Collision detection between non-linear objects

Question

I'm trying to do collision detection between two non linear objects that are moving in curve or non linear movement. how would one do that. I did it using linear movement, but I don't know the equation of two non linear objects.

double collision() // returns t>=0 the collision time or -1 if no collision
    {
    int i;
    double t,dt;
    double x,y,z,d0,d1;
    dt=_max_t;
    x=pos0[0]-pos1[0];
    y=pos0[1]-pos1[1];
    z=pos0[2]-pos1[2];
    d0=sqrt((x*x)+(y*y)+(z*z));
    x=pos0[0]-pos1[0]+(vel0[0]-vel1[0])*dt;
    y=pos0[1]-pos1[1]+(vel0[1]-vel1[1])*dt;
    z=pos0[2]-pos1[2]+(vel0[2]-vel1[2])*dt;
    d1=sqrt((x*x)+(y*y)+(z*z));
    if (d0<=_max_d) return  0.0; // collided now
    if (d0<=d1)     return -1.0; // never collide
    t=(_max_d-d0)*dt/(d1-d0);
    return t;
    }

Answer 1

For each time $t$, you have the positions of the two objects $(x_{p_1}(t),y_{p_1}(t),z_{p_1}(t))$ and $(x_{p_2}(t),y_{p_2}(t),z_{p_2}(t))$

Compute the distance between the two objects as $$d(t)=\sqrt{(x_{p_1}(t)-x_{p_2}(t))^2+(y_{p_1}(t)-y_{p_2}(t))^2+(z_{p_1}(t)-z_{p_2}(t))^2}$$

Then define a collision threshold, $d_{col}$.

The collision condition is therefore, for each time $t$, $$d(t)<d_{col}.$$

That is, if the above inequality holds we have a collision at time $t$, else, we don't.

Just make sure the time interval at which you check this condition and the value of $d_{col}$ are appropriate for the problem to look realistic.

Edit: C Implementation

//assumes both p1 and p2 have the same size 
unsigned int collision(double* p1, double* p2, unsigned int size, double dcol){ 
    double d, sum=0;
    unsigned int i;

    for( i=0; i<size; i++){
        sum+=(p1[i]-p2[i])*(p1[i]-p2[i]);  
    }
    d=sqrt(sum);

    if( d<dcol ) return 1; //collision produced
    else return 0; // no collision
}

//main flow of the algorithm
int main(){

    //Initialize the position vectors p1 and p2
    double p1={0,0,0}, p2={1,1,1};
    double t=0, dt=0.01, dcol=0.01; //set dt and dcol to whatever fits you

    while(1){
         updatePositions(...); //function that updates the positions of the objects
         if(collision(p1,p2,3,dcol)==1) break;
         else {t=t+dt;}
    }
    printf("Collision produced at t=%lf\n",t);
    return 0;
}