# Simple Sphere-Sphere Collision Detection and Collision Response

## Collision Detection

To determine if two spheres are colliding, we take the sum of the radiuses and compare it with the length from the centers of the spheres. If the length is smaller than the sum of the radiuses, we have a collision.

The difference vector (the length is the distance between those two spheres):

[pmath]vec{d} = s1.pos – s2.pos[/pmath]

Then the length is computed:

[pmath]distance = vec{d}.length = sqrt{vec{d}.x^2 + vec{d}.y^2 + vec{d}.z^2}[/pmath]

Sum of the radiuses:

[pmath]sumradius = s1.radius + s2.radius[/pmath]

If distance < sumradius the we have a collision to take care of.
This is an example of an elastic collision detection. More advanced algorithm includes calculations of collision time and direction.

**Related: **

Simple AABB vs AABB collision detection

Sphere vs AABB collision detection

## Collision Response

This is a little bit trickier, but with some basics explained, it should be pretty straight forward.

First, find the vector which will serve as a basis vector (x-axis), in an arbitrary direction. It has to be normalized to get realistic results.

[pmath]vec{x} = s1.pos – s2.pos[/pmath]

[pmath]vec{x}.normalize()[/pmath]

Then we calculate the x-direction velocity vector and the perpendicular y-vector.

[pmath]vec{v1} = s1.vel[/pmath]

[pmath]x1 = x.dot(vec{v1})[/pmath]

[pmath]vec{v1x} = vec{x} * x1[/pmath]

[pmath]vec{v1y} = vec{v1} – vec{v1x}[/pmath]

[pmath]m1 = s1.mass[/pmath]

Same procedure for the other sphere.

[pmath]vec{x} = vec{x}*-1[/pmath]

[pmath]vec{v2} = s2.vel[/pmath]

[pmath]x2 = vec{x}.dot(vec{v2})[/pmath]

[pmath]vec{v2x} = vec{x} * x2[/pmath]

[pmath]vec{v2y} = vec{v2} – vec{v2x}[/pmath]

[pmath]m2 = s2.mass[/pmath]

Then we mix and play around with some of Newton’s laws to obtain a formula for the speed (in vector format) after the collision.

[pmath]s1.vel = vec{v1x}{(m1-m2)/(m1+m2)} + vec{v2x}{(2*m2)/(m1+m2)} + vec{v1y}[/pmath]

[pmath]s2.vel = vec{v1x}{(2*m1)/(m1+m2)} + vec{v2x}{(m2-m1)/(m1+m2)} + vec{v2y}[/pmath]

And it actually works! ðŸ™‚

## Sample application:

Source code in subversion repository.

## Comments

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MarcoJanuary 6, 2009I was looking for a clear tutorial on collision response and i found it here.

great post, very usefull!

Thanks

JonasSeptember 15, 2009I had to refresh my dusty brain, and this was perfect for that!

Thanks!

EdgarOctober 29, 2009This information was very usefull in a demo that i’m trying to build in XNA. Thanks

Niklas RotherJuly 31, 2010I tried to adapt this in C#/XNA, but it seems that there is a mistake: x is a Vector3 (Vector with 3 Components) and v1 also. So x.dot(v1) is a float (scalar/normal number). But in the line you use the Cross product (or am I wrong?), and you canÂ´t make a Crossproduct of a Vector3 (x) and a float (x1).

Additional the SVN link seems down, so I couldt check the code…

kentAugust 4, 2010Niklas: I think it should be “vector multiplied by a scalar”, not cross product.

This goes without checking the code, which I’ll do later when I fix the repository.

kentAugust 15, 2010@Niklas

I have fixed the websvn interface (and svn interface).

Nick WiggillNovember 21, 2010This doesn’t actually appear to work when one of the circle-bodies is stationary — it will not impart any forces in this instance, at least in my use of this code.

Nick WiggillNovember 21, 2010No — ignore my prior comment — I am mistaken. There is something other bug in my application of your code.

Nick WiggillNovember 22, 2010I had issues with my vector rotation code. Your approach works fine. Many thanks.

kentNovember 23, 2010@Nick

I’m glad you found this useful!

Thanks.

RobertJanuary 28, 2011Super! All I need to to calculate the two angles that describe how to draw the line between sphere centers.. Can you help me there? How do I correctly write the code to get the two angles?

ChrisMarch 23, 2011A very nice tutorial for the collision response between two spheres. Unfortunately it doesn’t hold up when more than two spheres can collide at once.

Although I suppose the solution for that is seperate to this tutorial as you just figure out which collision occurs first in the current frame and then push the physics simulation back and forwards through the frame until all collisions have been resolved.

ElenaMarch 25, 2011Hi,

This helped a lot! I have been tormenting myself with books and other theoretical stuff for four days now and without any results. I still have a question. Does this work even when one object is stationary?

JulianMay 31, 2011Thanks! I’ve been looking for a nice and clear tut and i finally found it! You’re awesome!

pwSeptember 7, 2011great stuff. have converted this to c# xna with little much to change. x.dot() is vector3.dot(x,v1) and the final 2 lines omit the vector3 bracket: s1.Velocity = v1x…blah;

aramJanuary 26, 2012hi

i’m trying to write code of a game named peggle,maby you know it.

now i want to have collision functoin in my code & i hope you help me for that problem.

if it is possible for you send methis code as E-mail;

thank you…….!!!!!!!!

kentFebruary 1, 2012aram: you can see the source code in subversion repository (link at the end of the article)

AASNovember 17, 2013ThankYou! That logic for collision was very helpful.

GhassanFebruary 13, 2014Thank You, but i have a question. Is that elastic or inelastic collision ??? and what’s the difference between them in equations ??

kentFebruary 17, 2014This is an elastic collision. If you want inelastic, you would have to multiply the collision response with a number somewhere between 0.0 and up to 1.0.

MXXIVApril 12, 2016This actually doesn’t work. Heavy ball bounces off small ball. Also, the collision is hyper elastic. If you make row of balls and hit it with ball from one side, it acts as particle accelerator.

TheRabbidLemmingSeptember 16, 2016Great first steps into the topic. Thanks. Now to calculate for balls colliding with other balls! lol