Hubert curves are continuous curves which pass at least once through each point of a square or cube. They can be defined as the limit of a sequence of mappings of successively smaller dyadic subintervals of the unit interval to small subsquares or subcubes. These finite approximations are useful for coding images or volumes.

The animation starts with a circular tube, which deforms continuously through smooth piecewise-circular approximations to the 2D Hubert curve.The fourth order approximation is shown in figure i.The tube changes to a square cross section, and by the fifth order approximation, shown in figure 2, it is touching itself. The circular arcs then square off so that the approximation appears to cover the square, and the surfaces become partially transparent, to reveal the glowing volume density shown in figure 3. The curve is yellow, with its parametrization indicated by short segments in light green, purple, and dark green, repeated in eight cycles. The origin of the
curve as the anima-1 tion progresses, so that it looks like a crawling colored snake.This indicates the path of the curve even when the whole square is filled up. Volume rendering extends this color indication of the path to three dimensions, and the marble tile background helps in perceiving the volume opacity. The closed curve breaks between a purple and dark green band and the front part begins filling up the cube as a 3D Hilbert curve. By figure 4, the second purple band has started into the cube, and by figure 5, the whole curve has moved over to the cube, and again becomes closed.
Figures 3 through 5 show high order approximations, using a recursive rendering algorithm which only subdivides squares or cubes which are not of homogeneous color. Between any two frames in the animation, many tiny squares or cubes change color, so the motion is jerky. Figure 6 shows a switch to the third order approximation, where the color boundaries move smoothly.The surface is incised between cubes which are not adjacent on the approximation path, and in figure 7, the incisions have widened to reveal the squared off tube. This is the reverse of the process, not illustrated here, between figures 2 and 3. Finally in figure 8 the tube becomes rounded, to give the 3D version of figure i. In the animation sequence including figures 5 through 8 the camera is slowly rotated about the curve, so that motion parallax can help make the 3D structure more evident.