Whether it's your car crossing the finish line at 180 miles per hour, or a bullet tearing through the chest of your best friend, all games make use of collision detection for object interaction. This article describes some simple intersection tests for the most useful shapes: spheres and boxes.

Sweep Tests for Moving Objects

A common approach to collision detection is to simply test for whether two objects are overlapping at the end of each frame. The problem with this method is that quickly moving objects can pass through each other without detection. To avoid this problem, their trajectories can be subdivided and the objects checked for overlap at each point; however, this gets expensive if either object experienced a large displacement. On the other hand, a sweep test can efficiently determine a lower and upper bound for the time of overlap, which can then be used as more optimal starting points for the subdivision algorithm.

A Sphere-Plane Sweep Test

Figure 1 shows an example of a quickly moving sphere passing through a plane. It can be seen that C₀ is on the positive side of the plane and C₁ is on its negative side.

Figure 1. A sphere passes through a plane.

In general, if a sphere penetrated a plane at some point during the frame, then d₀>r and d₁<r, where r is the radius of the sphere and d₀ and d₁ are the signed distances from the plane to C₀ and C₁, respectively. The signed distance from a point C to a plane can be calculated with the formula

More efficiently, we can store the plane in the form {n, D}, where

The distance d is then calculated

The trajectory from C₀ to C₁ can be parameterized with a variable u, which may be thought of as normalized time, since its value is 0 at C₀ and 1 at C₁. The normalized time at which the sphere first intersects the plane is given by

The center of the sphere at this time can then be interpolated with an affine combination of C₀ and C₁

This formula interpolates C_i correctly as long as d₀ is not equal to d₁ (which is the case if displacement has occurred), even when r = 0 (the case of a line segment). If desired, the parameter u can also be used to linearly interpolate the orientation of an object at this point.

In this example, it was assumed that the sphere approached the plane from the positive side and that the sphere was not already penetrating the plane at C_0.. In the case that there could have been penetration on the previous frame, the condition |d₀|<=r should also be checked. Listing 1 gives an implementation of this sphere-plane sweep test.

Listing 1. A sphere-plane sweep test.

#include "vector.h"

class PLANE

{

PLANE( const VECTOR& p0, const VECTOR& n ): N(n), D(-N.dot(p0))
{}
//from 3 points

PLANE( const VECTOR& p0, const VECTOR& p1,
const VECTOR& p2 ): N((p1-p0).cross(p2-p0).unit()),
D(-N.dot(p0))
{}
//signed distance from the plane topoint 'p' along
//the unit normal

const SCALAR distanceToPoint( const VECTOR& p ) const
{

 

return N.dot(p) + D;

}

};

const bool SpherePlaneSweep
(
const SCALAR r, //sphere radius
const VECTOR& C0, //previous position of sphere
const VECTOR& C1, //current position of sphere
const PLANE& plane, //the plane
VECTOR& Ci, //position of sphere when it first touched the plane
SCALAR& u //normalized time of collision

)

{

const SCALAR d0 = plane.distanceToPoint( C0 );
const SCALAR d1 = plane.distanceToPoint( C1 );

//check if it was touching on previous frame
if( fabs(d0) <= r )

{

}

//check if the sphere penetrated during this frame
if( d0>r && d1<r )
{

}

return false;

}

For the definition of the VECTOR class, please see [3].