Let us consider a 2-D (latitude, longitude) point set. The points in the set are equispaced (approx.) with mutual distance 0.1 degree \times 0.1 degree . Each point in the set is the centre of a square grid of side length 0.1 degree (i.e., intersect point of two diagonals of the square). Each square is adjacent to the neighbouring squares.
Our goal is to get the coordinates of the outline polygon formed by the bounding sides of the square grids with given direction (will be illustrated with a figure). This polygon has no hole inside.
Let us consider a sample data set of size 10 (point set).
lat_x <- c(21.00749, 21.02675, 21.00396, 21.04602, 21.02317,
21.06524, 21.00008, 21.04247, 21.08454, 21.0192)
and
lon_y <- c(88.21993, 88.25369, 88.31292, 88.28740, 88.34669,
88.32118, 88.40608, 88.38045, 88.35494, 88.43984)
Here is the rough plot of the above points followed by some illustration,
The black points are the (lat,lon) points in the above sample.
The blue square boxes are the square grid.
The given directions (\theta) of the squares are $theta$=50 degree.
Our goal is to get the ordered (clockwise or counter clockwise) co-ordinates of the outline polygon (in yellow colour).
Note: This question is very similar to this with a nice answer given by #laune. There the goal is to get the outline polygon without direction (or 0 degree direction). But in the the present set up I need to include the direction (non-zero) while drawing the square grids and the resulted polygon.
I would gratefully appreciate any suggestion, java or R codes or helpful reference given by anyone solving the above problem.
I would do it like this:
may be some 2D array point grouping to match the grid
that should speed up all the following operations.
compute average grid sizes (img1 from left)
as two vectors
create blue points (img2)
as: gray_point (+/-) 0.5*blue_vector
create red points (img3)
as: blue_point (+/-) 0.5*red_vector
create list of gray lines (img4)
take all 2 original (gray) points that have distance close to average grid distance and add line for them
create list of red lines (img4)
take all 2 original (gray) points that have distance close to average grid distance and add line for them if it is not intersecting any line from gray lines
reorder line points to match polygon winding ...
angle
compute angle of red vector (via atan2)
compute angle of blue vector (via atan2)
return the one with smaller absolute value
[edit1] response to comments
grid size
find few points that are closest to each other so pick any point and find all closest points to it. The possible distances should be near:
sqrt(1.0)*d,sqrt(1+1)*d,sqrt(1+2)*d,sqrt(2+2)*d,...
where d is the grid size so compute d for few picked points. Remember the first smallest d found and throw away all that are not similar to smallest one. Make average of them and let call it d
grid vectors
Take any point A and find closest point B to it with distance near d. For example +/-10% comparison: |(|A-B|-d)|<=0.1*d Now the grid vector is (B-A). Find few of them (picking different A,B) and group them by sign of x,y coordinates into 4 groups.
Then join negative direction groups together by negating one group vectors so you will have 2 list of vectors (red,blue direction) and make average vectors from them (red,blue vectors)
shifting points
You take any point A and add or substract half of red or blue vector to it (not its size!!!) for example:
A.x+=0.5*red_vector.x;
A.y+=0.5*red_vector.y;
line lists
Make 2 nested fors per each 2 point combination A,B (original for gray lines,shifted red ones for red outline lines) and add condition for distance
|(|A-B|-d)|<=0.1*d
if it is true add line (A,B) to the list. Here pseudo C++ example:
int i,j,N=?; // N is number of input points in pnt[]
double x,y,d=?,dd=d*d,de=0.1*d; // d is the avg grid size
double pnt[N][2]=?; // your 2D points
for (i=0;i<N;i++) // i - all points
for (j=i+1;j<N;j++) // j - just the rest no need to test already tested combinations
{
x=pnt[i][0]-pnt[j][0];
y=pnt[i][1]-pnt[j][1];
if (fabs((x*x)+(y*y)-dd)<=de) ... // add line pnt[i],pnt[j] to the list...
}
Related
I am creating vertex array for a mesh with given points.So far I was able to create a continuous mesh with thickness.However there is a problem at the intersection of two line segments, the vectors between those segment's sizes needs to be bigger or smaller depending on the situation in order to have a continuous look.
What I have now:
With the given angles theta1 and theta2, how can I calculate the length of red vectors?
What I want:
How I structured my mesh:
You're probably making it more complicated than it needs to be.
Let's start by calculating the red arrows. For any line segment (p_i, p_j), we can calculate the segment's normal with:
dir = normalize(p_j - p_i)
normal = (-dir.y, dir.x) //negate if you want the other direction
At the connection point between two segments, we can just average (and re-normalize) the incident normals. This gives us the red arrows.
The only question that remains is how much you need to shift. The resulting offset for the line segment o_l given an offset of the vertex o_v is:
o_l = o_v * dot(normal_l, normal_v)
This means the following: Both normals are unit vectors. Hence, their dot product is at most one. This is the case when both line segments are parallel. Then, the entire offset of the vertex is transferred to the line. The smaller the angle becomes, the smaller becomes the transferred offset. E.g. if the angle between two consecutive line segments is 120°, then the dot product of normals is 0.5. If you shift the vertex by 1 unit along its normal, both line segment will have a thickness of 0.5.
So, in order to produce a specific line thickness (o_l), we need to shift the vertex by o_v:
o_v = o_l / dot(normal_l, normal_v)
The construction with averaging the line segments' normal for the vertex normal ensures that dot(normal_l1, normal_ v) = dot(normal_l2, normal_v), i.e. the resulting line thickness is equal for both lines in any case.
Does anybody know how to find out if a set of coordinates are within a triangle for which you have the coordinates for. i know how to work out length of sides, area and perimeter, but i have no idea where to begin working out the whereabouts within the triangle of other points.
Any advice would be appreciated
You can create a Polygon object.
Polygon triangle = new Polygon();
Add the vertexes of your triangle with the addPoint(int x, int y) method.
And then, you just need to check if the set of coordinates is inside your triangle using contains(double x, double y) method.
Use the contains method of the Polygon class as documented here.
For a solution without using the Polygon-class:
Assume that you have giving three points A,B,C the vertices of your polygon. Let P be the point you want to check. First calculate the vectors representing the edges of your triangle. Let us call them AB, BC, CA. Also calculate the three vectors PA, PB, PC.
Now calculate the cross product between the first two of the vectors from above.
The cross product of the first pair gives you the sin(alpha), where alpha is the angle between AB and PA, multiplied with a vector pendenpicular to AB and PA. Ignore this vector because we are interested in the angle and take a look at the sine (in the case of 2D vectors you can imagine it as the vector standing perpendicular to your screen).
The sine can take values between (let's say for the ease) betwenn 0 and 2*Pi. It's 0 exactly at 0 and Pi. For every value in between the sine is positive and for every value between Pi and 2*Pi it's negative.
So let's say your Point p is on the left hand side of AB, so the sine would be positive.
By taking the cross product of each pair from above, you could easily guess that the point P is on the left hand side of each edge from the triangle. This just means that it has to be inside the triangle.
Of course this method can even be used from calculating whether a point P is in a polygon. Be aware of the fact, that this method only works if the sides of the polygon are directed.
I have a series of points, which represent mobile devices within a room. Previously I have systematically emitted a ping from each and recorded the time at which it arrives at the others to calculate the distances.
Here's a simple diagram of an example network.
The bottom A node should have been a D instead
After recording the distances I have the distance information in hashes.
A = {B: 2, C: 1, D: 3}
B = {A: 2, C: 2, D: 2}
C = {A: 1, B: 2, D: 2}
D = {A: 3, B: 2, C: 2}
My maths is rusty, but I feel like I should be able to then draw circles using these values as the respective and then intersect the circles to calculate a relative graph of the nodes.
Every time I try to do it I start out with a series of circles drawn around the root node (in this case A) that looks something like this:
I know that the other nodes must lie on the lines that I have drawn around A, but without being able to position them, how do you draw their distances so that you may intersect the circles and create the graph?
Start with any one point say A. Now take the second point say B, and plot it somewhere on the circle with the center at A and radius as distance between A and B. Now take another point C.
Let the distance (A,C)=x and (B,C)=y. Find the point of intersection of the circles (A,x) and (B,y). Mark it as C.
where circle (P,q) specifies center at P and radius q.
If no such point exists, then the given data is incorrect.
Now take the 4th point and similarly find the point of intersection of circles with centers at first three points and radii as distance between 4th and other three points respectively. Apply this method until all the points are plotted.
Note that there can be infinitely many solutions as RobH pointed out. Since you need only a virtual representation, I guess anyone of the valid solutions suffices.
The above algorithm has an order of O(N^2). It may be inefficient if the number of points are greater than 10000.
Also note that, to find the point of intersection of k circles, you first need to find the point of intersection for any two circles and validate these points on the remaining circles. This is because k circles can at most intersect at two points, assuming all have distinct centers.
EDIT: At any stage if there are two valid plots for a point, we can choose anyone of them and yet we arrive at one of the valid solutions.
I'm trying to write a 2D game in Java that uses the Separating Axis Theorem for collision detection. In order to resolve collisions between two polygons, I need to know the Minimum Translation Vector of the collision, and I need to know which direction it points relative to the polygons (so that I can give one polygon a penalty force along that direction and the other a penalty force in the opposite direction). For reference, I'm trying to implement the algorithm here.
I'd like to guarantee that if I call my collision detection function collide(Polygon polygon1, Polygon polygon2) and it detects a collision, the returned MTV will always point away from polygon1, toward polygon2. In order to do this, I need to guarantee that the separating axes that I generate, which are the normals of the polygon edges, always point away from the polygon that generated them. (That way, I know to negate any axis from polygon2 before using it as the MTV).
Unfortunately, it seems that whether or not the normal I generate for a polygon edge points towards the interior of the polygon or the exterior depends on whether the polygon's points are declared in clockwise or counterclockwise order. I'm using the algorithm described here to generate normals, and assuming that I pick (x, y) => (y, -x) for the "perpendicular" method, the resulting normals will only point away from the polygon if I iterate over the vertices in clockwise order.
Given that I can't force the client to declare the points of the polygon in clockwise order (I'm using java.awt.Polygon, which just exposes two arrays for x and y coordinates), is there a mathematical way to guarantee that the direction of the normal vectors I generate is toward the exterior of the polygon? I'm not very good at vector math, so there may be an obvious solution to this that I'm missing. Most Internet resources about the SAT just assume that you can always iterate over the vertices of a polygon in clockwise order.
You can just calculate which direction each polygon is ordered, using, for example, the answer to this question, and then multiply your normal by -1 if the two polygons have different orders.
You could also check each polygon passed to your algorithm to see if it is ordered incorrectly, again using the algorithm above, and reverse the vertex order if necessary.
Note that when calculating the vertex order, some algorithms will work for all polygons and some just for convex polygons.
I finally figured it out, but the one answer posted was not the complete solution so I'm not going to accept it. I was able to determine the ordering of the polygon using the basic algorithm described in this SO answer (also described less clearly in David Norman's link), which is:
for each edge in polygon:
sum += (x2 - x1) * (y2 + y1)
However, there's an important caveat which none of these answers mention. Normally, you can decide that the polygon's vertices are clockwise if this sum is positive, and counterclockwise if the sum is negative. But the comparison is inverted in Java's 2D graphics system, and in fact in many graphics systems, because the positive y axis points downward. So in a normal, mathematical coordinate system, you can say
if sum > 0 then polygon is clockwise
but in a graphics coordinate system with an inverted y-axis, it's actually
if sum < 0 then polygon is clockwise
My actual code, using Java's Polygon, looked something like this:
//First, find the normals as if the polygon was clockwise
int sum = 0;
for(int i = 0; i < polygon.npoints; i++) {
int nextI = (i + 1 == polygon.npoints ? 0 : i + 1);
sum += (polygon.xpoints[nextI] - polygon.xpoints[i]) *
(polygon.ypoints[nextI] + polygon.ypoints[i]);
}
if(sum > 0) {
//reverse all the normals (multiply them by -1)
}
I have a small contest problem in which is given a set of points, in 2D, that form a triangle. This triangle may be subject to an arbitrary rotation, may be subject to an arbitrary translation (both in the 2D plane) and may be subject to a reflection on a mirror, but its dimensions were kept unchanged.
Then, they give me a set of points in the plane, and I have to find 3 points that form my triangle after one or more of those geometric operations.
Example:
5 15
8 5
20 10
6
5 17
5 20
20 5
10 5
15 20
15 10
Output:
5 17
10 5
15 20
I bet it's supposed to apply some known algorithm, but I don't know which. The most common are: convex hull, sweep plane, triangulation, etc.
Can someone give a tip? I don't need the code, only a push, please!
A triangle is uniquely defined (ignoring rotations, flips, and translations) by the lengths of its three sides. Label the vertices of your original triangle A,B,C. You're looking
for points D,E,F such that |AB| = |DE|, |AC| = |DF|, and |BC| = |EF|. The length is given by Pythagoras' formula (but you can save a square root operation at each test by comparing
the squares of the line segment lengths...)
The given triangle is defined by three lengths. You want to find three points in the list separated by exactly those lengths.
Square the given lengths to avoid bothering with sqrt.
Find the square of the distance between every pair of points in the list and only note those that coincide with the given lengths: O(V^2), but with a low coefficient because most lengths will not match.
Now you have a sparse graph with O(V) edges. Find every cycle of size 3 in O(V) time, and prune the matches. (Not sure of the best way, but here is one way with proper big-O.)
Total complexity: O(V^2) but finding the cycles in O(V) may be the limiting factor, depending on the number of points. Spatially sorting the list of points to avoid looking at all pairs should improve the asymptotic behavior, otherwise.
This is generally done with matrix math. This Wikipedia article covers the rotation, translation, and reflection matrices. Here's another site (with pictures).
Since the transformations are just rotation, scaling and mirroring, then you can find the points that form the transformed triangle by checking the dot product of two sides of the triangle:
For the original triangle A, B, C, calculate the dot product of AB.AC, BA.BC and CA.CB
For each set of three points D, E, F, calculate dot product of DE.DF and compare against the three dot products found in 1.
This works since AB.AC = |AB| x |AC| x cos(a), and two lengths and an angle between them defines a triangle.
EDIT: Yes, Jim is right, just one dot product isn't enough, you'll need to do all three, including ED.EF and FD.FE. So in the end, there's the same number of calculations in this as there is in the square distances method.