A Sphere-Sphere Sweep Test

Figure 2 shows two spheres that collided between frames. If these spheres experienced acceleration during the frame, their trajectories will be second or higher order curves; however, usually their paths can be accurately approximated as linear segments according to the equations

Since both spheres traveled for the same amount of time, u is the same for both trajectories. The square of the distance between the lines is

and to calculate when they first make contact, we must solve for u such that

This leads to the quadratic equation

The vector v_ba can be thought of as the displacement of B observed by A. This equation is quadratic in u, so there may be no solution (the spheres never collided), one solution (they just glanced each other), or two solutions (in which case the lesser solution is when they began to overlap and the greater is when they became disjoint again). Again, it is a good idea to check for overlap at the beginning of the frame, since this will handle the case of two stationary spheres. Listing 2 shows an implementation of the sphere-sphere sweep test.

Listing 2. The sphere-sphere sweep test.

#include "vector.h"

template< class T >

inline void SWAP( T& a, T& b )
//swap the values of a and b

{

}

// Return true if r1 and r2 are real
inline bool QuadraticFormula
(
const SCALAR a,
const SCALAR b,
const SCALAR c,
SCALAR& r1, //first
SCALAR& r2 //and second roots
)

{

const SCALAR q = b*b - 4*a*c;
if( q >= 0 )

{

 

 

const SCALAR sq = sqrt(q);
const SCALAR d = 1 / (2*a);
r1 = ( -b + sq ) * d;
r2 = ( -b - sq ) * d;
return true;//real roots

}

else

{

 

 

return false;//complex roots

}

}

const bool SphereSphereSweep
(
const SCALAR ra, //radius of sphere A
const VECTOR& A0, //previous position of sphere A
const VECTOR& A1, //current position of sphere A
const SCALAR rb, //radius of sphere B
const VECTOR& B0, //previous position of sphere B
const VECTOR& B1, //current position of sphere B
SCALAR& u0, //normalized time of first collision
SCALAR& u1 //normalized time of second collision

)

{

//check if they're currently overlapping
if( AB.dot(AB) <= rab*rab )

{

 

 

u0 = 0;
u1 = 0;
return true;

}

//check if they hit each other
// during the frame
if( QuadraticFormula( a, b, c, u0, u1 ) )
{

 

 

if( u0 > u1 )
SWAP( u0, u1 );
return true;

}

return false;

}