Editing Homotopy principle (section)

=== A car in the plane ===
As another simple example, consider a car moving in the plane. The position of a car in the plane is determined by three parameters: two coordinates <math>x</math> and <math>y</math> for the location (a good choice is the location of the midpoint between the back wheels) and an angle <math>\alpha</math> which describes the orientation of the car. The motion of the car satisfies the equation

:<math>\dot x \sin\alpha=\dot y\cos \alpha.</math>
since a non-skidding car must move in the direction of its wheels. In [[robotics]] terms, not all paths in the task space are [[holonomic (robotics)|holonomic]].

A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in the plane. In this case the non-holonomic solutions are not only [[homotopic]] to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space) – note that this approximates both the position and the angle of the car arbitrarily closely. This implies that, theoretically, it is possible to parallel park in any space longer than the length of your car. It also implies that, in a contact 3 manifold, any curve is <math>C^0</math>-close to a [[Legendrian knot|Legendrian]] curve.
This last property is stronger than the general h-principle; it is called the <math>C^0</math>-'''dense h-principle'''.

While this example is simple, compare to the [[Nash embedding theorem]], specifically the [[Nash–Kuiper theorem]], which says that any [[short map|short]] smooth (<math>C^\infty</math>) embedding or immersion of <math>M^m</math> in <math>\mathbf{R}^{m+1}</math> or larger can be arbitrarily well approximated by an isometric <math>C^1</math>-embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially similar "wrinkling" – or rather, circling – technique to the car in the plane, though it is much more involved.