If you want to get all pedantic about it, most sprites (that is, images representing a character or other entity in a 2D game) are 'composite' if they are larger than one hardware sprite (usually a tile of a multiple of 8 pixels). But what I am talking about here are collections of these sprites arranged in such a way as to give the illusion of a large, dynamically animated object (usually a vehicle, creature or scenery element). You've probably seen composite sprites if you have played a few games from the days before polygons became the norm, they are surprisingly common.

One of the simplest applications of composite sprites is to create a 'snake' from individual body segments. These segments can all look the same and have rotational symmetry, giving the impression of a tube or cylinder. The rules used to govern the motion of each segment in relation to the others can give different effects: by changing the size of each segment you can get a pulsating effect (Biohazard Battle).

By using the segments to plot points along a straight or curved line you can get the effect of a telescopic limb (Gunstar Heroes, Dynamite Headdy). Anyone who has seen diagrams of robotic limbs can imagine how this can be used. Many platform games use this effect to make a 'bendy pole' that the hero can jump on at different heights and angles to jump higher (Psycho Fox, Decap Attack, Rayman). And of course, we have swinging vines.

If your set of components has more than just spheres, you can create more complex effects. Treasure (probably the masters of the form) used composite sprites (small and large spheres and a selection of angled blocks) to create their transforming boss characters such as the Seven Force. If sprite scaling (or enough memory to store different sized versions of your components) is available, you can create the illusion of depth (Spinderella in Dynamite Headdy, the dragons in Magic Carpet and Space Harrier).

If memory isn't an issue, composite sprites can be constructed from a wide variety of specialised parts to take advantage of their ability to move independently (most of the bosses in the Metal Slug games used this, including the nice touch of moving some parts rapidly one pixel up and down to give the impression of engine vibration).

Composite sprites were seldom used exclusively in a game, with the exception of PF Magic's (?) Ballz which was a fighting game where all the fighters were composed of spheres (and looked a bit like artist's posable figures).

Although in the fancy 21st century you'd think that composite sprites would be no longer used, they do occasionally crop up due to their cheapness to draw organic looking things with - the hair of the creatures in Black & White and the Karmaramans in StarTopia both used this effect. Eventually this sort of use slopes off into the world of procedural texturing.

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