Sweeping of three-dimensional objects

The basic problem which this paper addresses is the following: given an arbitrary three-dimensional object, which moves along an arbitrary path (possibly rotating as it does so), to compute the volume swept out by the solid object as it moves, or in other words, to find the new solid volume which represents all of the points in space which the object has occupied at some time during the motion. The modelling of swept volumes is important for many areas of CAD and CAM. For example, it can provide a solution to many different types of interference problem [31], where it is necessary to tell whether a moving object will intersect or interfere with other objects as it moves. This is obviously very useful in mechanism design, for ensuring that all parts of a mechanism stay inside its casing as it operates, or that parts of the mechanism do not collide with each other. As an example, consider the problem of determining whether a car wheel collides with the wheel arch as it travels up and down on the suspension, and steers left and right. Another application of interference checking is in robot path planning [13,14,38], where a safe path is one that does not collide with any of the fixtures or other objects in the robot’s environment. When machining a solid object, interference checking can be useful to make sure that the cutter does not gouge the table or clamps. Calculation of swept volumes may also be used for cutter path verification [43,44]: if the volume swept out by the moving cutter is subtracted from the original blank, the result should represent the same shape as the part to be machined. A final use of swept shape calculation is as a creative design tool, where the sweeping of an initial shape along a desired path is used to produce new and useful solid shapes [17,37]. For example, the frame of a chair made from tubular steel could be represented by the path swept out by a moving sphere.