  Shape-constrained functional data encompass a wide array of application
fields especially in the life sciences, such as activity profiling, growth
curves, healthcare and mortality. Most existing methods for general functional
data analysis often ignore that such data are subject to inherent shape
constraints, while some specialized techniques rely on strict distributional
assumptions. We propose an approach for modeling such data that harnesses the
intrinsic geometry of functional trajectories by decomposing them into size and
shape components. We focus on the two most prevalent shape constraints,
positivity and monotonicity, and develop individual-level estimators for the
size and shape components. Furthermore, we demonstrate the applicability of our
approach by conducting subsequent analyses involving Fr\'{e}chet mean and
Fr\'{e}chet regression and establish rates of convergence for the empirical
estimators. Illustrative examples include simulations and data applications for
activity profiles for Mediterranean fruit flies during their entire lifespan
and for data from the Z\"{u}rich longitudinal growth study.
