Determine if Two Rotated Rectangles Overlap Each Other
The question is how to determine if two rectangles overlap each other. To simplify the question, we say the rectangles are on a 2 dimensional plane. The rectangles can be rotated.
The first problem is How to represent a rectangle.
Actually, using three points is sufficient, since the last one can be calculate by the first three. This solution use 8 integers to present a rectangle. Another solution is to use angle. First of all, we need the angle between the first (counter-clockwise) side and x-axis. We also need a start point, and the length of each side. Then the struct can be,
If the Edges are parallel to the coordinate axes, it is not that complicated, check this article.
http://articles.leetcode.com/2011/05/determine-if-two-rectangles-overlap.html
http://vijayt.com/Post/QuickSelect-works-in-linear-time
https://github.com/jaak-s/extraTrees/blob/master/src/main/java/org/extratrees/QuickSelect.java
Read full article from Determine if Two Rotated Rectangles Overlap Each Other
The question is how to determine if two rectangles overlap each other. To simplify the question, we say the rectangles are on a 2 dimensional plane. The rectangles can be rotated.
The first problem is How to represent a rectangle.
typedef struct point {
int x;
int y;
} Point;
typedef struct rectangle {
Point p[4];
} Rectangle;
Actually, using three points is sufficient, since the last one can be calculate by the first three. This solution use 8 integers to present a rectangle. Another solution is to use angle. First of all, we need the angle between the first (counter-clockwise) side and x-axis. We also need a start point, and the length of each side. Then the struct can be,
typedef struct point {
int x;
int y;
} Point;
typedef struct rectangle {
Point start;
int angle;
int side0, side1;
} Rectangle;
if the given rectangles are rotated by any angle, my solution is to check the corners of each rectangle. If one of them is inside of another rectangle, these two rectangles overlap. Otherwise, they are not.
The problem is how to determine if a point is in a rectangle. The idea is to use orientation of a three point triplet. The idea to exam whether the sides of each rectangle are intersected. If not, There will be two cases, these two rectangle are isolated, or on is in another. Then, we have to check if any points of a rectangle is in another. Suppose that the rectangle, r, has four points, {a, b, c, d}. The given point is p. We can observe that,
- If p is inside of r, (a, b, p), (b, c, p), (c, d, p) and (d, a, p) will take the same orientations.
- Otherwise, at least one of them have a different orientation.
typedef struct point {
int x;
int y;
} Point;
typedef struct rectangle {
Point p[4];
} Rectangle;
// Find the orientation of ordered triplet(p, q, r).
// 0 - colinear
// 1 - clockwise
// 2 - counterclockwise
int orientation(Point *p, Point *q, Point *r)
{
int val = (q->y - p->y) * (r->x - q->x) -
(q->x - p->x) * (r->y - q->y);
if (0 == val)
return 0;
return val > 0 ? 1: 2;
}
// Given three colinear points p, q, r
// check if point q lines on line segment(p, r)
// 0 - Not on the segment
// 1 - On the segment
int on_segment(Point *p, Point *q, Point *r)
{
if (q->x <= max(p->x, r->x) && q->x >= min(p->x, r->x) &&
q->y <= max(p->y, r->y) && q->y >= min(p->y, r->y))
return 1;
return 0;
}
// Exam if the line segment (p1, q1) and (p2, q2) are intersected
// 0 - Not intersected
// 1 - Intersected
int is_line_intersected(Point *p1, Point *q1, Point *p2, Point *q2)
{
int o1 = orientation(p1, q1, p2);
int o2 = orientation(p1, q1, q2);
int o3 = orientation(p2, q2, p1);
int o4 = orientation(p2, q2, q1);
if (o1 != o2 && o3 != o4)
return 1;
if (o1 == 0 && on_segment(p2, p2, q1))
return 1;
if (o2 == 0 && on_segment(p1, q2, q1))
return 1;
if (o3 == 0 && on_segment(p2, p1, q2))
return 1;
if (o4 == 0 && on_segment(p2, q1, q2))
return 1;
return 0;
}
// Exam if a point is in a rectangle
// 0 - Out
// 1 - In
int is_point_in_rectangle(Point *p, Rectangle *rec)
{
Point *rp = rec->p;
// orientaion
int o = orientation(rp + 0, rp + 1, p);
if (o == 0)
return 1;
return o == orientation(rp + 1, rp + 2, p) &&
o == orientation(rp + 2, rp + 3, p) &&
o == orientation(rp + 3, rp + 0, p);
}
// Exam if two rectangles are intersected
// 0 - Not intersected
// 1 - Intersected
int is_rectangle_intersected(Rectangle *rec0, Rectangle *rec1)
{
Point *p0 = rec0->p;
Point *p1 = rec1->p;
int i, j;
Point c0, c1;
// check sides of each rectangle
// if two are intersected, the rectagle should intersected
for (i = 0; i < 4; ++i) {
for (j = 0; j, j < 4; ++j) {
if (is_line_intersected(p0 + i, p0 + (i + 1) % 4, p1 + j, p1 + (j + 1) % 4)) {
return 1;
}
}
}
// check if one point of rectangle 0 is in rectangle 1
// or one point of rectangle 1 is in rectangle 0
if (is_point_in_rectangle(p1, rec0) ||
is_point_in_rectangle(p0, rec1))
return 1;
return 0;
}
If the Edges are parallel to the coordinate axes, it is not that complicated, check this article.
http://articles.leetcode.com/2011/05/determine-if-two-rectangles-overlap.html
http://vijayt.com/Post/QuickSelect-works-in-linear-time
https://github.com/jaak-s/extraTrees/blob/master/src/main/java/org/extratrees/QuickSelect.java
Read full article from Determine if Two Rotated Rectangles Overlap Each Other
