I'm building a heatmap-like rectangular array interface and I want the 'hot' location to be at the top left of the array, and the 'cold' location to be at the bottom right. Therefore, I need an array to be filled diagonally like this:
0 1 2 3
|----|----|----|----|
0 | 0 | 2 | 5 | 8 |
|----|----|----|----|
1 | 1 | 4 | 7 | 10 |
|----|----|----|----|
2 | 3 | 6 | 9 | 11 |
|----|----|----|----|
So actually, I need a function f(x,y) such that
f(0,0) = 0
f(2,1) = 7
f(1,2) = 6
f(3,2) = 11
(or, of course, a similar function f(n) where f(7) = 10, f(9) = 6, etc.).
Finally, yes, I know this question is similar to the ones asked here, here and here, but the solutions described there only traverse and don't fill a matrix.
Interesting problem if you are limited to go through the array row by row.
I divided the rectangle in three regions. The top left triangle, the bottom right triangle and the rhomboid in the middle.
For the top left triangle the values in the first column (x=0) can be calculated using the common arithmetic series 1 + 2 + 3 + .. + n = n*(n+1)/2. Fields in the that triangle with the same x+y value are in the same diagonal and there value is that sum from the first colum + x.
The same approach works for the bottom right triangle. But instead of x and y, w-x and h-y is used, where w is the width and h the height of rectangle. That value have to be subtracted from the highest value w*h-1 in the array.
There are two cases for the rhomboid in the middle. If the width of rectangle is greater than (or equal to) the height, then the bottom left field of the rectangle is the field with the lowest value in the rhomboid and can be calculated that sum from before for h-1. From there on you can imagine that the rhomboid is a rectangle with a x-value of x+y and a y-value of y from the original rectangle. So calculations of the remaining values in that new rectangle are easy.
In the other case when the height is greater than the width, then the field at x=w-1 and y=0 can be calculated using that arithmetic sum and the rhomboid can be imagined as a rectangle with x-value x and y-value y-(w-x-1).
The code can be optimised by precalculating values for example. I think there also is one formula for all that cases. Maybe i think about it later.
inline static int diagonalvalue(int x, int y, int w, int h) {
if (h > x+y+1 && w > x+y+1) {
// top/left triangle
return ((x+y)*(x+y+1)/2) + x;
} else if (y+x >= h && y+x >= w) {
// bottom/right triangle
return w*h - (((w-x-1)+(h-y-1))*((w-x-1)+(h-y-1)+1)/2) - (w-x-1) - 1;
}
// rhomboid in the middle
if (w >= h) {
return (h*(h+1)/2) + ((x+y+1)-h)*h - y - 1;
}
return (w*(w+1)/2) + ((x+y)-w)*w + x;
}
for (y=0; y<h; y++) {
for (x=0; x<w; x++) {
array[x][y] = diagonalvalue(x,y,w,h);
}
}
Of course if there is not such a limitation, something like that should be way faster:
n = w*h;
x = 0;
y = 0;
for (i=0; i<n; i++) {
array[x][y] = i;
if (y <= 0 || x+1 >= w) {
y = x+y+1;
if (y >= h) {
x = (y-h)+1;
y -= x;
} else {
x = 0;
}
} else {
x++;
y--;
}
}
What about this (having an NxN matrix):
count = 1;
for( int k = 0; k < 2*N-1; ++k ) {
int max_i = std::min(k,N-1);
int min_i = std::max(0,k-N+1);
for( int i = max_i, j = min_i; i >= min_i; --i, ++j ) {
M.at(i).at(j) = count++;
}
}
Follow the steps in the 3rd example -- this gives the indexes (in order to print out the slices) -- and just set the value with an incrementing counter:
int x[3][3];
int n = 3;
int pos = 1;
for (int slice = 0; slice < 2 * n - 1; ++slice) {
int z = slice < n ? 0 : slice - n + 1;
for (int j = z; j <= slice - z; ++j)
x[j][slice - j] = pos++;
}
At a M*N matrix, the values, when traversing like in your stated example, seem to increase by n, except for border cases, so
f(0,0)=0
f(1,0)=f(0,0)+2
f(2,0)=f(1,0)+3
...and so on up to f(N,0). Then
f(0,1)=1
f(0,2)=3
and then
f(m,n)=f(m-1,n)+N, where m,n are index variables
and
f(M,N)=f(M-1,N)+2, where M,N are the last indexes of the matrix
This is not conclusive, but it should give you something to work with. Note, that you only need the value of the preceding element in each row and a few starting values to begin.
If you want a simple function, you could use a recursive definition.
H = height
def get_point(x,y)
if x == 0
if y == 0
return 0
else
return get_point(y-1,0)+1
end
else
return get_point(x-1,y) + H
end
end
This takes advantage of the fact that any value is H+the value of the item to its left. If the item is already at the leftmost column, then you find the cell that is to its far upper right diagonal, and move left from there, and add 1.
This is a good chance to use dynamic programming, and "cache" or memoize the functions you've already accomplished.
If you want something "strictly" done by f(n), you could use the relationship:
n = ( n % W , n / H ) [integer division, with no remainder/decimal]
And work your function from there.
Alternatively, if you want a purely array-populating-by-rows method, with no recursion, you could follow these rules:
If you are on the first cell of the row, "remember" the item in the cell (R-1) (where R is your current row) of the first row, and add 1 to it.
Otherwise, simply add H to the cell you last computed (ie, the cell to your left).
Psuedo-Code: (Assuming array is indexed by arr[row,column])
arr[0,0] = 0
for R from 0 to H
if R > 0
arr[R,0] = arr[0,R-1] + 1
end
for C from 1 to W
arr[R,C] = arr[R,C-1]
end
end
I have a 2D matrix stored in a flat buffer along diagonals. For example a 4x4 matrix would have its indexes scattered like so:
0 2 5 9
1 4 8 12
3 7 11 14
6 10 13 15
With this representation, what is the most efficient way to calculate the index of a neighboring element given the original index and a X/Y offset? For example:
// return the index of a neighbor given an offset
int getNGonalNeighbor(const size_t index,
const int x_offset,
const int y_offset){
//...
}
// for the array above:
getNGonalNeighbor(15,-1,-1); // should return 11
getNGonalNeighbor(15, 0,-1); // should return 14
getNGonalNeighbor(15,-1, 0); // should return 13
getNGonalNeighbor(11,-2,-1); // should return 1
We assume here that overflow never occurs and there is no wrap-around.
I have a solution involving a lot of triangular number and triangular root calculations. It also contains a lot of branches, which I would prefer to replace with algebra if possible (this will run on GPUs where diverging control flow is expensive). My solution is working but very lengthy. I feel like there must be a much simpler and less compute intensive way of doing it.
Maybe it would help me if someone can put a name on this particular problem/representation.
I can post my full solution if anyone is interested, but as I said it is very long and relatively complicated for such a simple task. In a nutshell, my solution does:
translate the original index into a larger triangular matrix to avoid dealing with 2 triangles (for example 13 would become 17)
For the 4x4 matrix this would be:
0 2 5 9 14 20 27
1 4 8 13 19 26
3 7 12 18 25
6 11 17 24
10 16 23
15 22
21
calculate the index of the diagonal of the neighbor in this representation using the manhattan distance of the offset and the triangular root of the index.
calculate the position of the neighbor in this diagonal using the offset
translate back to the original representation by removing the padding.
For some reason this is the simplest solution i could come up with.
Edit:
having loop to accumulate the offset:
I realize that given the properties of the triangle numbers, it would be easier to split up the matrix in two triangles (let's call 0 to 9 'upper triangle' and 10 to 15 'lower triangle') and have a loop with a test inside to accumulate the offset by adding one while in the upper triangle and subtracting one in the lower (if that makes sense). But for my solution loops must be avoided at all cost, especially loops with unbalanced trip counts (again, very bad for GPUs).
So I am looking more for an algebraic solution rather than an algorithmic one.
Building a lookup table:
Again, because of the GPU, it is preferable to avoid building a lookup table and have random accesses in it (very expensive). An algebraic solution is preferable.
Properties of the matrix:
The size of the matrix is known.
For now I only consider square matrix, but a solution for rectangular ones as well would be nice.
as the name of the function in my example suggests, extending the solution to N-dimensional volumes (hence N-gonal flattening) would be a big plus too.
Table lookup
#include <stdio.h>
#define SIZE 16
#define SIDE 4 //sqrt(SIZE)
int table[SIZE];
int rtable[100];// {x,y| x<=99, y<=99 }
void setup(){
int i, x, y, xy, index;//xy = x + y
x=y=xy=0;
for(i=0;i<SIZE;++i){
table[i]= index= x*10 + y;
rtable[x*10+y]=i;
x = x + 1; y = y - 1;//right up
if(y < 0 || x >= SIDE){
++xy;
x = 0;
y = xy;;
while(y>=SIDE){
++x;
--y;
}
}
}
}
int getNGonalNeighbor(int index, int offsetX, int offsetY){
int x,y;
x=table[index] / 10 + offsetX;
y=table[index] % 10 + offsetY;
if(x < 0 || x >= SIDE || y < 0 || y >= SIDE) return -1; //ERROR
return rtable[ x*10+y ];
}
int main() {
int i;
setup();
printf("%d\n", getNGonalNeighbor(15,-1,-1));
printf("%d\n", getNGonalNeighbor(15, 0,-1));
printf("%d\n", getNGonalNeighbor(15,-1, 0));
printf("%d\n", getNGonalNeighbor(11,-2,-1));
printf("%d\n", getNGonalNeighbor(0, -1,-1));
return 0;
}
don't use table version.
#include <stdio.h>
#define SIZE 16
#define SIDE 4
void num2xy(int index, int *offsetX, int *offsetY){
int i, x, y, xy;//xy = x + y
x=y=xy=0;
for(i=0;i<SIZE;++i){
if(i == index){
*offsetX = x;
*offsetY = y;
return;
}
x = x + 1; y = y - 1;//right up
if(y < 0 || x >= SIDE){
++xy;
x = 0;
y = xy;;
while(y>=SIDE){
++x;
--y;
}
}
}
}
int xy2num(int offsetX, int offsetY){
int i, x, y, xy, index;//xy = x + y
x=y=xy=0;
for(i=0;i<SIZE;++i){
if(offsetX == x && offsetY == y) return i;
x = x + 1; y = y - 1;//right up
if(y < 0 || x >= SIDE){
++xy;
x = 0;
y = xy;;
while(y>=SIDE){
++x;
--y;
}
}
}
return -1;
}
int getNGonalNeighbor(int index, int offsetX, int offsetY){
int x,y;
num2xy(index, &x, &y);
return xy2num(x + offsetX, y + offsetY);
}
int main() {
printf("%d\n", getNGonalNeighbor(15,-1,-1));
printf("%d\n", getNGonalNeighbor(15, 0,-1));
printf("%d\n", getNGonalNeighbor(15,-1, 0));
printf("%d\n", getNGonalNeighbor(11,-2,-1));
printf("%d\n", getNGonalNeighbor(0, -1,-1));
return 0;
}
I actually already had the elements to solve it somewhere else in my code. As BLUEPIXY's solution hinted, I am using scatter/gather operations, which I had already implemented for layout transformation.
This solution basically rebuilds the original (x,y) index of the given element in the matrix, applies the index offset and translates the result back to the transformed layout. It splits the square in 2 triangles and adjust the computation depending on which triangle it belongs to.
It is an almost entirely algebraic transformation: it uses no loop and no table lookup, has a small memory footprint and little branching. The code can probably be optimized further.
Here is the draft of the code:
#include <stdio.h>
#include <math.h>
// size of the matrix
#define SIZE 4
// triangle number of X
#define TRIG(X) (((X) * ((X) + 1)) >> 1)
// triangle root of X
#define TRIROOT(X) ((int)(sqrt(8*(X)+1)-1)>>1);
// return the index of a neighbor given an offset
int getNGonalNeighbor(const size_t index,
const int x_offset,
const int y_offset){
// compute largest upper triangle index
const size_t upper_triangle = TRIG(SIZE);
// position of the actual element of index
unsigned int x = 0,y = 0;
// adjust the index depending of upper/lower triangle.
const size_t adjusted_index = index < upper_triangle ?
index :
SIZE * SIZE - index - 1;
// compute triangular root
const size_t triroot = TRIROOT(adjusted_index);
const size_t trig = TRIG(triroot);
const size_t offset = adjusted_index - trig;
// upper triangle
if(index < upper_triangle){
x = offset;
y = triroot-offset;
}
// lower triangle
else {
x = SIZE - offset - 1;
y = SIZE - (trig + triroot + 1 - adjusted_index);
}
// adjust the offset
x += x_offset;
y += y_offset;
// manhattan distance
const size_t man_dist = x+y;
// calculate index using triangular number
return TRIG(man_dist) +
(man_dist >= SIZE ? x - (man_dist - SIZE + 1) : x) -
(man_dist > SIZE ? 2* TRIG(man_dist - SIZE) : 0);
}
int main(){
printf("%d\n", getNGonalNeighbor(15,-1,-1)); // should return 11
printf("%d\n", getNGonalNeighbor(15, 0,-1)); // should return 14
printf("%d\n", getNGonalNeighbor(15,-1, 0)); // should return 13
printf("%d\n", getNGonalNeighbor(11,-2,-1)); // should return 1
}
And the output is indeed:
11
14
13
1
If you think this solution looks over complicated and inefficient, I remind you that the target here is GPU, where computation costs virtually nothing compared to memory accesses, and all index computations are computed at the same time using massively parallel architectures.
So recursion is not my strong point, and I have been challenged to make a recursive floodFill function that fills a vector of a vector of ints with 1's if the value is zero. My code keeps segfaulting for reasons beyond me. Perhaps my code will make that sound more clear.
This is the grid to be flood filled:
vector<vector<int> > grid_;
It belongs to an object I created called "Grid" that is basically a set of functions to help manipulate the vectors. The grid's values are initialized to all zeros.
This is my flood fill function:
void floodFill(int x, int y, Grid & G)
{
if (G.getValue(x,y))
{
G.setValue(x,y,1);
if(x < G.getColumns()-1 && x >= 0 && y < G.getRows()-1 && y >= 0)
floodFill(x+1,y,G);
if(x < G.getColumns()-1 && x >= 0 && y < G.getRows()-1 && y >= 0)
floodFill(x,y+1,G);
if(x < G.getColumns()-1 && x >= 0 && y < G.getRows()-1 && y >= 0)
floodFill(x-1,y,G);
if(x < G.getColumns()-1 && x >= 0 && y < G.getRows()-1 && y >= 0)
floodFill(x,y-1,G);
}
}
The intention here is to have the function check if a point's value is zero, and if it is, change it to one. Then it should check the one above it for the same. It does this until it either finds a 1 or hits the end of the vector. Then it tries another direction and keeps going until the same conditions as above and so on and so forth until its flood filled.
Can anyone help me fix this? Maybe tell me whats wrong?
Thanks!
if(x < G.getColumns()-1 && x >= 0 && y < G.getRows()-1 && y >= 0)
floodFill(x-1,y,G);
won't work, since you can access index -1 of the underlying vector if x == 0
Same goes for floodFill(x,y-1,G);
This code has a lot of problems. First of all you check with if(G.getValue(x,y)) whether the value at a position is 1, and if so, then you set it to 1 with G.setValue(x,y,1). Think about this for a second, this can't be right. When will you ever set non-zero values to 1?
Then, another more subtle point is that you shouldn't do the recursion into neighbors if they are already set to 1.
As it stands the code you have will likely run until you overflow the stack because just going to recurse forever on the 1's that are connected to wherever you start from.
How about this?
void floodFill(int x, int y, Grid &g) {
if(x >= g.getColumns() || y >= g.getRows()) {
return;
}
floodFill(x+1, y, g);
if( x == 0 ) {
floodFill(x, y+1, g);
}
g.setValue(x, y, 1)
}
I think that will fill the grid without every hitting the same coordinate multiple times, and whenever either index is out of bounds it just returns so no chance of a seg fault.
I am trying to write a C++ program that takes the following inputs from the user to construct rectangles (between 2 and 5): height, width, x-pos, y-pos. All of these rectangles will exist parallel to the x and the y axis, that is all of their edges will have slopes of 0 or infinity.
I've tried to implement what is mentioned in this question but I am not having very much luck.
My current implementation does the following:
// Gets all the vertices for Rectangle 1 and stores them in an array -> arrRect1
// point 1 x: arrRect1[0], point 1 y: arrRect1[1] and so on...
// Gets all the vertices for Rectangle 2 and stores them in an array -> arrRect2
// rotated edge of point a, rect 1
int rot_x, rot_y;
rot_x = -arrRect1[3];
rot_y = arrRect1[2];
// point on rotated edge
int pnt_x, pnt_y;
pnt_x = arrRect1[2];
pnt_y = arrRect1[3];
// test point, a from rect 2
int tst_x, tst_y;
tst_x = arrRect2[0];
tst_y = arrRect2[1];
int value;
value = (rot_x * (tst_x - pnt_x)) + (rot_y * (tst_y - pnt_y));
cout << "Value: " << value;
However I'm not quite sure if (a) I've implemented the algorithm I linked to correctly, or if I did exactly how to interpret this?
Any suggestions?
if (RectA.Left < RectB.Right && RectA.Right > RectB.Left &&
RectA.Top > RectB.Bottom && RectA.Bottom < RectB.Top )
or, using Cartesian coordinates
(With X1 being left coord, X2 being right coord, increasing from left to right and Y1 being Top coord, and Y2 being Bottom coord, increasing from bottom to top -- if this is not how your coordinate system [e.g. most computers have the Y direction reversed], swap the comparisons below) ...
if (RectA.X1 < RectB.X2 && RectA.X2 > RectB.X1 &&
RectA.Y1 > RectB.Y2 && RectA.Y2 < RectB.Y1)
Say you have Rect A, and Rect B.
Proof is by contradiction. Any one of four conditions guarantees that no overlap can exist:
Cond1. If A's left edge is to the right of the B's right edge,
- then A is Totally to right Of B
Cond2. If A's right edge is to the left of the B's left edge,
- then A is Totally to left Of B
Cond3. If A's top edge is below B's bottom edge,
- then A is Totally below B
Cond4. If A's bottom edge is above B's top edge,
- then A is Totally above B
So condition for Non-Overlap is
NON-Overlap => Cond1 Or Cond2 Or Cond3 Or Cond4
Therefore, a sufficient condition for Overlap is the opposite.
Overlap => NOT (Cond1 Or Cond2 Or Cond3 Or Cond4)
De Morgan's law says
Not (A or B or C or D) is the same as Not A And Not B And Not C And Not D
so using De Morgan, we have
Not Cond1 And Not Cond2 And Not Cond3 And Not Cond4
This is equivalent to:
A's Left Edge to left of B's right edge, [RectA.Left < RectB.Right], and
A's right edge to right of B's left edge, [RectA.Right > RectB.Left], and
A's top above B's bottom, [RectA.Top > RectB.Bottom], and
A's bottom below B's Top [RectA.Bottom < RectB.Top]
Note 1: It is fairly obvious this same principle can be extended to any number of dimensions.
Note 2: It should also be fairly obvious to count overlaps of just one pixel, change the < and/or the > on that boundary to a <= or a >=.
Note 3: This answer, when utilizing Cartesian coordinates (X, Y) is based on standard algebraic Cartesian coordinates (x increases left to right, and Y increases bottom to top). Obviously, where a computer system might mechanize screen coordinates differently, (e.g., increasing Y from top to bottom, or X From right to left), the syntax will need to be adjusted accordingly/
struct rect
{
int x;
int y;
int width;
int height;
};
bool valueInRange(int value, int min, int max)
{ return (value >= min) && (value <= max); }
bool rectOverlap(rect A, rect B)
{
bool xOverlap = valueInRange(A.x, B.x, B.x + B.width) ||
valueInRange(B.x, A.x, A.x + A.width);
bool yOverlap = valueInRange(A.y, B.y, B.y + B.height) ||
valueInRange(B.y, A.y, A.y + A.height);
return xOverlap && yOverlap;
}
struct Rect
{
Rect(int x1, int x2, int y1, int y2)
: x1(x1), x2(x2), y1(y1), y2(y2)
{
assert(x1 < x2);
assert(y1 < y2);
}
int x1, x2, y1, y2;
};
bool
overlap(const Rect &r1, const Rect &r2)
{
// The rectangles don't overlap if
// one rectangle's minimum in some dimension
// is greater than the other's maximum in
// that dimension.
bool noOverlap = r1.x1 > r2.x2 ||
r2.x1 > r1.x2 ||
r1.y1 > r2.y2 ||
r2.y1 > r1.y2;
return !noOverlap;
}
It is easier to check if a rectangle is completly outside the other, so if it is either
on the left...
(r1.x + r1.width < r2.x)
or on the right...
(r1.x > r2.x + r2.width)
or on top...
(r1.y + r1.height < r2.y)
or on the bottom...
(r1.y > r2.y + r2.height)
of the second rectangle, it cannot possibly collide with it. So to have a function that returns a Boolean saying weather the rectangles collide, we simply combine the conditions by logical ORs and negate the result:
function checkOverlap(r1, r2) : Boolean
{
return !(r1.x + r1.width < r2.x || r1.y + r1.height < r2.y || r1.x > r2.x + r2.width || r1.y > r2.y + r2.height);
}
To already receive a positive result when touching only, we can change the "<" and ">" by "<=" and ">=".
This is a very fast way to check with C++ if two rectangles overlap:
return std::max(rectA.left, rectB.left) < std::min(rectA.right, rectB.right)
&& std::max(rectA.top, rectB.top) < std::min(rectA.bottom, rectB.bottom);
It works by calculating the left and right borders of the intersecting rectangle, and then comparing them: if the right border is equal to or less than the left border, it means that the intersection is empty and therefore the rectangles do not overlap; otherwise, it tries again with the top and bottom borders.
What is the advantage of this method over the conventional alternative of 4 comparisons? It's about how modern processors are designed. They have something called branch prediction, which works well when the result of a comparison is always the same, but have a huge performance penalty otherwise. However, in the absence of branch instructions, the CPU performs quite well. By calculating the borders of the intersection instead of having two separate checks for each axis, we're saving two branches, one per pair.
It is possible that the four comparisons method outperforms this one, if the first comparison has a high chance of being false. That is very rare, though, because it means that the second rectangle is most often on the left side of the first rectangle, and not on the right side or overlapping it; and most often, you need to check rectangles on both sides of the first one, which normally voids the advantages of branch prediction.
This method can be improved even more, depending on the expected distribution of rectangles:
If you expect the checked rectangles to be predominantly to the left or right of each other, then the method above works best. This is probably the case, for example, when you're using the rectangle intersection to check collisions for a game, where the game objects are predominantly distributed horizontally (e.g. a SuperMarioBros-like game).
If you expect the checked rectangles to be predominantly to the top or bottom of each other, e.g. in an Icy Tower type of game, then checking top/bottom first and left/right last will probably be faster:
return std::max(rectA.top, rectB.top) < std::min(rectA.bottom, rectB.bottom)
&& std::max(rectA.left, rectB.left) < std::min(rectA.right, rectB.right);
If the probability of intersecting is close to the probability of not intersecting, however, it's better to have a completely branchless alternative:
return std::max(rectA.left, rectB.left) < std::min(rectA.right, rectB.right)
& std::max(rectA.top, rectB.top) < std::min(rectA.bottom, rectB.bottom);
(Note the change of && to a single &)
Suppose that you have defined the positions and sizes of the rectangles like this:
My C++ implementation is like this:
class Vector2D
{
public:
Vector2D(int x, int y) : x(x), y(y) {}
~Vector2D(){}
int x, y;
};
bool DoRectanglesOverlap( const Vector2D & Pos1,
const Vector2D & Size1,
const Vector2D & Pos2,
const Vector2D & Size2)
{
if ((Pos1.x < Pos2.x + Size2.x) &&
(Pos1.y < Pos2.y + Size2.y) &&
(Pos2.x < Pos1.x + Size1.x) &&
(Pos2.y < Pos1.y + Size1.y))
{
return true;
}
return false;
}
An example function call according to the given figure above:
DoRectanglesOverlap(Vector2D(3, 7),
Vector2D(8, 5),
Vector2D(6, 4),
Vector2D(9, 4));
The comparisons inside the if block will look like below:
if ((Pos1.x < Pos2.x + Size2.x) &&
(Pos1.y < Pos2.y + Size2.y) &&
(Pos2.x < Pos1.x + Size1.x) &&
(Pos2.y < Pos1.y + Size1.y))
↓
if (( 3 < 6 + 9 ) &&
( 7 < 4 + 4 ) &&
( 6 < 3 + 8 ) &&
( 4 < 7 + 5 ))
Ask yourself the opposite question: How can I determine if two rectangles do not intersect at all? Obviously, a rectangle A completely to the left of rectangle B does not intersect. Also if A is completely to the right. And similarly if A is completely above B or completely below B. In any other case A and B intersect.
What follows may have bugs, but I am pretty confident about the algorithm:
struct Rectangle { int x; int y; int width; int height; };
bool is_left_of(Rectangle const & a, Rectangle const & b) {
if (a.x + a.width <= b.x) return true;
return false;
}
bool is_right_of(Rectangle const & a, Rectangle const & b) {
return is_left_of(b, a);
}
bool not_intersect( Rectangle const & a, Rectangle const & b) {
if (is_left_of(a, b)) return true;
if (is_right_of(a, b)) return true;
// Do the same for top/bottom...
}
bool intersect(Rectangle const & a, Rectangle const & b) {
return !not_intersect(a, b);
}
In the question, you link to the maths for when rectangles are at arbitrary angles of rotation. If I understand the bit about angles in the question however, I interpret that all rectangles are perpendicular to one another.
A general knowing the area of overlap formula is:
Using the example:
1 2 3 4 5 6
1 +---+---+
| |
2 + A +---+---+
| | B |
3 + + +---+---+
| | | | |
4 +---+---+---+---+ +
| |
5 + C +
| |
6 +---+---+
1) collect all the x coordinates (both left and right) into a list, then sort it and remove duplicates
1 3 4 5 6
2) collect all the y coordinates (both top and bottom) into a list, then sort it and remove duplicates
1 2 3 4 6
3) create a 2D array by number of gaps between the unique x coordinates * number of gaps between the unique y coordinates.
4 * 4
4) paint all the rectangles into this grid, incrementing the count of each cell it occurs over:
1 3 4 5 6
1 +---+
| 1 | 0 0 0
2 +---+---+---+
| 1 | 1 | 1 | 0
3 +---+---+---+---+
| 1 | 1 | 2 | 1 |
4 +---+---+---+---+
0 0 | 1 | 1 |
6 +---+---+
5) As you paint the rectangles, its easy to intercept the overlaps.
Here's how it's done in the Java API:
public boolean intersects(Rectangle r) {
int tw = this.width;
int th = this.height;
int rw = r.width;
int rh = r.height;
if (rw <= 0 || rh <= 0 || tw <= 0 || th <= 0) {
return false;
}
int tx = this.x;
int ty = this.y;
int rx = r.x;
int ry = r.y;
rw += rx;
rh += ry;
tw += tx;
th += ty;
// overflow || intersect
return ((rw < rx || rw > tx) &&
(rh < ry || rh > ty) &&
(tw < tx || tw > rx) &&
(th < ty || th > ry));
}
struct Rect
{
Rect(int x1, int x2, int y1, int y2)
: x1(x1), x2(x2), y1(y1), y2(y2)
{
assert(x1 < x2);
assert(y1 < y2);
}
int x1, x2, y1, y2;
};
//some area of the r1 overlaps r2
bool overlap(const Rect &r1, const Rect &r2)
{
return r1.x1 < r2.x2 && r2.x1 < r1.x2 &&
r1.y1 < r2.y2 && r2.x1 < r1.y2;
}
//either the rectangles overlap or the edges touch
bool touch(const Rect &r1, const Rect &r2)
{
return r1.x1 <= r2.x2 && r2.x1 <= r1.x2 &&
r1.y1 <= r2.y2 && r2.x1 <= r1.y2;
}
Don't think of coordinates as indicating where pixels are. Think of them as being between the pixels. That way, the area of a 2x2 rectangle should be 4, not 9.
bool bOverlap = !((A.Left >= B.Right || B.Left >= A.Right)
&& (A.Bottom >= B.Top || B.Bottom >= A.Top));
Easiest way is
/**
* Check if two rectangles collide
* x_1, y_1, width_1, and height_1 define the boundaries of the first rectangle
* x_2, y_2, width_2, and height_2 define the boundaries of the second rectangle
*/
boolean rectangle_collision(float x_1, float y_1, float width_1, float height_1, float x_2, float y_2, float width_2, float height_2)
{
return !(x_1 > x_2+width_2 || x_1+width_1 < x_2 || y_1 > y_2+height_2 || y_1+height_1 < y_2);
}
first of all put it in to your mind that in computers the coordinates system is upside down. x-axis is same as in mathematics but y-axis increases downwards and decrease on going upward..
if rectangle are drawn from center.
if x1 coordinates is greater than x2 plus its its half of widht. then it means going half they will touch each other. and in the same manner going downward + half of its height. it will collide..
For those of you who are using center points and half sizes for their rectangle data, instead of the typical x,y,w,h, or x0,y0,x1,x1, here's how you can do it:
#include <cmath> // for fabsf(float)
struct Rectangle
{
float centerX, centerY, halfWidth, halfHeight;
};
bool isRectangleOverlapping(const Rectangle &a, const Rectangle &b)
{
return (fabsf(a.centerX - b.centerX) <= (a.halfWidth + b.halfWidth)) &&
(fabsf(a.centerY - b.centerY) <= (a.halfHeight + b.halfHeight));
}
Lets say the two rectangles are rectangle A and rectangle B. Let their centers be A1 and B1 (coordinates of A1 and B1 can be easily found out), let the heights be Ha and Hb, width be Wa and Wb, let dx be the width(x) distance between A1 and B1 and dy be the height(y) distance between A1 and B1.
Now we can say we can say A and B overlap: when
if(!(dx > Wa+Wb)||!(dy > Ha+Hb)) returns true
If the rectangles overlap then the overlap area will be greater than zero. Now let us find the overlap area:
If they overlap then the left edge of overlap-rect will be the max(r1.x1, r2.x1) and right edge will be min(r1.x2, r2.x2). So the length of the overlap will be min(r1.x2, r2.x2) - max(r1.x1, r2.x1)
So the area will be:
area = (max(r1.x1, r2.x1) - min(r1.x2, r2.x2)) * (max(r1.y1, r2.y1) - min(r1.y2, r2.y2))
If area = 0 then they don't overlap.
Simple isn't it?
I have implemented a C# version, it is easily converted to C++.
public bool Intersects ( Rectangle rect )
{
float ulx = Math.Max ( x, rect.x );
float uly = Math.Max ( y, rect.y );
float lrx = Math.Min ( x + width, rect.x + rect.width );
float lry = Math.Min ( y + height, rect.y + rect.height );
return ulx <= lrx && uly <= lry;
}
I have a very easy solution
let x1,y1 x2,y2 ,l1,b1,l2,be cordinates and lengths and breadths of them respectively
consider the condition ((x2
now the only way these rectangle will overlap is if the point diagonal to x1,y1 will lie inside the other rectangle or similarly the point diagonal to x2,y2 will lie inside the other rectangle. which is exactly the above condition implies.
A and B be two rectangle. C be their covering rectangle.
four points of A be (xAleft,yAtop),(xAleft,yAbottom),(xAright,yAtop),(xAright,yAbottom)
four points of A be (xBleft,yBtop),(xBleft,yBbottom),(xBright,yBtop),(xBright,yBbottom)
A.width = abs(xAleft-xAright);
A.height = abs(yAleft-yAright);
B.width = abs(xBleft-xBright);
B.height = abs(yBleft-yBright);
C.width = max(xAleft,xAright,xBleft,xBright)-min(xAleft,xAright,xBleft,xBright);
C.height = max(yAtop,yAbottom,yBtop,yBbottom)-min(yAtop,yAbottom,yBtop,yBbottom);
A and B does not overlap if
(C.width >= A.width + B.width )
OR
(C.height >= A.height + B.height)
It takes care all possible cases.
This is from exercise 3.28 from the book Introduction to Java Programming- Comprehensive Edition. The code tests whether the two rectangles are indenticle, whether one is inside the other and whether one is outside the other. If none of these condition are met then the two overlap.
**3.28 (Geometry: two rectangles) Write a program that prompts the user to enter the
center x-, y-coordinates, width, and height of two rectangles and determines
whether the second rectangle is inside the first or overlaps with the first, as shown
in Figure 3.9. Test your program to cover all cases.
Here are the sample runs:
Enter r1's center x-, y-coordinates, width, and height: 2.5 4 2.5 43
Enter r2's center x-, y-coordinates, width, and height: 1.5 5 0.5 3
r2 is inside r1
Enter r1's center x-, y-coordinates, width, and height: 1 2 3 5.5
Enter r2's center x-, y-coordinates, width, and height: 3 4 4.5 5
r2 overlaps r1
Enter r1's center x-, y-coordinates, width, and height: 1 2 3 3
Enter r2's center x-, y-coordinates, width, and height: 40 45 3 2
r2 does not overlap r1
import java.util.Scanner;
public class ProgrammingEx3_28 {
public static void main(String[] args) {
Scanner input = new Scanner(System.in);
System.out
.print("Enter r1's center x-, y-coordinates, width, and height:");
double x1 = input.nextDouble();
double y1 = input.nextDouble();
double w1 = input.nextDouble();
double h1 = input.nextDouble();
w1 = w1 / 2;
h1 = h1 / 2;
System.out
.print("Enter r2's center x-, y-coordinates, width, and height:");
double x2 = input.nextDouble();
double y2 = input.nextDouble();
double w2 = input.nextDouble();
double h2 = input.nextDouble();
w2 = w2 / 2;
h2 = h2 / 2;
// Calculating range of r1 and r2
double x1max = x1 + w1;
double y1max = y1 + h1;
double x1min = x1 - w1;
double y1min = y1 - h1;
double x2max = x2 + w2;
double y2max = y2 + h2;
double x2min = x2 - w2;
double y2min = y2 - h2;
if (x1max == x2max && x1min == x2min && y1max == y2max
&& y1min == y2min) {
// Check if the two are identicle
System.out.print("r1 and r2 are indentical");
} else if (x1max <= x2max && x1min >= x2min && y1max <= y2max
&& y1min >= y2min) {
// Check if r1 is in r2
System.out.print("r1 is inside r2");
} else if (x2max <= x1max && x2min >= x1min && y2max <= y1max
&& y2min >= y1min) {
// Check if r2 is in r1
System.out.print("r2 is inside r1");
} else if (x1max < x2min || x1min > x2max || y1max < y2min
|| y2min > y1max) {
// Check if the two overlap
System.out.print("r2 does not overlaps r1");
} else {
System.out.print("r2 overlaps r1");
}
}
}
bool Square::IsOverlappig(Square &other)
{
bool result1 = other.x >= x && other.y >= y && other.x <= (x + width) && other.y <= (y + height); // other's top left falls within this area
bool result2 = other.x >= x && other.y <= y && other.x <= (x + width) && (other.y + other.height) <= (y + height); // other's bottom left falls within this area
bool result3 = other.x <= x && other.y >= y && (other.x + other.width) <= (x + width) && other.y <= (y + height); // other's top right falls within this area
bool result4 = other.x <= x && other.y <= y && (other.x + other.width) >= x && (other.y + other.height) >= y; // other's bottom right falls within this area
return result1 | result2 | result3 | result4;
}
struct point { int x, y; };
struct rect { point tl, br; }; // top left and bottom right points
// return true if rectangles overlap
bool overlap(const rect &a, const rect &b)
{
return a.tl.x <= b.br.x && a.br.x >= b.tl.x &&
a.tl.y >= b.br.y && a.br.y <= b.tl.y;
}