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
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
|
/* See LICENSE file for copyright and license details. */
#include "common.h"
#ifndef TEST
static size_t
solve_cubic_bezier(double *restrict ts, double a, double b, double c, double d, double z)
{
/*
* B(t) = t(q - p) + p
* where
* p = t(v - u) + u
* q = t(w - v) + v
* where
* u = t(b - a) + a
* v = t(c - b) + b
* w = t(d - c) + c
*
* B(t) = t³(d - 3c + 3b - a) + t²(3c - 6b + 3a) + t(3b - 3a) + a
*
* z = t³(d - 3c + 3b - a) + t²(3c - 6b + 3a) + t(3b - 3a) + a
*
* t³(d - 3c + 3b - a) + t²(3c - 6b + 3a) + t(3b - 3a) + a - z = 0
*
* A := d - 3c + 3b - a
* B := 3c - 6b + 3a
* C := 3b - 3a
* D := a - z
*
* t³A + t²B + tC + D = 0
*/
double c3 = 3.0 * c;
double b3 = 3.0 * b;
double a3 = 3.0 * a;
double A = (d - c3) + (b3 - a);
double B = (c3 - b3) - (b3 - a3);
double C = b3 - a3;
double D = a - z;
if (A)
return solve_cubic(ts, A, B, C, D);
else if (B)
return solve_quadratic(ts, B, C, D);
else if (C)
return solve_linear(ts, C, D);
else
return 0;
}
static double
evaluate_cubic_bezier(double t, double a, double b, double c, double d)
{
double u = fma(t, b - a, a);
double v = fma(t, c - b, b);
double w = fma(t, d - c, c);
double p = fma(t, v - u, u);
double q = fma(t, w - v, v);
return fma(t, q - p, p);
}
static double
evaluate_cubic_bezier_derivative(double t, double a, double b, double c, double d)
{
double u = fma(t, b - a, a);
double v = fma(t, c - b, b);
double w = fma(t, d - c, c);
return fma(t, (w - v) + (u - v), v - u);
}
static void
draw_bounded_cubic_bezier(RASTER *restrict raster, double x1, double y1, double x2, double y2,
double x3, double y3, double x4, double y4, double t1, double t2)
{
double t = t1, x0, y0, dx, dy, x, y;
double w = (double)raster->width - 0.0001;
double h = (double)raster->height - 0.0001;
x0 = evaluate_cubic_bezier(t, x1, x2, x3, x4);
y0 = evaluate_cubic_bezier(t, y1, y2, y3, y4);
x0 = fmin(fmax(0, x0), w);
y0 = fmin(fmax(0, y0), h);
for (; t < t2; x0 = x, y0 = y) {
dx = evaluate_cubic_bezier_derivative(t, x1, x2, x3, x4);
dy = evaluate_cubic_bezier_derivative(t, y1, y2, y3, y4);
t += raster->draftness / hypot(dx, dy);
t = fmin(t, t2);
x = evaluate_cubic_bezier(t, x1, x2, x3, x4);
y = evaluate_cubic_bezier(t, y1, y2, y3, y4);
x = fmin(fmax(0, x), w);
y = fmin(fmax(0, y), h);
draw_bounded_line(raster, x0, y0, x, y);
}
}
void
rtgrpblib_draw_cubic_bezier(RASTER *restrict raster, double x1, double y1, double x2, double y2,
double x3, double y3, double x4, double y4)
{
double x, y, yt1, yt2, dy;
double ts[2+4*3], t, t1, t2;
size_t nts = 0;
size_t i;
#ifdef TODO /* untested */
/* Can we downgrade the curve to a quadratic bézier curve?
*
* B(t) = (1-t)²Q₁ + 2(1-t)tQ₂ + t²Q₃
* B(t) = (1-t)B(t) + tB(t) =
* = (1-t)((1-t)²Q₁ + 2(1-t)tQ₂ + t²Q₃) + t((1-t)²Q₁ + 2(1-t)tQ₂ + t²Q₃)
* = (1-t)³Q₁ + 2(1-t)²tQ₂ + (1-t)t²Q₃ + (1-t)²tQ₁ + 2(1-t)t²Q₂ + t³Q₃
* = (1-t)³Q₁ + 2(1-t)²tQ₂ + (1-t)²tQ₁ + 2(1-t)t²Q₂ + (1-t)t²Q₃ + t³Q₃
* = (1-t)³Q₁ + (1-t)²t(Q₁ + 2Q₂) + (1-t)t²(2Q₂ + Q₃) + t³Q₃
* = (1-t)³Q₁ + 3(1-t)²t(Q₁ + 2Q₂)/3 + 3(1-t)t²(2Q₂ + Q₃)/3 + t³Q₃
*
* B(t) = (1-t)³C₁ + 3(1-t)²tC₂ + 3(1-t)t²C₃ + t³C₄
*
* C₁ = Q₁, C₄ = Q₃
*
* B(t) = (1-t)³Q₁ + 3(1-t)²t(Q₁ + 2Q₂)/3 + 3(1-t)t²(2Q₂ + Q₃)/3 + t³Q₃
* B(t) = (1-t)³C₁ + 3(1-t)²tC₂ + 3(1-t)t²C₃ + t³C₄
*
* (1-t)³Q₁ + 3(1-t)²t(Q₁ + 2Q₂)/3 + 3(1-t)t²(2Q₂ + Q₃)/3 + t³Q₃ = (1-t)³C₁ + 3(1-t)²tC₂ + 3(1-t)t²C₃ + t³C₄w
* 3(1-t)²t(Q₁ + 2Q₂)/3 + 3(1-t)t²(2Q₂ + Q₃)/3 = 3(1-t)²tC₂ + 3(1-t)t²C₃
* (1-t)²t(Q₁ + 2Q₂)/3 + (1-t)t²(2Q₂ + Q₃)/3 = (1-t)²tC₂ + (1-t)t²C₃
* (1-t)t(Q₁ + 2Q₂)/3 + t²(2Q₂ + Q₃)/3 = (1-t)tC₂ + t²C₃
* (1-t)(Q₁ + 2Q₂)/3 + t(2Q₂ + Q₃)/3 = (1-t)C₂ + tC₃
* s(Q₁ + 2Q₂)/3 + t(2Q₂ + Q₃)/3 = sC₂ + tC₃
* (Q₁ + 2Q₂)/3 = C₂, (2Q₂ + Q₃)/3 = C₃
* Q₁ + 2Q₂ = 3C₂, 2Q₂ + Q₃ = 3C₃
* C₁ + 2Q₂ = 3C₂, 2Q₂ + C₄ = 3C₃
* 2Q₂ = 3C₂ - C₁, 2Q₂ = 3C₃ - C₄
* 2Q₂ = 3C₂ - C₁ = 3C₃ - C₄
*/
if (x2 + x2 + x2 - x1 == x3 + x3 + x3 - x4 && y2 + y2 + y2 - y1 == y3 + y3 + y3 - y4) {
rtgrpblib_draw_quadratic_bezier(raster, x1, y1, x2 + (x2 - x1) / 2.0, y2 + (y2 - y1) / 2.0, x4, y4);
return;
}
#endif
/* Beginning and end of curve */
ts[nts++] = 0.0;
ts[nts++] = 1.0;
/* Get points where the end intersects one of the edges of the raster */
nts += solve_cubic_bezier(&ts[nts], y1, y2, y3, y4, 0);
nts += solve_cubic_bezier(&ts[nts], y1, y2, y3, y4, (double)raster->height);
nts += solve_cubic_bezier(&ts[nts], x1, x2, x3, x4, 0);
nts += solve_cubic_bezier(&ts[nts], x1, x2, x3, x4, (double)raster->width);
/* Sort all points of interest */
qsort(ts, nts, sizeof(*ts), doublepcmp);
for (i = 0; i < nts - 1; i++) {
t1 = ts[i];
t2 = ts[i + 1];
if (t1 == t2)
continue;
t = 0.5 * (t1 + t2);
/* Remove any segments above or below the raster */
y = evaluate_cubic_bezier(t, y1, y2, y3, y4);
if (y < 0 || y >= (double)raster->height)
continue;
/* If the segment is inside the raster, draw it, */
x = evaluate_cubic_bezier(t, x1, x2, x3, x4);
if (0 <= x && x < (double)raster->width) {
draw_bounded_cubic_bezier(raster, x1, y1, x2, y2, x3, y3, x4, y4, t1, t2);
continue;
}
/* otherwise draw it flattened to the edge; however,
* we will reduce it to a from the start to the end
* and draw its project on the edge, which works
* because if the curve bends it will cancel it self
* out except within this project; this project will
* also preserv the direction of part of the curve
* that is not cancelled out */
yt1 = evaluate_cubic_bezier(t1, y1, y2, y3, y4);
yt2 = evaluate_cubic_bezier(t2, y1, y2, y3, y4);
dy = yt2 - yt1;
if (x < 0)
draw_vertical_line(raster, 0, yt1, yt2, SIGNUM(dy));
else
draw_vertical_line_opposite_only(raster, raster->width - 1, yt1, yt2, SIGNUM(dy));
}
}
#else
int
main(void)
{
return 0; /* TODO add test */
}
#endif
|
