How to set a motion trajectory for a material point in two-dimensional space?

Question

Mathematics

How to set a motion trajectory for a material point in two-dimensional space? The trajectory should be, for example, as in the picture.

Can set the trajectory using a system of equations? If it is possible, then (sorry for the stupid question) how to make such a system?

Answer 1

1. Piecewise: splines in 4-5-, etc. points. The idea is that not only the values ​​of the function itself, but also the values ​​of its first derivative coincide at the "sewing" points, then the stitching occurs imperceptibly. (And for this, in simple words, it is necessary to set a curve segment, for example, between two points, to "capture" into the calculations one more point on the left and right in order to even out the derivative).
2. Immediately all the way: from the parameter t in two axes: x=X(t) and y=Y(t), where X(t) and Y(t) are polynomials of the degree you need. You can find their coefficients by solving the system using N known points. Unlike the piecewise task, the method is not scalable, that is, adding the next waypoints forces you to recalculate both polynomials in their entirety, and there is no guarantee that
Actually option 1 is a truncated, but more practical case of option 2.
3. You can expand any function of time into a series in terms of basis functions. For example, Fourier. Accordingly, you will get two Fourier series ФХ(t) and ФY(t) - for the motion functions along X and along Y.

Answer 2

You need to build a table with 3 columns:
0. Time t
1. dependence of the X coordinate on t X(t)
2. dependence of the Y coordinate on t Y(t)
Next, you can interpolate the resulting tabular values.
For example, the Lagrange method (but not quite effective). Either splines or piecewise linear functions.
PS Actually, about this has already been written above.
PSS Read about table value interpolation.