  Monocular SLAM has long grappled with the challenge of accurately modeling 3D
geometries. Recent advances in Neural Radiance Fields (NeRF)-based monocular
SLAM have shown promise, yet these methods typically focus on novel view
synthesis rather than precise 3D geometry modeling. This focus results in a
significant disconnect between NeRF applications, i.e., novel-view synthesis
and the requirements of SLAM. We identify that the gap results from the
volumetric representations used in NeRF, which are often dense and noisy. In
this study, we propose a novel approach that reimagines volumetric
representations through the lens of quadric forms. We posit that most scene
components can be effectively represented as quadric planes. Leveraging this
assumption, we reshape the volumetric representations with million of cubes by
several quadric planes, which leads to more accurate and efficient modeling of
3D scenes in SLAM contexts. Our method involves two key steps: First, we use
the quadric assumption to enhance coarse depth estimations obtained from
tracking modules, e.g., Droid-SLAM. This step alone significantly improves
depth estimation accuracy. Second, in the subsequent mapping phase, we diverge
from previous NeRF-based SLAM methods that distribute sampling points across
the entire volume space. Instead, we concentrate sampling points around quadric
planes and aggregate them using a novel quadric-decomposed Transformer.
Additionally, we introduce an end-to-end joint optimization strategy that
synchronizes pose estimation with 3D reconstruction.
