I have to form a Convex Hull from large amount of points and I came across this article. Whole process of pruning is described and well explained, except for one part.
I don't know what this part means and how to convert it to code:
Since the space is two-dimensional, each point has two coordinates, x > and y. Every time we read a new point, we compute the following 4 > points:
A = (Ax, Ay) which maximizes x-y
B = (Bx, Xy) which maximizes x+y
C = (Cx, Cy) which minimizes x-y
D = (Dx, Dy) which minimizes x+y
Can anyone help me to calculate points A,B,C,D?
You're not so much calculating the points, you're selecting them from your input data:
A is the point in your input data where the value of x-y is larger than any other input data point.
B is the point in your input data where the value of x+y is larger than any other input data point.
C is the point in your input data where the value of x-y is smaller than any other input data point.
D is the point in your input data where the value of x+y is smaller than any other input data point.
Related
So lets imagine that I have a grid of 10x10 (can be any size, but just for the sake of example say 10), and with that grid there is 3 points marking vertexes of a triangle (again can be any amount of points delimiting any arbitrary shape).
So my question is.. Is there a way given just this information to determine programatically if any given coordinate is within that shape?
Lest say the coordinates are 3,2-7,3-5,5. Can I while iterating over the given grid pick out the cells that fall within these points?
Call P the point that you are checking, and S1,S2,...,Sn the n vertices of the shape.
Assume that P ≠ Si for all i.
Is P on the boundary?
If 1 is no, then randomly choose a line L that passes through P
Pick a point F that you know is outside the polygon
Follow the sequence of intersections of L with the shape from F until you hit P (Call the Sequence F, ..., P)
Count the sequence F, ..., P, store the value in M
If M is even, then P is in the polygon, Otherwise P is not in the polygon
NOTE: By introducing the starting point F, we change the parity mentioned in the point in polygon algorithm description on wikipedia
Recently I have started doing some research on the SAT (Separating Axis Theorem) for collision detection in a game I am making. I understand how the algorithm works and why it works, what I'm puzzled about is how it expects one to be able to so easily calculate the projection of the shape onto different axes.
I assume the projection of a polygon onto a vector is represented by line segment from point A to point B, so my best guess to find points A and B would be to find the angle of the line being projected onto and calculate the min and max x-values of the coordinates when the shape is rotated to the angle of the projection (i.e. such that it is parallel to the x-axis and the min and max values are simply the min and max values along the x-axis). But to do this for every projection would be a costly operation. Do any of you guys know a better solution, or could at least point me to a paper or document where a better solution is described?
Simple way to calculate the projection of the polygon on line is to calculate projection of all vertex onto the line and get the coordinates with min-max values like you suggested but you dont need to rotate the polygon to do so.
Here is algorithm to find projection of point on line :-
line : y = mx + c
point : (x1,y1)
projection is intersection of line perpendicular to given line and passing through (x1,y1)
perdenicular line :- y-y1 = -1/m(x-x1) slope of perpendicular line is -1/m
y = -1/m(x-x1) + y1
To find point of intersection solve the equation simultaneously :-
y = mx + c , y = -1/m(x-x1) + y1
mx + c = -1/m(x-x1) + y1
m^2*x + mc = x1-x + my1
(m^2+1)x = x1 + my1 - mc
x = (x1-my1 - mc)/(m^2+1)
y = mx + c = m(x1-my1-mc)/(m^2+1) + c
Time complexity : For each vertex it takes O(1) time so it is O(V) where V is no of vertex in the polygon
If your polygon is not convex, compute its convex hull first.
Given a convex polygon with n vertices, you can find its rotated minimum and maximum x-coordinate in n log n by binary search. You can always test whether a vertex is a minimum or a maximum by rotating an comparing it and the two adjacent vertices. Depending on the results of the comparison, you know whether to jump clockwise or counterclockwise. Jump by k vertices, each time decreasing k by half (at the start k=n/2).
This may or may not bring real speed improvement. If your typical polygon has a dozen or so vertices, it may make little sense to use binary search.
Input:
A set of points
Coordinates are non-negative integer type.
Integer k
Output:
A point P(x, y) (in or not in the given set) whose manhattan distance to closest is maximal and max(x, y) <= k
My (naive) solution:
For every (x, y) in the grid which contain given set
BFS to find closest point to (x, y)
...
return maximum;
But I feel it run very slow for a large grid, please help me to design a better algorithm (or the code / peseudo code) to solve this problem.
Should I instead of loop over every (x, y) in grid, just need to loop every median x, y
P.S: Sorry for my English
EDIT:
example:
Given P1(x1,y1), P2(x2,y2), P3(x3,y3). Find P(x,y) such that min{dist(P,P1), dist(P,P2),
dist(P,P3)} is maximal
Yes, you can do it better. I'm not sure if my solution is optimal, but it's better than yours.
Instead of doing separate BFS for every point in the grid. Do a 'cumulative' BFS from all the input points at once.
You start with 2-dimensional array dist[k][k] with cells initialized to +inf and zero if there is a point in the input for this cell, then from every point P in the input you try to go in every possible direction. The further you are from the start point the bigger integer you put in the array dist. If there is a value in dist for a specific cell, but you can get there with a smaller amount of steps (smaller integer) you overwrite it.
In the end, when no more moves can be done, you scan the array dist to find the cell with maximum value. This is your point.
I think this would work quite well in practice.
For k = 3, assuming 1 <= x,y <= k, P1 = (1,1), P2 = (1,3), P3 = (2,2)
dist would be equal in the beginning
0, +inf, +inf,
+inf, 0, +inf,
0, +inf, +inf,
in the next step it would be:
0, 1, +inf,
1, 0, 1,
0, 1, +inf,
and in the next step it would be:
0, 1, 2,
1, 0, 1,
0, 1, 2,
so the output is P = (3,1) or (3,3)
If K is not large enough and you need to find a point with integer coordinates, you should do, as another answer suggested - Calculate minimum distances for all points on the grid, using BFS, strarting from all given points at once.
Faster solution, for large K, and probably the only one which can find a point with float coordinates, is as following. It has complexity of O(n log n log k)
Search for resulting maximum distance using dihotomy. You have to check if there is any point inside the square [0, k] X [0, k] which is at least given distance away from all points in given set. Suppose, you can check that fast enough for any distance. It is obvious, that if there is such point for some distance R, there always will be some point for all smaller distances r < R. For example, the same point would go. Thus you can search for maximum distance using binary search procedure.
Now, how to fast check for existence (and also find) a point which is at least r units away from all given points. You should draw "Manhattan spheres of radius r" around all given points. These are set of points at most r units away from given point. They are tilted by 45 degrees squares with diagonal equal to 2r. Now turn the picture by 45 degrees, and all squares will be parallel to the axis. Now you can check for existence of any point outside such squares using sweeping line algorithm. You have to sort all vertical edges of squares, and then process them one by one from left to right. Left borders will add segment mark to sweeping line, Left borders will erase it. And you have to check if there is any non marked point on the line. You can implement it using segment tree. Then, you have to check if there is any non marked point on the line inside the initial square [0,k]X[0,k].
So, again, overall solution will be binary search for r. Inside of it you will have to check if there is any point at least r units away from all given points. Do that by constructing "manhattans spheres of radius r" and then scanning them with a diagonal line from left-top corner to right-bottom. While moving line you should store number of opened spheres at each point at the line in the segment tree. between opening and closing of any spheres, line does not change, and if there is any free point there, it means, that you found it for distance r.
Binary search contributes log k to complexity. Each checking procedure is n log n for sorting squares borders, and n log k (n log n?) for processing them all.
Voronoi diagram would be another fast solution and could also find non integer answer. But it is much much harder to implement even for Manhattan measure.
First try
We can turn a 2D problem into a 1D problem by projecting onto the lines y=x and y=-x. If the points are (x1,y1) and (x2,y2) then the manhattan distance is abs(x1-x2)+abs(y1-y2). Change coordinate to a u-v system with basis U = (1,1), V = (1,-1). Coords of the two points in this basis are u1 = (x1-y1)/sqrt(2), v1= (x1+y1), u2= (x1-y1), v2 = (x1+y1). And the manhatten distance is the largest of abs(u1-u2), abs(v1-v2).
How this helps. We can just work with the 1D u-values of each points. Sort by u-value, loop through points and find the largest difference between pains of points. Do the same of v-values.
Calculating u,v coords of O(n), quick sorting is O(n log n), looping through sorted list is O(n).
Alas does not work well. Fails if we have point (-10,0), (10,0), (0,-10), (0,10). Lets try a
Voronoi diagram
Construct a Voronoi diagram
using Manhattan distance. This can be calculate in O(n log n) using https://en.wikipedia.org/wiki/Fortune%27s_algorithm
The vertices in the diagram are points which have maximum distance from its nearest vertices. There is psudo-code for the algorithm on the wikipedia page. You might need to adapt this for Manhattan distance.
I'm having a hard time following the ray-plane intersection described in the following page.
SIGGRAPH Ray-Plane Intersection
Here is my understanding.
The plane is described as Ax + By + Cz + D = 0
or
The Vector ( A, B, C, D ), Where A, B, C define a normal plan. If A, B, and C define a unit normal, then the distance from the origin [0, 0, 0] to the plan is D.
My question is shouldn't D be a vector? Since it represents the distants between two points. I guess I just don't understand how you can represent the distance between to points as a non vector.
Any help is much appreciated.
Distance between two points is ALWAYS a scalar, a single number. Think of the vectors as points in space, right? So, when you say distance between two vectors, you are finding the distance between those two points which is a number. Distance between two vectors is the magnitude of the difference vector of the two vectors. So, you subtract the 2 vectors, get the difference vector and find its magnitude. That is your distance which is a SCALAR and NOT a vector.
Distance is a scalar value, not a vector. It is, in fact, the length of a vector.
You can think of a vector as a set of values describing a point in space in relation to the origin. In R3, you need a minimum of 3 pieces of information to describe the location of that point. These pieces of information give you a direction and a distance.
If you were to tell me that a city is 50 miles away, that would be you describing a distance. Of course, you will not have told me which direction that city was. When you give me 2 pieces of information, you have given me a vector, as opposed to scalar value.
Also recall the formula for distance:
D = sqrt(x^2 + y^2 + z^2)
Scalar value ;)
Picture a canvas that has a bunch of points randomly dispersed around it. Now pick one of those points. How would you find the closest 3 points to it such that if you drew a triangle connecting those points it would cover the chosen point?
Clarification: By "closest", I mean minimum sum of distances to the point.
This is mostly out of curiosity. I thought it would be a good way to estimate the "value" of a point if it is unknown, but the surrounding points are known. With 3 surrounding points you could extrapolate the value. I haven't heard of a problem like this before, doesn't seem very trivial so I thought it might be a fun exercise, even if it's not the best way to estimate something.
Your problem description is ambiguous. Which triangle are you after in this figure, the red one or the blue one?
The blue triangle is closer based on lexicographic comparison of the distances of the points, while the red triangle is closer based on the sum of the distances of the points.
Edit: you clarified it to make it clear that you want the sum of distances to be minimized (the red triangle).
So, how about this sketch algorithm?
Assume that the chosen point is at the origin (makes description of algorithm easy).
Sort the points by distance from the origin: P(1) is closest, P(n) is farthest.
Start with i = 3, s = ∞.
For each triple of points P(a), P(b), P(i) with a < b < i, if the triangle contains the origin, let s = min(s, |P(a)| + |P(b)| + |P(i)|).
If s ≤ |P(1)| + |P(2)| + |P(i)|, stop.
If i = n, stop.
Otherwise, increment i and go back to step 4.
Obviously this is O(n³) in the worst case.
Here's a sketch of another algorithm. Consider all pairs of points (A, B). For a third point to make a triangle containing the origin, it must lie in the grey shaded region in this figure:
By representing the points in polar coordinates (r, θ) and sorting them according to θ, it is straightforward to examine all these points and pick the closest one to the origin.
This is also O(n³) in the worst case, but a sensible order of visiting pairs (A, B) should yield an early exit in many problem instances.
Just a warning on the iterative method. You may find a triangle with 3 "near points" whose "length" is greater than another resulting by adding a more distant point to the set. Sorry, can't post this as a comment.
See Graph.
Red triangle has perimeter near 4 R while the black one has 3 Sqrt[3] -> 5.2 R
Like #thejh suggests, sort your points by distance from the chosen point.
Starting with the first 3 points, look for a triangle covering the chosen point.
If no triangle is found, expand you range to include the next closest point, and try all combinations.
Once a triangle is found, you don't necessarily have the final answer. However, you have now limited the final set of points to check. The furthest possible point to check would be at a distance equal to the sum of the distances of the first triangle found. Any further than this, and the sum of the distances is guaranteed to exceed the first triangle that was found.
Increase your range of points to include the last point whose distance <= the sum of the distances of the first triangle found.
Now check all combinations, and the answer is the triangle found from this set with the minimal sum of distances.
second shot
subsolution: (analytic geometry basics, skip if you are familiar with this) finding point of the opposite half-plane
Example: Let's have two points: A=[a,b]=[2,3] and B=[c,d]=[4,1]. Find vector u = A-B = (2-4,3-1) = (-2,2). This vector is parallel to AB line, so is the vector (-1,1). The equation for this line is defined by vector u and point in AB (i.e. A):
X = 2 -1*t
Y = 3 +1*t
Where t is any real number. Get rid of t:
t = 2 - X
Y = 3 + t = 3 + (2 - X) = 5 - X
X + Y - 5 = 0
Any point that fits in this equation is in the line.
Now let's have another point to define the half-plane, i.e. C=[1,1], we get:
X + Y - 5 = 1 + 1 - 5 < 0
Any point with opposite non-equation sign is in another half-plane, which are these points:
X + Y - 5 > 0
solution: finding the minimum triangle that fits the point S
Find the closest point P as min(sqrt( (Xp - Xs)^2 + (Yp - Ys)^2 ))
Find perpendicular vector to SP as u = (-Yp+Ys,Xp-Xs)
Find two closest points A, B from the opposite half-plane to sigma = pP where p = Su (see subsolution), such as A is on the different site of line q = SP (see final part of the subsolution)
Now we have triangle ABP that covers S: calculate sum of distances |SP|+|SA|+|SB|
Find the second closest point to S and continue from 1. If the sum of distances is smaller than that in previous steps, remember it. Stop if |SP| is greater than the smallest sum of distances or no more points are available.
I hope this diagram makes it clear.
This is my first shot:
split the space into quadrants
with picked point at the [0,0]
coords
find the closest point
from each quadrant (so you have 4
points)
any triangle from these
points should be small enough (but not necesarilly the smallest)
Take the closest N=3 points. Check whether the triange fits. If not, increment N by one and try out all combinations. Do that until something fits or nothing does.