Linear and Quadratic Leap-and-Land Trajectory Tracking Algorithms

Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the other. In this note we first go over the basic procedure of moving forward tangentially from an already placed point then orthogonally returning to the curve. Next, we further consider higher order forward stepping procedures for greater accuracy. We note, however, that higher order methods, desirable for greater accuracy, may harbor latent instabilities. This note suggests ways of holding such instabilities in check, to have stable and highly accurate tracing methods. The note has several supporting numerical examples, including the rounding of a dynamical “snap-through” point.


Introduction
Pursuing a smooth implicit trajectory such as an energy level line is of great interest in numerical analysis and has attracted considerable attention [1] [2] [3] [4] [5] in recent years. In principle, such tracing entails marking close points on an implicit curve. Once a point is secured on the curve, a short distance forward leap is executed to place an initial point in the vicinity of the curve. Then, a returning iterative procedure is carried out to come back and land on the curve. To stay close to the curve, this stepping-out from the curve to the next initial guess is plausibly made by a tangential move at a prescribed step size. Orthogonal [2] iterative corrections are then carried out to bring the initially placed point ever closer to the curve.
The method, by dint of its orthogonal descent, is successful in rounding sharp bends as it follows the twists and turns of contorted trajectories. Still, the predictor leap ignores all previous information about the curving of the trajectory.
We propose here to improve the accuracy of the leap-out with a higher order parametrization of the trajectory by Bezier-like polynomial approximations. With their ability to bend, these approximations are likely to deposit an initial guess closer to the traced implicit curve for a hopefully quicker correction. We have shown that positioning an initial guess closer to the traced trajectory may well have both advantages of reducing the initial error in the following Newton-Raphson [6] correction, and also of improving the orthogonality of the landing approach along a gradient closer to the true one at landfall. , A x y . The total differential d d d

Linear Leap
x y z f x f y = + , for differentials d ,d x y , is reduced along the level line to where the first vector in the dot product is the gradient f and we propose to leap forward from point A and place predicted point B on the tangent line at distance ε . Evidently, for any such d ,d Stepping out from point A to point B is what we term a linear tangential leap.

Linearized Orthogonal Landing
We consider point ( ) 1 1 , B x y as situated on the curve , having the tangent line This system of equations is approximately solved with the differential linearization in which function f and its partial derivatives See also [7].

A Detailed Numerical Example
Consider the implicit function The tangent line to this curve at point B is or specifically The line orthogonal to this second tangent line is All as seen in Figure 2.
Next we apply the landing correction ( ) by the constrained linearization

Higher Order Leaps
For a quadratic leap, we start with three points already placed on the curve, say ( ) , , P x y . Parametrization of the interpolant to the trajectory over the three points is achieved with with parameter 0, 1 t t = = and 2 t = , at 1 2 , P P and 3 P , respectively. At 3 t = , Equation (31) predicts the coordinates

Bézier Curve Leaps
Use of partial derivatives allows the construction of a quadratic Bézier-like approximation to the trajectory [8] over only two points. Let , P x y be two such points on the trajectory. The quadratic parametric approximation to both x and y between 1 P and 2 P is of the general form where the coefficients of the approximation are to be determined from the initial and end conditions at 1 P and 2 P . Let x and y denote the derivatives of x and y with respect to parameter t. At any point t on the trajectory x The coordinates of point P are corrected thereby to 1 0 d and the procedure may be repeated.   2  4  3  1  2  3  1  2  3  1  2  3   3  5  1  1  3  2  4  2  2   with a turning-back that tends to become critically sharp. The computation described in Figure 4

Conclusion
We have considered in this paper the basic tangential leap followed by an orthogonal landing method for tracing implicit curves. We have then suggested a more accurate quadratic forward leap. However, such a leap may become unstable and we have shown how to restrain these possible instabilities to have a stable and accurate method. It would be of interest to consider next how to economize on the differentiations to have a more efficient method.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.