1. Scientific Principles & Anamorphic Projection in Biomedical Imaging
The Biomedical Polygram Laboratory investigates the mathematical convergence of projective geometry, spatial depth illusions, and computational biomedical imaging. Anamorphosis is an optical phenomenon wherein an image is distorted and fragmented across three-dimensional Euclidean coordinates in such a manner that it can only be reconstituted into an undistorted, coherent planar form when viewed from one unique, mathematically predetermined vantage point in space.
In modern computational medicine, biological structures are inherently multi-planar and volumetric. Modalities such as high-resolution Fundus Photography, Micro-Computed Tomography ($\mu\text{CT}$), Magnetic Resonance Imaging (MRI), and Scala Tympani Cochlear Array insertion mappings require clinicians and biomechanical engineers to mentally or algorithmically synthesize fragmented spatial cross-sections into coherent continuous structures. When cross-sectional slices or scattered functional signals are viewed out of axial alignment, severe parallax errors occur, masking the underlying physiological topology.
This simulator operates as an interactive didactic instrument where planar biological specimens are dissected into a discrete spatial lattice. By dispersing individual planar shards along the longitudinal optical axis ($Z$) while counter-scaling their physical dimensions proportionally to camera distance, the simulation constructs a precise homography that collapses into an ideal continuous image exclusively at the canonical focal orientation.
3. Mathematical Formulation & Projective Geometry Architecture
Let the virtual observer's viewpoint be positioned along the optical $Z$-axis at focal distance $d_{\text{cam}} = (0, 0, d)$. A point $\mathbf{P} = (x, y, z)$ in world space projects onto the normalized projection plane at $(x', y')$ defined by:
$$x' = x \cdot \left(\frac{d}{d - z}\right), \quad y' = y \cdot \left(\frac{d}{d - z}\right)$$
To achieve seamless visual reconstruction when viewed from the canonical front orientation ($\mathbf{R} = \mathbf{I}_{3\times 3}$), each shard $(i, j)$ at $(x_{\text{aligned}}, y_{\text{aligned}}, 0)$ is displaced by depth offset $\Delta z \in [-\frac{Z_{\text{spread}}}{2}, +\frac{Z_{\text{spread}}}{2}]$ and scaled inversely by factor $S$:
$$S = \frac{d - \Delta z}{d}$$
$$x_{\text{scatter}} = x_{\text{aligned}} \cdot S, \quad y_{\text{scatter}} = y_{\text{aligned}} \cdot S$$
$$\text{Width}_{\text{shard}} = W_0 \cdot S, \quad \text{Height}_{\text{shard}} = H_0 \cdot S$$
Rotational deviation is calculated via the geodesic metric across symmetry quaternions:
$$\Delta \theta = \min_{\mathbf{q}^* \in \mathcal{Q}^*} 2 \arccos\left(\min\left(1, |\mathbf{q} \cdot \mathbf{q}^*|\right)\right)$$