Exact overlap of two Bézier curves isn't possible as the original picture shows, i.e., over a finite interval, whereas the curves diverge outside the interval. See Lemma 1 in (Hu et al., 1997) or Theorem 5.9.1 in (Patrikalakis and Maekawa, 2003).

Overlapping curvesNevertheless, an overlap bounded by endpoints, as depicted, is possible when two Bézier curves b(u), c(v) are simply segments of the same (untrimmed) polynomial curve. Sánchez-Reyes (2014) describes a procedure for detecting and computing this potential overlap (for properly parameterized curves). As a necessary condition, b(u), c(v) must have parametrizations related by a linear transformation, which can be determined from the last non-vanishing derivatives. The overlap occurs precisely over the overlapping portions of their domains, so the overlapping condition amounts to the coincidence of the Bézier polygons for such subdomains.

REFERENCES

Patrikalakis, N.M., Maekawa, T., 2003. Shape Interrogation for Computer Aided Design and Manufacturing. Springer.

Hu, C.-Y., Maekawa, T., Patrikalakis, N.M., Ye, X., 1997. Robust interval algorithm for surface intersection. Comput.-Aided Des. 29 (9), 617-627. https://doi.org/10.1016/S0010-4485(96)00099-1

Sánchez-Reyes, J., 2014. The conditions for the coincidence or overlapping of two Bézier curves. Applied Mathematics and Computation 248, 625-630. https://doi.org/10.1016/j.amc.2014.10.008