1/*
2http://stackoverflow.com/questions/2009160/how-do-i-convert-the-2-control-points-of-a-cubic-curve-to-the-single-control-poi
3*/
4
5/*
6Let's call the control points of the cubic Q0..Q3 and the control points of the quadratic P0..P2.
7Then for degree elevation, the equations are:
8
9Q0 = P0
10Q1 = 1/3 P0 + 2/3 P1
11Q2 = 2/3 P1 + 1/3 P2
12Q3 = P2
13In your case you have Q0..Q3 and you're solving for P0..P2. There are two ways to compute P1 from
14 the equations above:
15
16P1 = 3/2 Q1 - 1/2 Q0
17P1 = 3/2 Q2 - 1/2 Q3
18If this is a degree-elevated cubic, then both equations will give the same answer for P1. Since
19 it's likely not, your best bet is to average them. So,
20
21P1 = -1/4 Q0 + 3/4 Q1 + 3/4 Q2 - 1/4 Q3
22
23
24Cubic defined by: P1/2 - anchor points, C1/C2 control points
25|x| is the euclidean norm of x
26mid-point approx of cubic: a quad that shares the same anchors with the cubic and has the
27 control point at C = (3·C2 - P2 + 3·C1 - P1)/4
28
29Algorithm
30
31pick an absolute precision (prec)
32Compute the Tdiv as the root of (cubic) equation
33sqrt(3)/18 · |P2 - 3·C2 + 3·C1 - P1|/2 · Tdiv ^ 3 = prec
34if Tdiv < 0.5 divide the cubic at Tdiv. First segment [0..Tdiv] can be approximated 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)
370.5<=Tdiv<1 - simply divide the cubic in two. The two halves can be approximated by the mid-point
38 approximation
39Tdiv>=1 - the entire cubic can be approximated by the mid-point approximation
40
41confirmed by (maybe stolen from)
42http://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
53const bool AVERAGE_END_POINTS = true; // results in better fitting curves
54
55#define USE_CUBIC_END_POINTS 1
56
57static 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 = &sub;
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
80void demote_cubic_to_quad(const Cubic& cubic, Quadratic& quad) {
81    quad[0] = cubic[0];
82if (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
93int cubic_to_quadratics(const Cubic& cubic, double precision, SkTDArray<Quadratic>& 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
116static 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
128static 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 are needed
141// FIXME: when called from recursive intersect 2, this could take the original cubic
142// and do a more precise job when calling chop at and sub divide by computing the fractional ts.
143// it would still take the prechopped cubic for reduce order and find cubic inflections
144void 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]) * (--inflections - 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_NoQuadraticsAllowed,
183                kReduceOrder_TreatAsFill);
184        if (orderP1 < 2) {
185            --inflections;
186        } else {
187            int orderP2 = reduceOrder(pair.second(), reduced, kReduceOrder_NoQuadraticsAllowed,
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}
218