1+
2+
3+ export Line
4+
5+
6+
7+
8+ # # Standard interval
9+
10+
11+
12+ # angle is π*a where a is (false==0) and (true==1)
13+ # or ranges from (-1,1]. We use 1 as 1==true.
14+
15+ """
16+ Line{a}(c)
17+
18+ represents the line at angle `a` in the complex plane, centred at `c`.
19+ """
20+ struct Line{angle,T<: Number } <: SegmentDomain{T}
21+ center:: T
22+ α:: Float64
23+ β:: Float64
24+
25+ # TODO get this inner constructor working again.
26+ Line {angle,T} (c,α,β) where {angle,T} = new {angle,T} (c,α,β)
27+ Line {angle,T} (c) where {angle,T} = new {angle,T} (c,- 1. ,- 1. )
28+ Line {angle,T} () where {angle,T} = new {angle,T} (zero (T),- 1. ,- 1. )
29+ end
30+
31+ const RealLine{T} = Union{Line{false ,T},Line{true ,T}}
32+
33+ Line {a} (c,α,β) where {a} = Line {a,typeof(c)} (c,α,β)
34+ Line {a} (c:: Number ) where {a} = Line {a,typeof(c)} (c)
35+ Line {a} () where {a} = Line {a,Float64} ()
36+
37+ angle (d:: Line{a} ) where {a} = a* π
38+
39+ reverseorientation (d:: Line{true} ) = Line {false} (d. center,d. β,d. α)
40+ reverseorientation (d:: Line{false} ) = Line {true} (d. center,d. β,d. α)
41+ reverseorientation (d:: Line{a} ) where {a} = Line {a-1} (d. center,d. β,d. α)
42+
43+ # ensure the angle is always in (-1,1]
44+ Line (c,a,α,β) = Line {mod(a/π-1,-2)+1,typeof(c)} (c,α,β)
45+ Line (c,a) = Line (c,a,- 1. ,- 1. )
46+ # true is negative orientation, false is positive orientation
47+ # this is because false==0 and we take angle=0
48+ Line (b:: Bool ) = Line {b} ()
49+ Line () = Line (false )
50+
51+
52+ isambiguous (d:: Line )= isnan (d. center)
53+ convert (:: Type{Domain{T}} ,d:: Line{a} ) where {a,T<: Number } = Line {a,T} (d. center, d. α, d. β)
54+ convert (:: Type{Line{a,T}} ,:: AnyDomain ) where {a,T<: Number } = Line {a,T} (NaN )
55+ convert (:: Type{IT} ,:: AnyDomain ) where {IT<: Line }= Line (NaN ,NaN )
56+
57+ # # Map interval
58+
59+
60+ # #TODO non-1 alpha,beta
61+
62+ isempty (:: Line ) = false
63+
64+ function line_tocanonical (α,β,x)
65+ @assert α== β== - 1. || α== β== - .5
66+
67+ if α== β== - 1.
68+ 2 x/ (1 + sqrt (1 + 4 x^ 2 ))
69+ elseif α== β== - .5
70+ x/ sqrt (1 + x^ 2 )
71+ end
72+ end
73+
74+ function line_tocanonicalD (α,β,x)
75+ @assert α== β== - 1. || α== β== - .5
76+
77+ if α== β== - 1.
78+ 2 / (1 + 4 x^ 2 + sqrt (1 + 4 x^ 2 ))
79+ elseif α== β== - 0.5
80+ (1 + x^ 2 )^ (- 3 / 2 )
81+ end
82+ end
83+ function line_fromcanonical (α,β,x)
84+ # TODO : why is this consistent?
85+ if α== β== - 1.
86+ x/ (1 - x^ 2 )
87+ else
88+ x* (1 + x)^ α* (1 - x)^ β
89+ end
90+ end
91+ function line_fromcanonicalD (α,β,x)
92+ if α== β== - 1.
93+ (1 + x^ 2 )/ (1 - x^ 2 )^ 2
94+ else
95+ (1 - (β- α)x - (β+ α+ 1 )x^ 2 )* (1 + x)^ (α- 1 )* (1 - x)^ (β- 1 )
96+ end
97+ end
98+
99+ function line_invfromcanonicalD (α,β,x)
100+ if α== β== - 1.
101+ (1 - x^ 2 )^ 2 / (1 + x^ 2 )
102+ else
103+ 1 / (1 - (β- α)x - (β+ α+ 1 )x^ 2 )* (1 + x)^ (1 - α)* (1 - x)^ (1 - β)
104+ end
105+ end
106+
107+
108+ tocanonical (d:: Line ,x) = line_tocanonical (d. α,d. β,cis (- angle (d)).* (x- d. center))
109+ tocanonical (d:: Line{false} ,x) = line_tocanonical (d. α,d. β,x- d. center)
110+ tocanonical (d:: Line{true} ,x) = line_tocanonical (d. α,d. β,d. center- x)
111+
112+ tocanonicalD (d:: Line ,x) = cis (- angle (d)).* line_tocanonicalD (d. α,d. β,cis (- angle (d)).* (x- d. center))
113+ tocanonicalD (d:: Line{false} ,x) = line_tocanonicalD (d. α,d. β,x- d. center)
114+ tocanonicalD (d:: Line{true} ,x) = - line_tocanonicalD (d. α,d. β,d. center- x)
115+
116+ fromcanonical (d:: Line ,x) = cis (angle (d))* line_fromcanonical (d. α,d. β,x)+ d. center
117+ fromcanonical (d:: Line{false} ,x) = line_fromcanonical (d. α,d. β,x)+ d. center
118+ fromcanonical (d:: Line{true} ,x) = - line_fromcanonical (d. α,d. β,x)+ d. center
119+
120+ fromcanonicalD (d:: Line ,x) = cis (angle (d))* line_fromcanonicalD (d. α,d. β,x)
121+ fromcanonicalD (d:: Line{false} ,x) = line_fromcanonicalD (d. α,d. β,x)
122+ fromcanonicalD (d:: Line{true} ,x) = - line_fromcanonicalD (d. α,d. β,x)
123+
124+ invfromcanonicalD (d:: Line ,x) = cis (- angle (d))* line_invfromcanonicalD (d. α,d. β,x)
125+ invfromcanonicalD (d:: Line{false} ,x) = line_invfromcanonicalD (d. α,d. β,x)
126+ invfromcanonicalD (d:: Line{true} ,x) = - line_invfromcanonicalD (d. α,d. β,x)
127+
128+
129+
130+
131+
132+
133+ == (d:: Line{a} ,m:: Line{a} ) where {a} = d. center == m. center && d. β == m. β && d. α == m. α
134+
135+
136+
137+ # algebra
138+ * (c:: Real ,d:: Line{false} ) = Line {sign(c)>0 ? false : true} (isapprox (d. center,0 ) ? d. center : c* d. center,d. α,d. β)
139+ * (c:: Real ,d:: Line{true} ) = Line {sign(c)>0 ? true : false} (isapprox (d. center,0 ) ? d. center : c* d. center,d. α,d. β)
140+ * (c:: Number ,d:: Line ) = Line (isapprox (d. center,0 ) ? d. center : c* d. center,angle (d)+ angle (c),d. α,d. β)
141+ * (d:: Line ,c:: Number ) = c* d
142+ for OP in (:+ ,:- )
143+ @eval begin
144+ $ OP (c:: Number ,d:: Line{a} ) where {a} = Line {a} ($ OP (c,d. center),d. α,d. β)
145+ $ OP (d:: Line{a} ,c:: Number ) where {a} = Line {a} ($ OP (d. center,c),d. α,d. β)
146+ end
147+ end
148+
149+
150+
151+
152+
153+
154+ # algebra
155+ arclength (d:: Line ) = Inf
156+ leftendpoint (d:: Line ) = - Inf
157+ rightendpoint (d:: Line ) = Inf
158+ complexlength (d:: Line ) = Inf
159+
160+ # # vectorized
161+
162+
163+ function convert (:: Type{Line} ,d:: ClosedInterval )
164+ a,b= d. left,d. right
165+ @assert abs (a) == abs (b) == Inf
166+
167+ if isa (a,Real) && isa (b,Real)
168+ if a== Inf
169+ @assert b== - Inf
170+ Line (true )
171+ else
172+ @assert a== - Inf && b== Inf
173+ Line (false )
174+ end
175+ elseif abs (real (a)) < Inf
176+ @assert real (a)== real (b)
177+ @assert sign (imag (a))== - sign (imag (b))
178+
179+ Line (real (b),angle (b))
180+ elseif isnan (real (a)) && isnan (real (b)) # hack for -im*Inf
181+ Line ([imag (a)* im,imag (b)* im])
182+ elseif isnan (real (a)) # hack for -im*Inf
183+ Line ([real (b)+ imag (a)* im,b])
184+ elseif isnan (real (b)) # hack for -im*Inf
185+ Line ([a,real (a)+ imag (b)* im])
186+ elseif abs (imag (a)) < Inf
187+ @assert imag (a)== imag (b)
188+ @assert sign (real (a))== - sign (real (b))
189+
190+ Line (imag (b),angle (b))
191+ else
192+ @assert angle (b) == - angle (a)
193+
194+ Line (0. ,angle (b))
195+ end
196+ end
197+
198+
199+
200+ issubset (a:: IntervalOrSegment , b:: Line ) = leftendpoint (a)∈ b && rightendpoint (a)∈ b
201+ issubset (a:: Ray{angle} , b:: Line{angle} ) where angle = leftendpoint (a) ∈ b
202+ issubset (a:: Ray{true} , b:: Line{false} ) = true
203+ issubset (a:: Ray{false} , b:: Line{true} ) = true
204+
205+ function intersect (a:: Union{Interval,Segment,Ray} ,b:: Line )
206+ @assert a ⊆ b
207+ a
208+ end
209+
210+ function union (a:: Union{Interval,Segment,Ray} ,b:: Line )
211+ @assert a ⊆ b
212+ b
213+ end
214+
215+ intersect (b:: Line ,a:: Union{Interval,Segment,Ray} ) = intersect (a,b)
216+ union (b:: Line ,a:: Union{Interval,Segment,Ray} ) = union (a,b)
217+
218+
219+ function setdiff (b:: Line ,a:: Segment )
220+ @assert a ⊆ b
221+ if leftendpoint (a)> rightendpoint (a)
222+ b\ reverseorientation (a)
223+ else
224+ Ray ([leftendpoint (b),leftendpoint (a)]) ∪ Ray ([rightendpoint (a),rightendpoint (b)])
225+ end
226+ end
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