Is there a fast way to do this? Searching online shows convexity of functions or single polygons. But I need the ability to check this for the whole model. An object can have convex faces but can be concave as a whole like a torus.
Kneejerk: if you build a leafy BSP tree and end up with all your geometry at one node, the object is convex.
Slightly smarter way to approach the same solution: for each polygon, get the hyperplane. Make sure every vertex in the model is behind that hyperplane.
Equivalently: check the line segment between every pair of vertices; if it doesn't intersect any faces then the object is convex.
I guess you could also get the convex hull, via quickhull or whatever, and compare it to the original object. Or, similarly, get the convex hull and check that every vertex of the original object lies on a face of the hull.
For every face, compute the equation of the plane of support and check that all vertices* yield the same sign when plugged in the plane equation.
Will take time O(F.V), for F faces and V vertices.
*For safety, disregard the vertices of the face being processed.
Alternatively, compute the 3D convex hull, in time O(V.Log(V)). If at any stage in the algorithm a vertex gets discarded, then the polyhedron was not convex.
bool IsConvex(std::vector<vec3> &points, std::vector<int> &triangles, float threshold = 0.001)
{
for (unsigned long i = 0; i < triangles.size() / 3; i++)
{
vec3 Atmp = points[triangles[i * 3 + 0]];
vec3 Btmp = points[triangles[i * 3 + 1]];
vec3 Ctmp = points[triangles[i * 3 + 2]];
btVector3 A(Atmp.x, Atmp.y, Atmp.z);
btVector3 B(Btmp.x, Btmp.y, Btmp.z);
btVector3 C(Ctmp.x, Ctmp.y, Ctmp.z);
B -= A;
C -= A;
btVector3 BCNorm = B.cross(C).normalized();
float checkPoint = btVector3(points[0].x - A.x(), points[0].y - A.y(), points[0].z - A.z()).dot(BCNorm);
for (unsigned long j = 0; j < points.size(); j++)
{
float dist = btVector3(points[j].x - A.x(), points[j].y - A.y(), points[j].z - A.z()).dot(BCNorm);
if((std::abs(checkPoint) > threshold) && (std::abs(dist) > threshold) && (checkPoint * dist < 0))
{
return false;
}
}
}
return true;
}
trimesh is a Python library that can load a 3D mesh and evaluates if mesh is convex or not.
import trimesh
mesh = trimesh.load('my_mesh_file')
print(mesh.is_convex)
Code is given here.
It can be run from a command line with the following instructions:
python -m pip install trimesh
python -c "import trimesh; mesh = trimesh.load('my_mesh_file'); print(mesh.is_convex)"
You can accelerate the plane-vertex tests by adding all vertices to a tree structure first, so you can reject whole leaves if their bounds don't intersect the plane.
The BSP idea should actually be identical to testing all triangle planes, as no BSP leaf will be able to subdivide the set of vertices for a convex object.
You probably want to include an epsilon for your plane tests, as both floating point precision and modelling precision for manually created meshes can result in vertices slightly above a plane.
Related
I have a list of consecutive points and I need to find the coordinates of a polygon some size larger. I can calculate each of the points in the new polygon if it has convex angles, but I'm not sure how to adjust for when the angles are concave.
Concave angles can be treated in exactly the same way as convex ones: For each vertex you generate lines that are parallel to the two original segments but shifted by your offset value. Then the vertex is replaced with the intersection of these two lines.
The difficulty is that the resulting polygon can have intersections if the original one has one or more concave angles. There are different ways to handle these intersections. Generally they can produce inner contours (holes in the polygon) but maybe you are only interested in the outer contour.
In any case you have to find the intersection points first. If you don't find any, you are finished.
Otherwise find a start point of which you can be sure that it is on the outer contour. In many cases you can take the one with smallest X coordinate for that. Then trace the polygon contour until you get to the first intersection. Add the intersection to the polygon. If you are only interested in the outer contour, then skip all following vertices until you get back to the intersection point. Then continue to add the vertexes to the resulting polygon until you get to the next intersection and so on.
If you also need the inner contours (holes) it gets a bit more complicated, but I guess you can figure this out.
I also need to add that you should pe prepared for special cases like (almost) duplicate edges that cause numerical problems. Generally this is not a trivial task, so if possible, try to find a suitable polygon library.
For this problem I found a relatively simple solution for figuring out whether the calculated point was inside or outside the original polygon. Check to see whether the newly formed line intersects the original polygon's line. A formula can be found here http://www.geeksforgeeks.org/orientation-3-ordered-points/.
Suppose your polygon is given in counter-clockwise order. Let P1=(x1,y1), P2=(x2,y2) and P3=(x3,y3) be consecutive vertices. You want to know if the angle at P2 is “concave” i.e. more than 180 degrees. Let V1=(x4,y4)=P2-P1 and V2=(x5,y5)=P3-P2. Compute the “cross product” V1 x V2 = (x4.y5-x5.y4). This is negative iff the angle is concave.
Here is a code in C# that receives a list of Vector2D representing the ordered points of a polygon and returns a list with the angles of each vertex. It first checks if the points are clockwise or counterclockwise, and then it loops through the points calculating the sign of the cross product (z) for each triple of angles, and compare the value of the cross product to the clockwise function result to check if the calculated angle to that point needs to be the calculated angle or adjusted to 360-angle. The IsClockwise function was obtained in this discussion: How to determine if a list of polygon points are in clockwise order?
public bool IsClockwise(List<Vector2> vertices)
{
double sum = 0.0;
for (int i = 0; i < vertices.Count; i++)
{
Vector2 v1 = vertices[i];
Vector2 v2 = vertices[(i + 1) % vertices.Count];
sum += (v2.x - v1.x) * (v2.y + v1.y);
}
return sum > 0.0;
}
List<float> estimatePolygonAngles(List<Vector2> vertices)
{
if (vertices.Count < 3)
return null;
//1. check if the points are clockwise or counterclockwise:
int clockwise = (IsClockwise(vertices) ? 1 : -1);
List<float> angles = new List<float>();
List<float> crossProductsSigns = new List<float>();
Vector2 v1, v2;
//2. calculate the angles between each triple of vertices (first and last angles are computed separetely because index of the array):
v1 = vertices[vertices.Count - 1] - vertices[0];
v2 = vertices[1] - vertices[0];
angles.Add(Vector2.Angle(v1, v2));
crossProductsSigns.Add(Vector3.Cross(v1, v2).z > 0 ? 1 : -1);
for (int i = 1; i < vertices.Count-1; i++)
{
v1 = vertices[i-1] - vertices[i];
v2 = vertices[i+1] - vertices[i];
angles.Add(Vector2.Angle(v1, v2));
crossProductsSigns.Add(Vector3.Cross(v1, v2).z > 0 ? 1 : -1);
}
v1 = vertices[vertices.Count - 2] - vertices[vertices.Count - 1];
v2 = vertices[0] - vertices[vertices.Count - 1];
angles.Add(Vector2.Angle(v1, v2));
crossProductsSigns.Add(Vector3.Cross(v1, v2).z > 0 ? 1 : -1);
//3. for each computed angle, check if the cross product is the same as the as the direction provided by the clockwise function, if dont, the angle must be adjusted to 360-angle
for (int i = 0; i < vertices.Count; i++)
{
if (crossProductsSigns[i] != clockwise)
angles[i] = 360.0f - angles[i];
}
return angles;
}
I've been looking at and trying to understand the following bit of code
float sdBox( vec3 p, vec3 b )
{
vec3 d = abs(p) - b;
return min(max(d.x,max(d.y,d.z)),0.0) +
length(max(d,0.0));
}
I understand that length(d) handles the SDF case where the point is off to the 'corner' (ie. all components of d are positive) and that max(d.x, d.y, d.z) gives us the proper distance in all other cases. What I don't understand is how these two are combined here without the use of an if statement to check the signs of d's components.
When all of the d components are positive, the return expression can be reduced to length(d) because of the way min/max will evaluate - and when all of the d components are negative, we get max(d.x, d.y, d.z). But how am I supposed to understand the in-between cases? The ones where the components of d have mixed signs?
I've been trying to graph it out to no avail. I would really appreciate it if someone could explain this to me in geometrical/mathematical terms. Thanks.
If you like to know how It works It's better do the following steps:
1.first of all you should know definitions of shapes
2.It's always better to consider 2D shape of them, because three dimensions may be complex for you.
so let me to explain some shapes:
Circle
A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the center.
You can use distance(), length() or sqrt() to calculate the distance to the center of the billboard.
The book of shaders - Chapter 7
Square
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles).
I describe 2D shapes In before section now let me to describe 3D definition.
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.
Like a circle, which geometrically is an object in two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in three-dimensional space.
Refrence - Wikipedia
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
Refrence : Wikipedia
Modeling with distance functions
now It's time to understanding modeling with distance functions
Sphere
As mentioned In last sections.In below code length() used to calculate the distance to the center of the billboard , and you can scale this shape by s parameter.
//Sphere - signed - exact
/// <param name="p">Position.</param>
/// <param name="s">Scale.</param>
float sdSphere( vec3 p, float s )
{
return length(p)-s;
}
Box
// Box - unsigned - exact
/// <param name="p">Position.</param>
/// <param name="b">Bound(Scale).</param>
float udBox( vec3 p, vec3 b )
{
return length(max(abs(p)-b,0.0));
}
length() used like previous example.
next we have max(x,0) It called Positive and negative parts
this is mean below code is equivalent:
float udBox( vec3 p, vec3 b )
{
vec3 value = abs(p)-b;
if(value.x<0.){
value.x = 0.;
}
if(value.y<0.){
value.y = 0.;
}
if(value.z<0.){
value.z = 0.;
}
return length(value);
}
step 1
if(value.x<0.){
value.x = 0.;
}
step 2
if(value.y<0.){
value.y = 0.;
}
step 3
if(value.z<0.){
value.z = 0.;
}
step 4
next we have absolution function.It used to remove additional parts.
Absolution Steps
Absolution step 1
if(value.x < -1.){
value.x = 1.;
}
Absolution step 2
if(value.y < -1.){
value.y = 1.;
}
Absolution step 3
if(value.z < -1.){
value.z = 1.;
}
Also you can make any shape by using Constructive solid geometry.
CSG is built on 3 primitive operations: intersection ( ∩ ), union ( ∪ ), and difference ( - ).
It turns out these operations are all concisely expressible when combining two surfaces expressed as SDFs.
float intersectSDF(float distA, float distB) {
return max(distA, distB);
}
float unionSDF(float distA, float distB) {
return min(distA, distB);
}
float differenceSDF(float distA, float distB) {
return max(distA, -distB);
}
I figured it out a while ago and wrote about this extensively in a blog post here: http://fabricecastel.github.io/blog/2016-02-11/main.html
Here's an excerpt (see the full post for a full explanation):
Consider the four points, A, B, C and D. Let's crudely reduce the distance function to try and get rid of the min/max functions in order to understand their effect (since that's what's puzzling about this function). The notation below is a little sloppy, I'm using square brackets to denote 2D vectors.
// 2D version of the function
d(p) = min(max(p.x, p.y), 0)
+ length(max(p, 0))
---
d(A) = min(max(-1, -1), 0)
+ length(max([-1, -1], 0))
d(A) = -1 + length[0, 0]
---
d(B) = min(max(1, 1), 0)
+ length(max([1, 1], 0))
d(B) = 0 + length[1, 1]
Ok, so far nothing special. When A is inside the square, we essentially get our first distance function based on planes/lines and when B is in the area where our first distance function is inaccurate, it gets zeroed out and we get the second distance function (the length). The trick lies in the other two cases C and D. Let's work them out.
d(C) = min(max(-1, 1), 0)
+ length(max([-1, 1], 0))
d(C) = 0 + length[0, 1]
---
d(D) = min(max(1, -1), 0)
+ length(max([-1, 1], 0))
d(D) = 0 + length[1, 0]
If you look back to the graph above, you'll note C' and D'. Those points have coordinates [0,1] and [1,0], respectively. This method uses the fact that both distance fields intersect on the axes - that D and D' lie at the same distance from the square.
If we zero out all negative component of a vector and take its length we will get the proper distance between the point and the square (for points outside of the square only). This is what max(d,0.0) does; a component-wise max operation. So long as the vector has at least one positive component, min(max(d.x,d.y),0.0) will resolve to 0 leaving us with only the second part of the equation. In the event that the point is inside the square, we want to return the first part of the equation (since it represents our first distance function). If all components of the vector are negative it's easy to see our condition will be met.
This understanding should tranlsate back into 3D seamlessly once you wrap your head around it. You may or may not have to draw a few graphs by hand to really "get" it - I know I did and would encourage you to do so if you're dissatisfied with my explanation.
Working this implementation into our own code, we get this:
float distanceToNearestSurface(vec3 p){
float s = 1.0;
vec3 d = abs(p) - vec3(s);
return min(max(d.x, max(d.y,d.z)), 0.0)
+ length(max(d,0.0));
}
And there you have it.
I have an array of points (x0,y0)... (xn,yn) monotonic in x and wish to draw the "best" curve through these using Bezier curves. This curve should not be too "jaggy" (e.g. similar to joining the dots) and not too sinuous (and definitely not "go backwards"). I have created a prototype but wonder whether there is an objectively "best solution".
I need to find control points for all segments xi,y1 x(i+1)y(i+1). My current approach (except for the endpoints) for a segment x(i), x(i+1) is:
find the vector x(i-1)...x(i+1) , normalize, and scale it by factor * len(i,i+1) to give the vector for the leading control point
find the vector x(i+2)...x(i) , normalize, and scale it by factor * len(i,i+1) to give the vector for the trailing control point.
I have tried factor=0.1 (too jaggy), 0.33 (too curvy) and 0.20 - about right. But is there a better approach which (say) makes 2nd and 3nd derivatives as smooth as possible. (I assume such an algorithm is implemented in graphics packages)?
I can post pseudo/code if requested. Here are the three images (0.1/0.2/0.33). The control points are shown by straight lines: black (trailing) and red (leading)
Here's the current code. It's aimed at plotting Y against X (monotonic X) without close-ing. I have built my own library for creating SVG (preferred output); this code creates triples of x,y in coordArray for each curve segment (control1, xcontrol2, end). Start is assumed by last operation (Move or Curve). It's Java but should be easy to interpret (CurvePrimitive maps to cubic, "d" is the String representation of the complete path in SVG).
List<SVGPathPrimitive> primitiveList = new ArrayList<SVGPathPrimitive>();
primitiveList.add(new MovePrimitive(real2Array.get(0)));
for(int i = 0; i < real2Array.size()-1; i++) {
// create path 12
Real2 p0 = (i == 0) ? null : real2Array.get(i-1);
Real2 p1 = real2Array.get(i);
Real2 p2 = real2Array.get(i+1);
Real2 p3 = (i == real2Array.size()-2) ? null : real2Array.get(i+2);
Real2Array coordArray = plotSegment(factor, p0, p1, p2, p3);
SVGPathPrimitive primitive = new CurvePrimitive(coordArray);
primitiveList.add(primitive);
}
String d = SVGPath.constructDString(primitiveList);
SVGPath path1 = new SVGPath(d);
svg.appendChild(path1);
/**
*
* #param factor to scale control points by
* #param p0 previous point (null at start)
* #param p1 start of segment
* #param p2 end of segment
* #param p3 following point (null at end)
* #return
*/
private Real2Array plotSegment(double factor, Real2 p0, Real2 p1, Real2 p2, Real2 p3) {
// create p1-p2 curve
double len12 = p1.getDistance(p2) * factor;
Vector2 vStart = (p0 == null) ? new Vector2(p2.subtract(p1)) : new Vector2(p2.subtract(p0));
vStart = new Vector2(vStart.getUnitVector().multiplyBy(len12));
Vector2 vEnd = (p3 == null) ? new Vector2(p2.subtract(p1)) : new Vector2(p3.subtract(p1));
vEnd = new Vector2(vEnd.getUnitVector().multiplyBy(len12));
Real2Array coordArray = new Real2Array();
Real2 controlStart = p1.plus(vStart);
coordArray.add(controlStart);
Real2 controlEnd = p2.subtract(vEnd);
coordArray.add(controlEnd);
coordArray.add(p2);
// plot controls
SVGLine line12 = new SVGLine(p1, controlStart);
line12.setStroke("red");
svg.appendChild(line12);
SVGLine line21 = new SVGLine(p2, controlEnd);
svg.appendChild(line21);
return coordArray;
}
A Bezier curve requires the data points, along with the slope and curvature at each point. In a graphics program, the slope is set by the slope of the control-line, and the curvature is visualized by the length.
When you don't have such control-lines input by the user, you need to estimate the gradient and curvature at each point. The wikipedia page http://en.wikipedia.org/wiki/Cubic_Hermite_spline, and in particular the 'interpolating a data set' section has a formula that takes these values directly.
Typically, estimating these values from points is done using a finite difference - so you use the values of the points on either side to help estimate. The only choice here is how to deal with the end points where there is only one adjacent point: you can set the curvature to zero, or if the curve is periodic you can 'wrap around' and use the value of the last point.
The wikipedia page I referenced also has other schemes, but most others introduce some other 'free parameter' that you will need to find a way of setting, so in the absence of more information to help you decide how to set other parameters, I'd go for the simple scheme and see if you like the results.
Let me know if the wikipedia article is not clear enough, and I'll knock up some code.
One other point to be aware of: what 'sort' of Bezier interpolation are you after? Most graphics programs do cubic bezier in 2 dimensions (ie you can draw a circle-like curve), but your sample images look like it could be 1d functions approximation (as in for every x there is only one y value). The graphics program type curve is not really mentioned on the page I referenced. The maths involved for converting estimate of slope and curvature into a control vector of the form illustrated on http://en.wikipedia.org/wiki/B%C3%A9zier_curve (Cubic Bezier) would take some working out, but the idea is similar.
Below is a picture and algorithm for a possible scheme, assuming your only input is the three points P1, P2, P3
Construct a line (C1,P1,C2) such that the angles (P3,P1,C1) and (P2,P1,C2) are equal. In a similar fashion construct the other dark-grey lines. The intersections of these dark-grey lines (marked C1, C2 and C3) become the control-points as in the same sense as the images on the Bezier Curve wikipedia site. So each red curve, such as (P3,P1), is a quadratic bezier curve defined by the points (P3, C1, P1). The construction of the red curve is the same as given on the wikipedia site.
However, I notice that the control-vector on the Bezier Curve wikipedia page doesn't seem to match the sort of control vector you are using, so you might have to figure out how to equate the two approaches.
I tried this with quadratic splines instead of cubic ones which simplifies the selection of control points (you just choose the gradient at each point to be a weighted average of the mean gradients of the neighbouring intervals, and then draw tangents to the curve at the data points and stick the control points where those tangents intersect), but I couldn't find a sensible policy for setting the gradients of the end points. So I opted for Lagrange fitting instead:
function lagrange(points) { //points is [ [x1,y1], [x2,y2], ... ]
// See: http://www.codecogs.com/library/maths/approximation/interpolation/lagrange.php
var j,n = points.length;
var p = [];
for (j=0;j<n;j++) {
p[j] = function (x,j) { //have to pass j cos JS is lame at currying
var k, res = 1;
for (k=0;k<n;k++)
res*=( k==j ? points[j][1] : ((x-points[k][0])/(points[j][0]-points[k][0])) );
return res;
}
}
return function(x) {
var i, res = 0;
for (i=0;i<n;i++)
res += p[i](x,i);
return res;
}
}
With that, I just make lots of samples and join them with straight lines.
This is still wrong if your data (like mine) consists of real world measurements. These are subject to random errors and if you use a technique that forces the curve to hit them all precisely, then you can get silly valleys and hills between the points. In cases like these, you should ask yourself what order of polynomial the data should fit and ... well ... that's what I'm about to go figure out.
is there some good and better way to find centroid of contour in opencv, without using built in functions?
While Sonaten's answer is perfectly correct, there is a simple way to do it: Use the dedicated opencv function for that: moments()
http://opencv.itseez.com/modules/imgproc/doc/structural_analysis_and_shape_descriptors.html?highlight=moments#moments
It does not only returns the centroid, but some more statistics about your shape. And you can send it a contour or a raster shape (binary image), whatever best fits your need.
EDIT
example (modified) from "Learning OpenCV", by gary bradsky
CvMoments moments;
double M00, M01, M10;
cvMoments(contour,&moments);
M00 = cvGetSpatialMoment(&moments,0,0);
M10 = cvGetSpatialMoment(&moments,1,0);
M01 = cvGetSpatialMoment(&moments,0,1);
centers[i].x = (int)(M10/M00);
centers[i].y = (int)(M01/M00);
What you get in your current piece of code is of course the centroid of your bounding box.
"If you have a bunch of points(2d vectors), you should be able to get the centroid by averaging those points: create a point to add all the other points' positions into and then divide the components of that point with accumulated positions by the total number of points." - George Profenza mentions
This is indeed the right approach for the exact centroid of any given object in two-dimentionalspace.
On wikipedia we have some general forms for finding the centroid of an object.
http://en.wikipedia.org/wiki/Centroid
Personally, I would ask myself what I needed from this program. Do I want a thorough but performance heavy operation, or do I want to make some approximations? I might even be able to find an OpenCV function that deals with this correct and efficiently.
Don't have a working example, so I'm writing this in pseudocode on a simple 5 pixel example on a thorough method.
x_centroid = (pixel1_x + pixel2_x + pixel3_x + pixel4_x +pixel5_x)/5
y_centroid = (pixel1_y + pixel2_y + pixel3_y + pixel4_y +pixel5_y)/5
centroidPoint(x_centroid, y_centroid)
Looped for x pixels
Loop j times *sample (for (int i=0, i < j, i++))*
{
x_centroid = pixel[j]_x + x_centroid
y_centroid = pixel[j]_x + x_centroid
}
x_centroid = x_centroid/j
y_centroid = y_centroid/j
centroidPoint(x_centroid, y_centroid)
Essentially, you have the vector contours of the type
vector<vector<point>>
in OpenCV 2.3. I believe you have something similar in earlier versions, and you should be able to go through each blob on your picture with the first index of this "double vector", and go through each pixel in the inner vector.
Here is a link to documentation on the contour function
http://opencv.itseez.com/modules/imgproc/doc/structural_analysis_and_shape_descriptors.html?highlight=contours#cv.DrawContours
note: you've tagged your question as c++ visual. I'd suggest that you use the c++ syntax in OpenCV 2.3 instead of c. The first and good reason to use 2.3 is that it is more class based, which in this case means that the class Mat (instead of IplImage) does leak memory. One does not have to write destroy commands all the live long day :)
I hope this shed some light on your problem. Enjoy.
I've used Joseph O'Rourke excellent polygon centroid algorithm to great success.
See http://maven.smith.edu/~orourke/Code/centroid.c
Essentially:
For each point in the contour, find the triangle area from the current index polygon xy to the next 2 polygon xy points e.g.: Math.Abs(((X1 - X0) * (Y2 - Y0) - (X2 - X0) * (Y1 - Y0)) / 2)
Add this triangle area to a list TriAreas
Sum the triangle area, and store in SumT
Find the centroid CTx and CTy from this current triangle: CTx = (X0 + X1 + X2) / 3 and CTy = (Y0 + Y1 + Y2) / 3;
Store these 2 centroid values in 2 other lists CTxs CTys.
Finally after performing this with all points in the contour, find the contours centroid x and y using the 2 triangle x and y lists in 5 which is a weighted sum of signed triangle areas, weighted by the centroid of each triangle:
for (Int32 Index = 0; Index < CTxs.Count; Index++)
{
CentroidPointRet.X += CTxs[Index] * (TriAreas[Index] / SumT);
}
// now find centroid Y value
for (Int32 Index = 0; Index < CTys.Count; Index++)
{
CentroidPointRet.Y += CTys[Index] * (TriAreas[Index] / SumT);
}
With the help of the Stack Overflow community I've written a pretty basic-but fun physics simulator.
You click and drag the mouse to launch a ball. It will bounce around and eventually stop on the "floor".
My next big feature I want to add in is ball to ball collision. The ball's movement is broken up into a x and y speed vector. I have gravity (small reduction of the y vector each step), I have friction (small reduction of both vectors each collision with a wall). The balls honestly move around in a surprisingly realistic way.
I guess my question has two parts:
What is the best method to detect ball to ball collision?
Do I just have an O(n^2) loop that iterates over each ball and checks every other ball to see if it's radius overlaps?
What equations do I use to handle the ball to ball collisions? Physics 101
How does it effect the two balls speed x/y vectors? What is the resulting direction the two balls head off in? How do I apply this to each ball?
Handling the collision detection of the "walls" and the resulting vector changes were easy but I see more complications with ball-ball collisions. With walls I simply had to take the negative of the appropriate x or y vector and off it would go in the correct direction. With balls I don't think it is that way.
Some quick clarifications: for simplicity I'm ok with a perfectly elastic collision for now, also all my balls have the same mass right now, but I might change that in the future.
Edit: Resources I have found useful
2d Ball physics with vectors: 2-Dimensional Collisions Without Trigonometry.pdf
2d Ball collision detection example: Adding Collision Detection
Success!
I have the ball collision detection and response working great!
Relevant code:
Collision Detection:
for (int i = 0; i < ballCount; i++)
{
for (int j = i + 1; j < ballCount; j++)
{
if (balls[i].colliding(balls[j]))
{
balls[i].resolveCollision(balls[j]);
}
}
}
This will check for collisions between every ball but skip redundant checks (if you have to check if ball 1 collides with ball 2 then you don't need to check if ball 2 collides with ball 1. Also, it skips checking for collisions with itself).
Then, in my ball class I have my colliding() and resolveCollision() methods:
public boolean colliding(Ball ball)
{
float xd = position.getX() - ball.position.getX();
float yd = position.getY() - ball.position.getY();
float sumRadius = getRadius() + ball.getRadius();
float sqrRadius = sumRadius * sumRadius;
float distSqr = (xd * xd) + (yd * yd);
if (distSqr <= sqrRadius)
{
return true;
}
return false;
}
public void resolveCollision(Ball ball)
{
// get the mtd
Vector2d delta = (position.subtract(ball.position));
float d = delta.getLength();
// minimum translation distance to push balls apart after intersecting
Vector2d mtd = delta.multiply(((getRadius() + ball.getRadius())-d)/d);
// resolve intersection --
// inverse mass quantities
float im1 = 1 / getMass();
float im2 = 1 / ball.getMass();
// push-pull them apart based off their mass
position = position.add(mtd.multiply(im1 / (im1 + im2)));
ball.position = ball.position.subtract(mtd.multiply(im2 / (im1 + im2)));
// impact speed
Vector2d v = (this.velocity.subtract(ball.velocity));
float vn = v.dot(mtd.normalize());
// sphere intersecting but moving away from each other already
if (vn > 0.0f) return;
// collision impulse
float i = (-(1.0f + Constants.restitution) * vn) / (im1 + im2);
Vector2d impulse = mtd.normalize().multiply(i);
// change in momentum
this.velocity = this.velocity.add(impulse.multiply(im1));
ball.velocity = ball.velocity.subtract(impulse.multiply(im2));
}
Source Code: Complete source for ball to ball collider.
If anyone has some suggestions for how to improve this basic physics simulator let me know! One thing I have yet to add is angular momentum so the balls will roll more realistically. Any other suggestions? Leave a comment!
To detect whether two balls collide, just check whether the distance between their centers is less than two times the radius. To do a perfectly elastic collision between the balls, you only need to worry about the component of the velocity that is in the direction of the collision. The other component (tangent to the collision) will stay the same for both balls. You can get the collision components by creating a unit vector pointing in the direction from one ball to the other, then taking the dot product with the velocity vectors of the balls. You can then plug these components into a 1D perfectly elastic collision equation.
Wikipedia has a pretty good summary of the whole process. For balls of any mass, the new velocities can be calculated using the equations (where v1 and v2 are the velocities after the collision, and u1, u2 are from before):
If the balls have the same mass then the velocities are simply switched. Here's some code I wrote which does something similar:
void Simulation::collide(Storage::Iterator a, Storage::Iterator b)
{
// Check whether there actually was a collision
if (a == b)
return;
Vector collision = a.position() - b.position();
double distance = collision.length();
if (distance == 0.0) { // hack to avoid div by zero
collision = Vector(1.0, 0.0);
distance = 1.0;
}
if (distance > 1.0)
return;
// Get the components of the velocity vectors which are parallel to the collision.
// The perpendicular component remains the same for both fish
collision = collision / distance;
double aci = a.velocity().dot(collision);
double bci = b.velocity().dot(collision);
// Solve for the new velocities using the 1-dimensional elastic collision equations.
// Turns out it's really simple when the masses are the same.
double acf = bci;
double bcf = aci;
// Replace the collision velocity components with the new ones
a.velocity() += (acf - aci) * collision;
b.velocity() += (bcf - bci) * collision;
}
As for efficiency, Ryan Fox is right, you should consider dividing up the region into sections, then doing collision detection within each section. Keep in mind that balls can collide with other balls on the boundaries of a section, so this may make your code much more complicated. Efficiency probably won't matter until you have several hundred balls though. For bonus points, you can run each section on a different core, or split up the processing of collisions within each section.
Well, years ago I made the program like you presented here.
There is one hidden problem (or many, depends on point of view):
If the speed of the ball is too
high, you can miss the collision.
And also, almost in 100% cases your new speeds will be wrong. Well, not speeds, but positions. You have to calculate new speeds precisely in the correct place. Otherwise you just shift balls on some small "error" amount, which is available from the previous discrete step.
The solution is obvious: you have to split the timestep so, that first you shift to correct place, then collide, then shift for the rest of the time you have.
You should use space partitioning to solve this problem.
Read up on
Binary Space Partitioning
and
Quadtrees
As a clarification to the suggestion by Ryan Fox to split the screen into regions, and only checking for collisions within regions...
e.g. split the play area up into a grid of squares (which will will arbitrarily say are of 1 unit length per side), and check for collisions within each grid square.
That's absolutely the correct solution. The only problem with it (as another poster pointed out) is that collisions across boundaries are a problem.
The solution to this is to overlay a second grid at a 0.5 unit vertical and horizontal offset to the first one.
Then, any collisions that would be across boundaries in the first grid (and hence not detected) will be within grid squares in the second grid. As long as you keep track of the collisions you've already handled (as there is likely to be some overlap) you don't have to worry about handling edge cases. All collisions will be within a grid square on one of the grids.
A good way of reducing the number of collision checks is to split the screen into different sections. You then only compare each ball to the balls in the same section.
One thing I see here to optimize.
While I do agree that the balls hit when the distance is the sum of their radii one should never actually calculate this distance! Rather, calculate it's square and work with it that way. There's no reason for that expensive square root operation.
Also, once you have found a collision you have to continue to evaluate collisions until no more remain. The problem is that the first one might cause others that have to be resolved before you get an accurate picture. Consider what happens if the ball hits a ball at the edge? The second ball hits the edge and immediately rebounds into the first ball. If you bang into a pile of balls in the corner you could have quite a few collisions that have to be resolved before you can iterate the next cycle.
As for the O(n^2), all you can do is minimize the cost of rejecting ones that miss:
1) A ball that is not moving can't hit anything. If there are a reasonable number of balls lying around on the floor this could save a lot of tests. (Note that you must still check if something hit the stationary ball.)
2) Something that might be worth doing: Divide the screen into a number of zones but the lines should be fuzzy--balls at the edge of a zone are listed as being in all the relevant (could be 4) zones. I would use a 4x4 grid, store the zones as bits. If an AND of the zones of two balls zones returns zero, end of test.
3) As I mentioned, don't do the square root.
I found an excellent page with information on collision detection and response in 2D.
http://www.metanetsoftware.com/technique.html (web.archive.org)
They try to explain how it's done from an academic point of view. They start with the simple object-to-object collision detection, and move on to collision response and how to scale it up.
Edit: Updated link
You have two easy ways to do this. Jay has covered the accurate way of checking from the center of the ball.
The easier way is to use a rectangle bounding box, set the size of your box to be 80% the size of the ball, and you'll simulate collision pretty well.
Add a method to your ball class:
public Rectangle getBoundingRect()
{
int ballHeight = (int)Ball.Height * 0.80f;
int ballWidth = (int)Ball.Width * 0.80f;
int x = Ball.X - ballWidth / 2;
int y = Ball.Y - ballHeight / 2;
return new Rectangle(x,y,ballHeight,ballWidth);
}
Then, in your loop:
// Checks every ball against every other ball.
// For best results, split it into quadrants like Ryan suggested.
// I didn't do that for simplicity here.
for (int i = 0; i < balls.count; i++)
{
Rectangle r1 = balls[i].getBoundingRect();
for (int k = 0; k < balls.count; k++)
{
if (balls[i] != balls[k])
{
Rectangle r2 = balls[k].getBoundingRect();
if (r1.Intersects(r2))
{
// balls[i] collided with balls[k]
}
}
}
}
I see it hinted here and there, but you could also do a faster calculation first, like, compare the bounding boxes for overlap, and THEN do a radius-based overlap if that first test passes.
The addition/difference math is much faster for a bounding box than all the trig for the radius, and most times, the bounding box test will dismiss the possibility of a collision. But if you then re-test with trig, you're getting the accurate results that you're seeking.
Yes, it's two tests, but it will be faster overall.
This KineticModel is an implementation of the cited approach in Java.
I implemented this code in JavaScript using the HTML Canvas element, and it produced wonderful simulations at 60 frames per second. I started the simulation off with a collection of a dozen balls at random positions and velocities. I found that at higher velocities, a glancing collision between a small ball and a much larger one caused the small ball to appear to STICK to the edge of the larger ball, and moved up to around 90 degrees around the larger ball before separating. (I wonder if anyone else observed this behavior.)
Some logging of the calculations showed that the Minimum Translation Distance in these cases was not large enough to prevent the same balls from colliding in the very next time step. I did some experimenting and found that I could solve this problem by scaling up the MTD based on the relative velocities:
dot_velocity = ball_1.velocity.dot(ball_2.velocity);
mtd_factor = 1. + 0.5 * Math.abs(dot_velocity * Math.sin(collision_angle));
mtd.multplyScalar(mtd_factor);
I verified that before and after this fix, the total kinetic energy was conserved for every collision. The 0.5 value in the mtd_factor was the approximately the minumum value found to always cause the balls to separate after a collision.
Although this fix introduces a small amount of error in the exact physics of the system, the tradeoff is that now very fast balls can be simulated in a browser without decreasing the time step size.
Improving the solution to detect circle with circle collision detection given within the question:
float dx = circle1.x - circle2.x,
dy = circle1.y - circle2.y,
r = circle1.r + circle2.r;
return (dx * dx + dy * dy <= r * r);
It avoids the unnecessary "if with two returns" and the use of more variables than necessary.
After some trial and error, I used this document's method for 2D collisions : https://www.vobarian.com/collisions/2dcollisions2.pdf
(that OP linked to)
I applied this within a JavaScript program using p5js, and it works perfectly. I had previously attempted to use trigonometrical equations and while they do work for specific collisions, I could not find one that worked for every collision no matter the angle at the which it happened.
The method explained in this document uses no trigonometrical functions whatsoever, it's just plain vector operations, I recommend this to anyone trying to implement ball to ball collision, trigonometrical functions in my experience are hard to generalize. I asked a Physicist at my university to show me how to do it and he told me not to bother with trigonometrical functions and showed me a method that is analogous to the one linked in the document.
NB : My masses are all equal, but this can be generalised to different masses using the equations presented in the document.
Here's my code for calculating the resulting speed vectors after collision :
//you just need a ball object with a speed and position vector.
class TBall {
constructor(x, y, vx, vy) {
this.r = [x, y];
this.v = [0, 0];
}
}
//throw two balls into this function and it'll update their speed vectors
//if they collide, you need to call this in your main loop for every pair of
//balls.
function collision(ball1, ball2) {
n = [ (ball1.r)[0] - (ball2.r)[0], (ball1.r)[1] - (ball2.r)[1] ];
un = [n[0] / vecNorm(n), n[1] / vecNorm(n) ] ;
ut = [ -un[1], un[0] ];
v1n = dotProd(un, (ball1.v));
v1t = dotProd(ut, (ball1.v) );
v2n = dotProd(un, (ball2.v) );
v2t = dotProd(ut, (ball2.v) );
v1t_p = v1t; v2t_p = v2t;
v1n_p = v2n; v2n_p = v1n;
v1n_pvec = [v1n_p * un[0], v1n_p * un[1] ];
v1t_pvec = [v1t_p * ut[0], v1t_p * ut[1] ];
v2n_pvec = [v2n_p * un[0], v2n_p * un[1] ];
v2t_pvec = [v2t_p * ut[0], v2t_p * ut[1] ];
ball1.v = vecSum(v1n_pvec, v1t_pvec); ball2.v = vecSum(v2n_pvec, v2t_pvec);
}
I would consider using a quadtree if you have a large number of balls. For deciding the direction of bounce, just use simple conservation of energy formulas based on the collision normal. Elasticity, weight, and velocity would make it a bit more realistic.
Here is a simple example that supports mass.
private void CollideBalls(Transform ball1, Transform ball2, ref Vector3 vel1, ref Vector3 vel2, float radius1, float radius2)
{
var vec = ball1.position - ball2.position;
float dis = vec.magnitude;
if (dis < radius1 + radius2)
{
var n = vec.normalized;
ReflectVelocity(ref vel1, ref vel2, ballMass1, ballMass2, n);
var c = Vector3.Lerp(ball1.position, ball2.position, radius1 / (radius1 + radius2));
ball1.position = c + (n * radius1);
ball2.position = c - (n * radius2);
}
}
public static void ReflectVelocity(ref Vector3 vel1, ref Vector3 vel2, float mass1, float mass2, Vector3 intersectionNormal)
{
float velImpact1 = Vector3.Dot(vel1, intersectionNormal);
float velImpact2 = Vector3.Dot(vel2, intersectionNormal);
float totalMass = mass1 + mass2;
float massTransfure1 = mass1 / totalMass;
float massTransfure2 = mass2 / totalMass;
vel1 += ((velImpact2 * massTransfure2) - (velImpact1 * massTransfure2)) * intersectionNormal;
vel2 += ((velImpact1 * massTransfure1) - (velImpact2 * massTransfure1)) * intersectionNormal;
}