Here, the curvature of Swan's motion trajectory is given by an equation in a user program; obviously, these motions are in the curvature mode. An automobile's motion is controlled in the curvature mode, where the position of its steering wheel determines the path curvature.

In this motion category, the curvature *k* is expressed as *k* = *k*(*s*), where *s* is the arc length. This equation is called a *natural equation* in differential geometry. Three simple examples are presented here:

(1) Clothoid curve (or Cornu spiral): The first subclass is the set of Clothoid curves:

*κ*= *κ*(*s*) = *As* + *B*, where *A* and *B* are constants.

The curvature *k* is a linear function of the arc length *s*. Traditionally, Clothoid curves have been heavily used in *highway and railroad design* because of their favorable property of *curvature continuity*. Watch the video, **Clothoid Curve**. See the trajectory. The actual equation of this specific Clothoid curve is

*κ* = *κ*(*s*)= 0.00005 *s* - 0.05, with 0 ≤*s* ≤ 2000.

The total arc length is 2000 (cm). At *s* = 1000; the curvature becomes 0; this point is called the* inflection point*.

(2) Open serpentine: A second subclass is the set of curves in which the curvature is a function of the cosine function:

*κ* = *κ*(*s*)= *A* cos(*Bs*), where *A* and *B* are constants.

See an example video demonstration, **Open Serpentine**. Daniel, one of my grandsons, is interacting with Swan. Swan makes three complete cycles. See the trajectory. We call this set of motions *open serpentines*. The specific natural equation of this motion is

*κ*= *κ*(*s*) = 0.05 cos(*s*/40), 0 ≤ *s* ≤ 240*π*.

Although this curve is generated by the cosine-curvature natural equation, the generated trajectory itself is far from the cosine curve, i.e.,* y* = *a* cos(*bx*). If this natural equation is changed into 0.03 cos(*s*/40) or 0.1 cos(*s*/40), the trajectory becomes totally different.

(3) Closed serpentine: A third subclass is the set of curves in which the curvature is a function of the cosine function with an offset:

*κ* = *A* cos(*Bs*) + *C*, where *A*, *B* and *C* are constants.
The principle of this curve is as follows: (1) the constant curvature *C* part generates a circle, (2) the term *A* cos(*Bs*) part generates an open serpentine, and (3) they are combined to make a "perturbed circle." Watch the video, **Closed Serpentine**. Swan makes six complete cosine cycles and comes back to the starting frame. See the trajectory. The specific natural equation of this motion is

*κ* = 0.04 cos(*s*/40) + 1/240, 0 ≤ *s* ≤ 480*π*.
We call this set of motions *closed serpentines*.

Because the user can define any natural equation for a motion, he or she can create an infinite number of new curvature-controlled motions.