here are a number of useful tips to avoid monotony in motion. Perhaps the most generic piece of advice that can be given in this regard is: eliminate linearity. Although a static straight line has a certain aesthetic appeal due to its geometric perfection, a monotone linear motion is much less likely to work satisfactorily - in most cases, it's simply boring. However, being nonlinear in motion does not necessarily mean pushing your objects along a Bezier curve (although this might be a good idea in some cases, too). See what other options exist:

• Accelerate or decelerate all instances of motion. Constant speed adds a mechanic feeling to any movement. Imagine that every element in your composition has a certain mass, and therefore cannot start or stop moving too abruptly. When you need some object to move and then stop, or to stay for some time and then start moving, gradual decelerating or accelerating is a must; but even for the cases where an object just keeps moving without stopping (for example, until it gets out of sight), adding some second derivative (which is the mathematical equivalent for the physical concept of acceleration) will make its movement more natural and engaging.

• Use curvilinear motion paths instead of linear ones where possible, even if the curve is a simple circle. When moving an object along a linear path, rotating it at some degree during the shift will help to enliven it. Using rotation in combination with acceleration or deceleration will make a motion much more organic.

• Adding a third dimension to your motion may compensate for its too linear character. By this I do not mean real 3D, only a simple trick of enlarging or reducing an object dynamically thus giving it an impression of "emerging" or "sinking" relative to the plane of the screen. Again, this technique can be used in combination with linear motion, accelerating/decelerating and rotation.

• Color and texture effects can, to some extent, mask the linearity of a motion. Combined with a gradual darkening/lightening, transparency variation, or a moving highlight such as that of Fig. 2, even an absolutely monotonous progression may turn into an interesting experience.

• Similarly, about any transformation of an object, such as shape morphing or tweening, forces us to watch it with much closer attention, even if the object is immobile or involved in a simple linear motion. Transformations are something computers can do amazingly well, and creative opportunities are enormous.

• With linear motions, try, at the very least, to make them as short as possible; avoid frustrating the viewer by pushing an object too far in a too monotonous manner. For short bits of motion, it is much more difficult for the eye to notice their linearity.

• Furthermore, use a number of short linear movements of different objects instead of a single object's complex motion path. With a vector format such as Flash, you can use multiple instances of the same object moving simultaneously in different points and directions to create an impressive "swarming" effect with minimum bandwidth. Generally, always attempt to compress your movie's timespan by partially overlapping consecutive animation stages so that the entire clip looks more dynamic and the possible deficiencies of each single object's behavior are less noticeable.

Nonlinearity in animation is not as geometrically simple a concept as it sounds. For example, the animation shown on Fig. 3, b has a good deal of nonlinearity about it, too - although it does not use any rotation or acceleration. Note that this short movie consists of three obviously distinct stages, with two of them dominated by exploding and imploding circles, and the third intermediate stage featuring a left-to-right motion theme. As there are no clear boundaries between these stages, we feel that the entire movie's character changes at least twice during the cycle - and it is this change that contributes a nonlinearity component to the motion, making it interesting enough to watch from beginning to end.

In mathematical terms, nonlinearity means that the derivatives of the second, third, and higher orders for some variable - be it an object's size, spatial coordinate, character of motion or any other aspect - are non-zero. To find an analog for this animation concept, consider this fact from the science of acoustics. A sound containing only a first order sinusoid, without any overtones (harmonics), is very sharp, shallow, mechanistic. To get a rich, lively sound of an interesting timbre, you should add some higher order components to it. Analogously, to make a motion interesting, you should expand its linear base by some nonlinear components of gradual unfolding, growth, or inflexion.