Nonlinear Design. Curvature range
esides being visually multifarious, Bezier curves are capable of expressing a surprisingly wide range of emotions. Even when looking at simple examples of curves with no design context surrounding them, you can't suppress the feeling that some of them are elegant and some are awkward, some are sharply dynamic and some are languid and sluggish, some are intriguing and others are plain dull. This may look like a mystery - it's so simple an object, just where in it can one hide so many meanings and connotations?
The fact is, Bezier's simplicity is deceptive. Such a curve may look graphically obvious, but even from the mathematical viewpoint, four points and a third order equation that define a Bezier curve are considerably more complex than what is used in plain and bloodless geometric primitives. However, if we don't happen to be mathematicians, do we have any method to sensibly classify the curves we use, to figure out what defines a curve's aesthetic impact, and to get any idea of what sort of curve to use and when?
The very word "curve" suggests that the concept of curvature may have some important implications for this geometric object. Curvature (or, to be precise, curvature radius) has, too, its precise definition in mathematical terms, but for our purposes it is more important that this parameter is intuitively obvious, too: we can see at once what curves are more curved and what are closer to a straight line.
In fact, curvature is a property not of a curve, but of a point on it, as different parts of a curve may have different curvature, from very low (almost linear fragments) to very high (acute bends). Therefore, even more important for design is the concept of curvature range. The greater the difference between the most and the least curved fragments of a line, the bigger its curvature range is. From this viewpoint, a straight line and a circle are similar, as they both have zero curvature range (in both cases curvature is constant, although for a straight line it is zero and for a circle it is non-zero).
It is easy to see that a noticeable curvature range is what brings life, dynamics, expression to a curve (Fig. 3). That's why, for example, ellipses, whose curvature is smoothly changing along the contour, are often more interesting for perception than plain circles whose constant curvature is, for a designer, almost "linear" looking in its dull and immaculate symmetry.
|Fig. 3: Extending the curvature range: from calm to storm|
|Those interested in mathematics might notice that, if we regard the curve as a function graph, its curvature range is roughly equivalent to the maximum value of that function's third derivative. Indeed, the first derivative represents the slant of the line in each point, the second corresponds to curvature, and the third is the rate of change (i.e. velocity of increasing or decreasing) of the line's curvature. The wider is the range of curvature changes, the higher the value that the third derivative can achieve.|
Revised: Feb. 12, 1999