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1 /* | |
2 http://stackoverflow.com/questions/2009160/how-do-i-convert-the-2-control-points
-of-a-cubic-curve-to-the-single-control-poi | |
3 */ | |
4 | |
5 /* | |
6 Let's call the control points of the cubic Q0..Q3 and the control points of the
quadratic P0..P2. | |
7 Then for degree elevation, the equations are: | |
8 | |
9 Q0 = P0 | |
10 Q1 = 1/3 P0 + 2/3 P1 | |
11 Q2 = 2/3 P1 + 1/3 P2 | |
12 Q3 = P2 | |
13 In your case you have Q0..Q3 and you're solving for P0..P2. There are two ways t
o compute P1 from | |
14 the equations above: | |
15 | |
16 P1 = 3/2 Q1 - 1/2 Q0 | |
17 P1 = 3/2 Q2 - 1/2 Q3 | |
18 If this is a degree-elevated cubic, then both equations will give the same answe
r for P1. Since | |
19 it's likely not, your best bet is to average them. So, | |
20 | |
21 P1 = -1/4 Q0 + 3/4 Q1 + 3/4 Q2 - 1/4 Q3 | |
22 | |
23 | |
24 Cubic defined by: P1/2 - anchor points, C1/C2 control points | |
25 |x| is the euclidean norm of x | |
26 mid-point approx of cubic: a quad that shares the same anchors with the cubic an
d has the | |
27 control point at C = (3·C2 - P2 + 3·C1 - P1)/4 | |
28 | |
29 Algorithm | |
30 | |
31 pick an absolute precision (prec) | |
32 Compute the Tdiv as the root of (cubic) equation | |
33 sqrt(3)/18 · |P2 - 3·C2 + 3·C1 - P1|/2 · Tdiv ^ 3 = prec | |
34 if Tdiv < 0.5 divide the cubic at Tdiv. First segment [0..Tdiv] can be approxima
ted with by a | |
35 quadratic, with a defect less than prec, by the mid-point approximation. | |
36 Repeat from step 2 with the second resulted segment (corresponding to 1-Tdiv) | |
37 0.5<=Tdiv<1 - simply divide the cubic in two. The two halves can be approximated
by the mid-point | |
38 approximation | |
39 Tdiv>=1 - the entire cubic can be approximated by the mid-point approximation | |
40 | |
41 confirmed by (maybe stolen from) | |
42 http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html | |
43 // maybe in turn derived from http://www.cccg.ca/proceedings/2004/36.pdf | |
44 // also stored at http://www.cis.usouthal.edu/~hain/general/Publications/Bezier/
bezier%20cccg04%20paper.pdf | |
45 | |
46 */ | |
47 | |
48 #include "CubicUtilities.h" | |
49 #include "CurveIntersection.h" | |
50 #include "LineIntersection.h" | |
51 #include "TSearch.h" | |
52 | |
53 const bool AVERAGE_END_POINTS = true; // results in better fitting curves | |
54 | |
55 #define USE_CUBIC_END_POINTS 1 | |
56 | |
57 static double calcTDiv(const Cubic& cubic, double precision, double start) { | |
58 const double adjust = sqrt(3) / 36; | |
59 Cubic sub; | |
60 const Cubic* cPtr; | |
61 if (start == 0) { | |
62 cPtr = &cubic; | |
63 } else { | |
64 // OPTIMIZE: special-case half-split ? | |
65 sub_divide(cubic, start, 1, sub); | |
66 cPtr = ⊂ | |
67 } | |
68 const Cubic& c = *cPtr; | |
69 double dx = c[3].x - 3 * (c[2].x - c[1].x) - c[0].x; | |
70 double dy = c[3].y - 3 * (c[2].y - c[1].y) - c[0].y; | |
71 double dist = sqrt(dx * dx + dy * dy); | |
72 double tDiv3 = precision / (adjust * dist); | |
73 double t = cube_root(tDiv3); | |
74 if (start > 0) { | |
75 t = start + (1 - start) * t; | |
76 } | |
77 return t; | |
78 } | |
79 | |
80 void demote_cubic_to_quad(const Cubic& cubic, Quadratic& quad) { | |
81 quad[0] = cubic[0]; | |
82 if (AVERAGE_END_POINTS) { | |
83 const _Point fromC1 = { (3 * cubic[1].x - cubic[0].x) / 2, (3 * cubic[1].y -
cubic[0].y) / 2 }; | |
84 const _Point fromC2 = { (3 * cubic[2].x - cubic[3].x) / 2, (3 * cubic[2].y -
cubic[3].y) / 2 }; | |
85 quad[1].x = (fromC1.x + fromC2.x) / 2; | |
86 quad[1].y = (fromC1.y + fromC2.y) / 2; | |
87 } else { | |
88 lineIntersect((const _Line&) cubic[0], (const _Line&) cubic[2], quad[1]); | |
89 } | |
90 quad[2] = cubic[3]; | |
91 } | |
92 | |
93 int cubic_to_quadratics(const Cubic& cubic, double precision, SkTDArray<Quadrati
c>& quadratics) { | |
94 SkTDArray<double> ts; | |
95 cubic_to_quadratics(cubic, precision, ts); | |
96 int tsCount = ts.count(); | |
97 double t1Start = 0; | |
98 int order = 0; | |
99 for (int idx = 0; idx <= tsCount; ++idx) { | |
100 double t1 = idx < tsCount ? ts[idx] : 1; | |
101 Cubic part; | |
102 sub_divide(cubic, t1Start, t1, part); | |
103 Quadratic q1; | |
104 demote_cubic_to_quad(part, q1); | |
105 Quadratic s1; | |
106 int o1 = reduceOrder(q1, s1, kReduceOrder_TreatAsFill); | |
107 if (order < o1) { | |
108 order = o1; | |
109 } | |
110 memcpy(quadratics.append(), o1 < 2 ? s1 : q1, sizeof(Quadratic)); | |
111 t1Start = t1; | |
112 } | |
113 return order; | |
114 } | |
115 | |
116 static bool addSimpleTs(const Cubic& cubic, double precision, SkTDArray<double>&
ts) { | |
117 double tDiv = calcTDiv(cubic, precision, 0); | |
118 if (tDiv >= 1) { | |
119 return true; | |
120 } | |
121 if (tDiv >= 0.5) { | |
122 *ts.append() = 0.5; | |
123 return true; | |
124 } | |
125 return false; | |
126 } | |
127 | |
128 static void addTs(const Cubic& cubic, double precision, double start, double end
, | |
129 SkTDArray<double>& ts) { | |
130 double tDiv = calcTDiv(cubic, precision, 0); | |
131 double parts = ceil(1.0 / tDiv); | |
132 for (double index = 0; index < parts; ++index) { | |
133 double newT = start + (index / parts) * (end - start); | |
134 if (newT > 0 && newT < 1) { | |
135 *ts.append() = newT; | |
136 } | |
137 } | |
138 } | |
139 | |
140 // flavor that returns T values only, deferring computing the quads until they a
re needed | |
141 // FIXME: when called from recursive intersect 2, this could take the original c
ubic | |
142 // and do a more precise job when calling chop at and sub divide by computing th
e fractional ts. | |
143 // it would still take the prechopped cubic for reduce order and find cubic infl
ections | |
144 void cubic_to_quadratics(const Cubic& cubic, double precision, SkTDArray<double>
& ts) { | |
145 Cubic reduced; | |
146 int order = reduceOrder(cubic, reduced, kReduceOrder_QuadraticsAllowed, | |
147 kReduceOrder_TreatAsFill); | |
148 if (order < 3) { | |
149 return; | |
150 } | |
151 double inflectT[5]; | |
152 int inflections = find_cubic_inflections(cubic, inflectT); | |
153 SkASSERT(inflections <= 2); | |
154 if (!ends_are_extrema_in_x_or_y(cubic)) { | |
155 inflections += find_cubic_max_curvature(cubic, &inflectT[inflections]); | |
156 SkASSERT(inflections <= 5); | |
157 } | |
158 QSort<double>(inflectT, &inflectT[inflections - 1]); | |
159 // OPTIMIZATION: is this filtering common enough that it needs to be pulled
out into its | |
160 // own subroutine? | |
161 while (inflections && approximately_less_than_zero(inflectT[0])) { | |
162 memcpy(inflectT, &inflectT[1], sizeof(inflectT[0]) * --inflections); | |
163 } | |
164 int start = 0; | |
165 do { | |
166 int next = start + 1; | |
167 if (next >= inflections) { | |
168 break; | |
169 } | |
170 if (!approximately_equal(inflectT[start], inflectT[next])) { | |
171 ++start; | |
172 continue; | |
173 } | |
174 memcpy(&inflectT[start], &inflectT[next], sizeof(inflectT[0]) * (--infle
ctions - start)); | |
175 } while (true); | |
176 while (inflections && approximately_greater_than_one(inflectT[inflections -
1])) { | |
177 --inflections; | |
178 } | |
179 CubicPair pair; | |
180 if (inflections == 1) { | |
181 chop_at(cubic, pair, inflectT[0]); | |
182 int orderP1 = reduceOrder(pair.first(), reduced, kReduceOrder_NoQuadrati
csAllowed, | |
183 kReduceOrder_TreatAsFill); | |
184 if (orderP1 < 2) { | |
185 --inflections; | |
186 } else { | |
187 int orderP2 = reduceOrder(pair.second(), reduced, kReduceOrder_NoQua
draticsAllowed, | |
188 kReduceOrder_TreatAsFill); | |
189 if (orderP2 < 2) { | |
190 --inflections; | |
191 } | |
192 } | |
193 } | |
194 if (inflections == 0 && addSimpleTs(cubic, precision, ts)) { | |
195 return; | |
196 } | |
197 if (inflections == 1) { | |
198 chop_at(cubic, pair, inflectT[0]); | |
199 addTs(pair.first(), precision, 0, inflectT[0], ts); | |
200 addTs(pair.second(), precision, inflectT[0], 1, ts); | |
201 return; | |
202 } | |
203 if (inflections > 1) { | |
204 Cubic part; | |
205 sub_divide(cubic, 0, inflectT[0], part); | |
206 addTs(part, precision, 0, inflectT[0], ts); | |
207 int last = inflections - 1; | |
208 for (int idx = 0; idx < last; ++idx) { | |
209 sub_divide(cubic, inflectT[idx], inflectT[idx + 1], part); | |
210 addTs(part, precision, inflectT[idx], inflectT[idx + 1], ts); | |
211 } | |
212 sub_divide(cubic, inflectT[last], 1, part); | |
213 addTs(part, precision, inflectT[last], 1, ts); | |
214 return; | |
215 } | |
216 addTs(cubic, precision, 0, 1, ts); | |
217 } | |
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