# Problem 165: Intersections

A segment is uniquely defined by its two endpoints.
By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.

Moreover when two segments have exactly one point in common it might be the case that that common point is an endpoint of either one of the segments or of both.
If a common point of two segments is not an endpoint of either of the segments it is an interior point of both segments.
We will call a common point T of two segments L_1 and L_2 a true intersection point of L_1 and L_2 if T is the only common point of L_1 and L_2 and T is an interior point of both segments.

Consider the three segments L_1, L_2, and L_3:

L_1: (27, 44) to (12, 32)
L_2: (46, 53) to (17, 62)
L_3: (46, 70) to (22, 40)

It can be verified that line segments L_2 and L_3 have a true intersection point.
We note that as the one of the end points of L_3: (22,40) lies on L_1 this is not considered to be a true point of intersection.
L_1 and L_2 have no common point. So among the three line segments, we find one true intersection point.

Now let us do the same for 5000 line segments. To this end, we generate 20000 numbers using the so-called "Blum Blum Shub" pseudo-random number generator.

s_0 = 290797
s_{n+1} == s_n * s_n mod 50515093
t_n == s_n mod 500

To create each line segment, we use four consecutive numbers tn. That is, the first line segment is given by:

(t_1, t_2) to (t_3, t_4)

The first four numbers computed according to the above generator should be: 27, 144, 12 and 232. The first segment would thus be (27,144) to (12,232).

How many distinct true intersection points are found among the 5000 line segments?

# My Algorithm

I studied Computer Graphics at university and had no problem coming up with an intersection algorithm.
Wikipedia's explanations of the use of the determinant are somehow weird and I would have trouble unterstanding it - but I already knew that from classes at university.
The main issue inside my intersect code is numerical stability.
These three lines reduce the number of digits so that the correct result is found:
const auto Precision = 0.00000001;
where.x = round(where.x / Precision) * Precision;
where.y = round(where.y / Precision) * Precision;

The constant Precision was found by trial-and-error.

The pseudo-random Blum Blum shub algorithm can be converted to a few lines of code (see next).

My code provides two simple classes Point and Segment to keep things organized.
Point needs to support comparisons operators for std::sort and std::unique (weeding out duplicate intersections).

main finds all (!) intersections and defers checking for duplicates until the end.
Duplicates are eliminated as follows:

• sort all intersections → identical intersections will be neighboring elements in the sorted set
• call std::unique to move all duplicates to the end of the data container ("garbage")
• remove all garbage points

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):

This is equivalent to
echo 10 | ./165

Output:

Note: the original problem's input 5000 cannot be entered
because just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       #include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>

// generate pseudo-random numbers (Blum Blum Shub algorithm)
unsigned int next()
{
static unsigned long long seed = 290797;
seed *= seed;
seed %= 50515093;
return seed % 500;
}

// a 2D point
struct Point
{
double x, y;

// needed for std::unique
bool operator==(const Point& other) const
{
return x == other.x && y == other.y;
}
// needed for std::sort
bool operator< (const Point& other) const
{
if (x != other.x)
return x < other.x;
else
return y < other.y;
}
};

// define a segment
struct Segment
{
Point from, to;
};

// find intersection of two segments, out parameter "where" is only valid if function returns true
bool intersect(const Segment& segment1, const Segment& segment2, Point& where)
{
// shorter names for the four endpoints
auto a = segment1.from;
auto b = segment1.to;
auto c = segment2.from;
auto d = segment2.to;

// store slope in a Point (just because I'm lazy and don't want to introduce another data type)
Point slope1, slope2;
slope1.x = b.x - a.x;
slope1.y = b.y - a.y;
slope2.x = d.x - c.x;
slope2.y = d.y - c.y;

auto determinant = slope1.x * slope2.y - slope2.x * slope1.y;
// parallel ?
if (determinant == 0)
return false;

// now the lines intersect, but not necessarily the segments
auto s = (slope1.x * (a.y - c.y) - slope1.y * (a.x - c.x)) / determinant;
auto t = (slope2.x * (a.y - c.y) - slope2.y * (a.x - c.x)) / determinant;

// parameters s and t must be in (0 ... 1)
// borders (=endpoints) are not true intersections according to problem statement
if (s <= 0 || s >= 1 || t <= 0 || t >= 1)
return false;

// yes, intersection found (might be a duplicate, though !)
where.x = a.x + t * slope1.x;
where.y = a.y + t * slope1.y;

// cut off a few digits to avoid rounding issues
const auto Precision = 0.00000001;
where.x = round(where.x / Precision) * Precision;
where.y = round(where.y / Precision) * Precision;

return true;
}

int main()
{
std::vector<Segment> segments;
std::vector<Point>   intersections;

unsigned int limit = 5000;
std::cin >> limit;

for (unsigned int i = 0; i < limit; i++)
{
// create "random" segment
Segment current;
current.from.x = next();
current.from.y = next();
current.to  .x = next();
current.to  .y = next();

// try to intersect with all other segments
Point where;
for (auto compare : segments)
if (intersect(current, compare, where))
intersections.push_back(where);

// add current segment to list of segments
segments.push_back(current);
}

// eliminate duplicate intersection points
std::sort(intersections.begin(), intersections.end());
auto garbage = std::unique(intersections.begin(), intersections.end());
intersections.erase(garbage, intersections.end());

// display result
std::cout << intersections.size() << std::endl;
return 0;
}


This solution contains 19 empty lines, 19 comments and 4 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.6 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 69 MByte.

(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

June 20, 2017 submitted solution

# Hackerrank

My code solves 11 out of 11 test cases (score: 100%)

# Difficulty

Project Euler ranks this problem at 65% (out of 100%).

Hackerrank describes this problem as easy.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
 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
 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500
 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600
 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692
The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !