as Units>::unit(3))
        ));
        assert!(!is_ccw(
            &ZZ::zero(),
            (&<ZZ as Units>::unit(0), &<ZZ as Units>::unit(0))
        ));
        assert!(!is_ccw(
            &ZZ::zero(),
            (&<ZZ as Units>::unit(1), &<ZZ as Units>::unit(1))
        ));
        assert!(!is_ccw(
            &ZZ::zero(),
            (&<ZZ as Units>::unit(1), &<ZZ as Units>::unit(0))
        ));
        assert!(!is_ccw(
            &ZZ::zero(),
            (&<ZZ as Units>::unit(6), &<ZZ as Units>::unit(5))
        ));
    }

    #[test]
    fn test_angle_between() {
        type ZZ = ZZ12;
        // test inside
        assert!(angle_between(
            &<ZZ as Units>::unit(1),
            (&<ZZ as Units>::unit(0), &<ZZ as Units>::unit(2))
        ));
        assert!(angle_between(
            &<ZZ as Units>::unit(-2),
            (&<ZZ as Units>::unit(-3), &<ZZ as Units>::unit(1))
        ));
        // test outside
        assert!(!angle_between(
            &<ZZ as Units>::unit(1),
            (&<ZZ as Units>::unit(-1), &<ZZ as Units>::unit(0))
        ));
        // bound inclusive
        assert!(angle_between(
            &<ZZ as Units>::unit(1),
            (&<ZZ as Units>::unit(1), &<ZZ as Units>::unit(2))
        ));
        assert!(angle_between(
            &<ZZ as Units>::unit(2),
            (&<ZZ as Units>::unit(1), &<ZZ as Units>::unit(2))
        ));
    }
}