OCTOPUS: A Versatile, User-Friendly, and Extensible Public Code for General-Relativistic Ray-Tracing in Spherically Symmetric and Static Spacetimes

This paper presents OCTOPUS, a relativistic ray-tracing algorithm developed within a Fortran-based, OpenMP-accelerated framework, designed for asymptotically flat, spherically symmetric curved spacetimes. The code efficiently and accurately computes key relativistic features -- including the black hole event horizon, photon rings, critical curves, and innermost stable circular orbits -- and simulates black hole shadows, redshift factor distributions, accretion disk images, toroidal images, as well as gravitational lensing, light curves, and gravitational radiation from hot-spots. OCTOPUS provides an automated, modular solution for qualitative studies of black hole observables and multi-messenger correlations between electromagnetic and gravitational signals in curved spacetime. Its implementation requires only the metric potential and its first-, second-, and third-order radial derivatives as input, ensuring low user barriers while remaining highly extensible and adaptable. Using a Schwarzschild black hole surrounded by a Dehnen-type dark matter halo, we thoroughly validate the algorithm's precision, efficiency, and functionality, and investigate how dark matter halo parameters affect observational signatures. Our results demonstrate that increasing the scale and density of the dark matter halo strengthens the spacetime's gravitational field, an effect clearly reflected in black hole images and supported by hot-spot light curve signatures. A future version of OCTOPUS, with expanded capabilities for axisymmetric spacetimes, is planned for release.