#flatearth

Abstract

This paper introduces an optical model explaining the apparent disappearance of distant objects and the visual curvature of the horizon on a flat Earth. It combines Perspective Compression, Grazing Occlusion, and Foreground Texture Scaling to account for how depth cues emerge from nonlinear reductions in visual angle with distance. Treating Earth as a continuous flat plane, the model attributes horizon dip and curvature to non-uniform angular reduction rather than spherical geometry.

Introduction

Observations of distant horizons reveal that objects diminish in size and vanish progressively downward. Traditional models attribute this to Earth’s curvature, but here we propose an alternative: optical effects from viewing vast, flat surfaces. The model explains horizon curvature and bottom-up occlusion through perspective transformations and textural occlusion, eliminating the need for physical curvature.

Perspective Compression

For an object of height hh at distance LL, its visual angle is approximately tan⁡(θ)≈hLtan(θ)≈hL. Since h/Lh/L is hyperbolic, height decreases nonlinearly with distance, causing a curved reduction in both height and width. This dual compression creates the illusion of the ground rising toward the horizon.

Grazing Occlusion

At low sighting angles, foreground textures (e.g., soil grains, waves) become large in angular size. These elements physically block the observer’s line of sight to distant objects, causing the lowest parts to disappear first. This grazing-angle occlusion explains why distant objects vanish bottom-up.

Foreground Texture and Roughness

The horizon emerges from the cumulative effect of countless local occlusions. Nearby features dominate angular space, merging into a continuous horizon line. The surface remains flat, but the horizon appears curved due to rapid angular compression at great distances.

Non-Uniform Scaling and Horizon Curvature

As distance increases, angular reduction accelerates, producing a curved scaling of the visual field. At maximum distance (LmaxLmax​), the ground plane compresses into a near-infinitesimal line, creating the Horizon Dip Angle—a result of perspective transformation, not physical curvature.

Explaining Anomalous Observations

Hidden Objects (Occultation): Bottom-up disappearance occurs when foreground occlusion and perspective compression combine. The base of distant objects vanishes first due to extreme angular reduction.

Horizon Curvature: Rapid angular compression at large distances distorts the horizon into a convex curve, mimicking a physical dip.

Foreground Texture and Roughness

Local occlusions merge across distance to form the horizon. The horizon’s continuity arises from the integration of countless foreground textures, while the surface itself remains flat.

Non-Uniform Scaling and Horizon Dip

Observer elevation alters the visual mapping, shifting the angular field and producing the dip angle. This dip reflects perspective origin changes, not surface curvature.

Object at Point XX: Angular Reduction

The visual angle θXθX​ for a distant object at point XX is given by:θX=2arctan⁡(hX2LX2),θX​=2arctan(LX2​hX2​​),

where hXhX​ and LXLX​ are height and distance. As LXLX​ increases, θXθX​ decreases, causing visual contraction and eventual disappearance.

Conclusion

The Flat-Planar Perspective Compression Model unifies horizon curvature, dip angle, and bottom-up occlusion as optical effects from perspective transformations and foreground occlusion. These phenomena arise from nonlinear angular reduction on a flat Earth, challenging traditional curvature-based explanations. Further research is needed to refine mathematical models of angular scaling and visual perception.