-
Notifications
You must be signed in to change notification settings - Fork 5
/
geometryhelpers.py
342 lines (282 loc) · 9.66 KB
/
geometryhelpers.py
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
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
import math
def intersection(cline1, cline2):
"""Return intersection (x,y) of 2 clines expressed as (a,b,c) coeff."""
a, b, c = cline1
d, e, f = cline2
i = b*f-c*e
j = c*d-a*f
k = a*e-b*d
if k:
return (i/k, j/k)
return None
def cnvrt_2pts_to_coef(pt1, pt2):
"""Return (a,b,c) coefficients of cline defined by 2 (x,y) pts."""
x1, y1 = pt1
x2, y2 = pt2
a = y2 - y1
b = x1 - x2
c = x2*y1-x1*y2
return (a, b, c)
def proj_pt_on_line(cline, pt):
"""Return point which is the projection of pt on cline."""
a, b, c = cline
x, y = pt
denom = a**2 + b**2
if not denom:
return pt
xp = (b**2 * x - a*b*y - a*c) / denom
yp = (a**2 * y - a*b*x - b*c) / denom
return (xp, yp)
def pnt_in_box_p(pnt, box):
'''Point in box predicate: Return True if pnt is in box.'''
x, y = pnt
x1, y1, x2, y2 = box
return bool(x1 < x < x2 and y1 < y < y2)
def midpoint(p1, p2, f=.5):
"""Return point part way (f=.5 by def) between points p1 and p2."""
return (((p2[0]-p1[0])*f)+p1[0], ((p2[1]-p1[1])*f)+p1[1])
def p2p_dist(p1, p2):
"""Return the distance between two points"""
x, y = p1
u, v = p2
return math.sqrt((x-u)**2 + (y-v)**2)
def p2p_angle(p0, p1):
"""Return angle (degrees) from p0 to p1."""
return math.atan2(p1[1]-p0[1], p1[0]-p0[0])*180/math.pi
def add_pt(p0, p1):
return (p0[0]+p1[0], p0[1]+p1[1])
def sub_pt(p0, p1):
return (p0[0]-p1[0], p0[1]-p1[1])
def line_circ_inters(x1, y1, x2, y2, xc, yc, r):
'''Return list of intersection pts of line defined by pts x1,y1 and x2,y2
and circle (cntr xc,yc and radius r).
Uses algorithm from Paul Bourke's web page.'''
intpnts = []
num = (xc - x1)*(x2 - x1) + (yc - y1)*(y2 - y1)
denom = (x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1)
if denom == 0:
return
u = num / denom
xp = x1 + u*(x2-x1)
yp = y1 + u*(y2-y1)
a = (x2 - x1)**2 + (y2 - y1)**2
b = 2*((x2-x1)*(x1-xc) + (y2-y1)*(y1-yc))
c = xc**2+yc**2+x1**2+y1**2-2*(xc*x1+yc*y1)-r**2
q = b**2 - 4*a*c
if q == 0:
intpnts.append((xp, yp))
elif q:
u1 = (-b+math.sqrt(abs(q)))/(2*a)
u2 = (-b-math.sqrt(abs(q)))/(2*a)
intpnts.append(((x1 + u1*(x2-x1)), (y1 + u1*(y2-y1))))
intpnts.append(((x1 + u2*(x2-x1)), (y1 + u2*(y2-y1))))
return intpnts
def circ_circ_inters(x1, y1, r1, x2, y2, r2):
'''Return list of intersection pts of 2 circles.
Uses algorithm from Robert S. Wilson's web page.'''
pts = []
D = (x2-x1)**2 + (y2-y1)**2
if not D:
return pts # circles have same cntr; no intersection
q = math.sqrt(abs(((r1+r2)**2-D)*(D-(r2-r1)**2)))
pts = [((x2+x1)/2+(x2-x1)*(r1**2-r2**2)/(2*D)+(y2-y1)*q/(2*D),
(y2+y1)/2+(y2-y1)*(r1**2-r2**2)/(2*D)-(x2-x1)*q/(2*D)),
((x2+x1)/2+(x2-x1)*(r1**2-r2**2)/(2*D)-(y2-y1)*q/(2*D),
(y2+y1)/2+(y2-y1)*(r1**2-r2**2)/(2*D)+(x2-x1)*q/(2*D))]
if same_pt_p(pts[0], pts[1]):
pts.pop() # circles are tangent
return pts
def same_pt_p(p1, p2):
'''Return True if p1 and p2 are within 1e-10 of each other.'''
return bool(p2p_dist(p1, p2) < 1e-6)
def cline_box_intrsctn(cline, box):
"""Return tuple of pts where line intersects edges of box."""
x0, y0, x1, y1 = box
pts = []
segments = [((x0, y0), (x1, y0)),
((x1, y0), (x1, y1)),
((x1, y1), (x0, y1)),
((x0, y1), (x0, y0))]
for seg in segments:
pt = intersection(cline, cnvrt_2pts_to_coef(seg[0], seg[1]))
if pt:
if p2p_dist(pt, seg[0]) <= p2p_dist(seg[0], seg[1]) and \
p2p_dist(pt, seg[1]) <= p2p_dist(seg[0], seg[1]):
if pt not in pts:
pts.append(pt)
return tuple(pts)
def para_line(cline, pt):
"""Return coeff of newline thru pt and parallel to cline."""
a, b, c = cline
x, y = pt
cnew = -(a*x + b*y)
return (a, b, cnew)
def para_lines(cline, d):
"""Return 2 parallel lines straddling line, offset d."""
a, b, c = cline
c1 = math.sqrt(a**2 + b**2)*d
cline1 = (a, b, c + c1)
cline2 = (a, b, c - c1)
return (cline1, cline2)
def perp_line(cline, pt):
"""Return coeff of newline thru pt and perpend to cline."""
a, b, c = cline
x, y = pt
cnew = a*y - b*x
return (b, -a, cnew)
def closer(p0, p1, p2):
"""Return closer of p1 or p2 to point p0."""
d1 = (p1[0] - p0[0])**2 + (p1[1] - p0[1])**2
d2 = (p2[0] - p0[0])**2 + (p2[1] - p0[1])**2
if d1 < d2:
return p1
return p2
def farther(p0, p1, p2):
"""Return farther of p1 or p2 from point p0."""
d1 = (p1[0] - p0[0])**2 + (p1[1] - p0[1])**2
d2 = (p2[0] - p0[0])**2 + (p2[1] - p0[1])**2
if d1 > d2:
return p1
return p2
def find_fillet_pts(r, commonpt, end1, end2):
"""Return ctr of fillet (radius r) and tangent pts for corner
defined by a common pt, and two adjacent corner pts."""
line1 = cnvrt_2pts_to_coef(commonpt, end1)
line2 = cnvrt_2pts_to_coef(commonpt, end2)
# find 'interior' clines
cl1a, cl1b = para_lines(line1, r)
p2a = proj_pt_on_line(cl1a, end2)
p2b = proj_pt_on_line(cl1b, end2)
da = p2p_dist(p2a, end2)
db = p2p_dist(p2b, end2)
if da <= db:
cl1 = cl1a
else:
cl1 = cl1b
cl2a, cl2b = para_lines(line2, r)
p1a = proj_pt_on_line(cl2a, end1)
p1b = proj_pt_on_line(cl2b, end1)
da = p2p_dist(p1a, end1)
db = p2p_dist(p1b, end1)
if da <= db:
cl2 = cl2a
else:
cl2 = cl2b
pc = intersection(cl1, cl2)
p1 = proj_pt_on_line(line1, pc)
p2 = proj_pt_on_line(line2, pc)
return (pc, p1, p2)
def find_common_pt(apair, bpair):
"""Return (common pt, other pt from a, other pt from b), where a and b
are coordinate pt pairs in (p1, p2) format."""
p0, p1 = apair
p2, p3 = bpair
if same_pt_p(p0, p2):
cp = p0 # common pt
opa = p1 # other pt a
opb = p3 # other pt b
elif same_pt_p(p0, p3):
cp = p0
opa = p1
opb = p2
elif same_pt_p(p1, p2):
cp = p1
opa = p0
opb = p3
elif same_pt_p(p1, p3):
cp = p1
opa = p0
opb = p2
else:
return
return (cp, opa, opb)
def cr_from_3p(p1, p2, p3):
"""Return ctr pt and radius of circle on which 3 pts reside.
From Paul Bourke's web page."""
chord1 = cnvrt_2pts_to_coef(p1, p2)
chord2 = cnvrt_2pts_to_coef(p2, p3)
radial_line1 = perp_line(chord1, midpoint(p1, p2))
radial_line2 = perp_line(chord2, midpoint(p2, p3))
ctr = intersection(radial_line1, radial_line2)
if ctr:
radius = p2p_dist(p1, ctr)
return (ctr, radius)
def extendline(p0, p1, d):
"""Return point which lies on extension of line segment p0-p1,
beyond p1 by distance d."""
pts = line_circ_inters(p0[0], p0[1], p1[0], p1[1], p1[0], p1[1], d)
if pts:
return farther(p0, pts[0], pts[1])
return
def shortenline(p0, p1, d):
"""Return point which lies on line segment p0-p1,
short of p1 by distance d."""
pts = line_circ_inters(p0[0], p0[1], p1[0], p1[1], p1[0], p1[1], d)
if pts:
return closer(p0, pts[0], pts[1])
return
def line_tan_to_circ(circ, p):
"""Return tan pts on circ of line through p."""
c, r = circ
d = p2p_dist(c, p)
ang0 = p2p_angle(c, p)*math.pi/180
theta = math.asin(r/d)
ang1 = ang0+math.pi/2-theta
ang2 = ang0-math.pi/2+theta
p1 = (c[0]+(r*math.cos(ang1)), c[1]+(r*math.sin(ang1)))
p2 = (c[0]+(r*math.cos(ang2)), c[1]+(r*math.sin(ang2)))
return (p1, p2)
def line_tan_to_2circs(circ1, circ2):
"""Return tangent pts on line tangent to 2 circles.
Order of circle picks determines which tangent line."""
c1, r1 = circ1
c2, r2 = circ2
d = p2p_dist(c1, c2) # distance between centers
ang_loc = p2p_angle(c2, c1)*math.pi/180 # angle of line of centers
f = (r2/r1-1)/d # reciprocal dist from c1 to intersection of loc & tan line
theta = math.asin(r1*f) # angle between loc and tangent line
ang1 = (ang_loc + math.pi/2 - theta)
# ang2 = (ang_loc - math.pi/2 + theta) # unused
p1 = (c1[0]+(r1*math.cos(ang1)), c1[1]+(r1*math.sin(ang1)))
p2 = (c2[0]+(r2*math.cos(ang1)), c2[1]+(r2*math.sin(ang1)))
return (p1, p2)
def angled_cline(pt, angle):
"""Return cline through pt at angle (degrees)"""
ang = angle * math.pi / 180
dx = math.cos(ang)
dy = math.sin(ang)
p2 = (pt[0]+dx, pt[1]+dy)
cline = cnvrt_2pts_to_coef(pt, p2)
return cline
def ang_bisector(p0, p1, p2, f=0.5):
"""Return cline coefficients of line through vertex p0, factor=f
between p1 and p2."""
ang1 = math.atan2(p1[1]-p0[1], p1[0]-p0[0])
ang2 = math.atan2(p2[1]-p0[1], p2[0]-p0[0])
deltang = ang2 - ang1
ang3 = (f * deltang + ang1)*180/math.pi
return angled_cline(p0, ang3)
def pt_on_RHS_p(pt, p0, p1):
"""Return True if pt is on right hand side going from p0 to p1."""
angline = p2p_angle(p0, p1)
angpt = p2p_angle(p0, pt)
if angline >= 0:
if angline > angpt > angline-180:
return True
else:
angline += 360
if angpt < 0:
angpt += 360
if angline > angpt > angline-180:
return True
def rotate_pt(pt, ang, ctr):
"""Return coordinates of pt rotated ang (deg) CCW about ctr.
This is a 3-step process:
1. translate to place ctr at origin.
2. rotate about origin (CCW version of Paul Bourke's algorithm.
3. apply inverse translation. """
x, y = sub_pt(pt, ctr)
A = ang * math.pi / 180
u = x * math.cos(A) - y * math.sin(A)
v = y * math.cos(A) + x * math.sin(A)
return add_pt((u, v), ctr)