Parametric surface design for static solar concentration: Enhancing plastic waste pyrolysis in sustainable housing

This paper develops a mathematical framework for the design of static parabolic solar concentrators with purely parametric geometries, aimed at replacing mechanically tracked systems. The central problem is to translate the apparent solar motion into a continuous parabolic surface whose focal line evolves consistently with the Sun’s trajectory. We formulate the astronomical parameters (solar declination, hour angle, azimuth-elevation) and show how each parabola, defined in its local plane, is rotated and assembled into a complete surface through parametric transformations. To ensure smoothness and predictable reflection properties, the resulting point cloud is interpolated using spline-based methods that guarantee C2 continuity, enabling reliable computation of tangents and normals. Practical implementation is achieved through an embedded device that verifies point placement during construction, bridging theoretical design with reproducible fabrication. While the broader motivation of this study is to address plastic waste through solar pyrolysis, the focus here is on the mathematical formulation that underpins the geometry and interpolation techniques necessary for building resilient, low-cost, and efficient static concentrators.