We consider the collision-detection problem for a three-dimensional solid object moving among polyhedral obstacles. The configuration space for this problem is six-dimensional, and the traditional representation of the space uses three translational parameters and three angles (typically Euler angles). The constraints between the object and obstacles then involve trigonometric functions. We show that a quaternion representation of rotation yields constraints which are purely algebraic in a seven-dimensional space. By simple manipulation, the constraints may be projected down into a six-dimensional space with no increase in complexity. The algebraic form of the constraints greatly simplifies computation of collision points, and allows us to derive an efficient exact intersection test for an object which is translating and rotating among obstacles.