In astronomy and geodesy, the visible horizon at a place is defined as the circle of contact between the Earth and which cone of visual rays?

Introduction / Context:
In astronomy, navigation, and surveying, several types of horizons are defined, such as true horizon, visible horizon, and celestial horizon. These horizons are based on geometric relationships between the observer, the Earth, and lines of sight. This question asks you to identify the correct geometric definition of the visible horizon in terms of a cone of visual rays touching the Earth surface.

Given Data / Assumptions:

• The visible horizon is the boundary where the sky appears to meet the Earth or sea for an observer at a given height above the surface.
• The options describe different cones of visual rays, some passing through the meridian or the point of observation and some not.
• The observer is located at a specific point on or above the Earth surface, and visual rays originate at the observer and extend outward.

Concept / Approach:
The visible horizon for an observer is determined by all the lines of sight from the observer that just graze the Earth surface, meaning they are tangent to the Earth. These lines of sight form a tangential cone whose vertex is at the observer point. Where this cone touches the Earth, it creates a circle of tangency. The curve of this circle, as projected onto the surface, is the visible horizon. Because the rays start from the observer, the cone necessarily passes through the point of observation. The reference to the local meridian (a great circle passing through the geographical north and south and the observer) is not essential to the definition. Therefore, the best description of the visible horizon is the circle of contact of the Earth and the cone of visual rays that pass through the point of observation and just graze the surface.

Step-by-Step Solution:

Verification / Alternative check:
Geodesy and navigation texts define the visible horizon as the intersection of the Earth surface with the cone of rays drawn from the observer that just touch (are tangent to) the Earth. This is different from the true or geometric horizon, which is often defined with respect to a plane tangent to the Earth at the observer location and can extend beyond physical obstructions. The visible horizon is limited by local obstacles and the curvature of the Earth, but the tangency concept remains central. No mention is usually made of the local meridian in the core definition of the visible horizon, reinforcing that options focusing on the meridian are not precise definitions of this horizon.

Why Other Options Are Wrong:

The cone of visual rays passing through the meridian of the place only is wrong because the visible horizon surrounds the observer in all directions, not just along the meridian plane.

The cone of visual rays that does not pass through the meridian of the place is incorrect because the meridian is irrelevant to the definition; excluding it does not capture the key idea of tangency from the observer.

The cone of visual rays that do not pass through the point of observation at all is clearly wrong, since visual rays, by definition, start at the observer and extend outward.

Common Pitfalls:
Students may get distracted by technical sounding terms such as meridian and think that any definition referring to the meridian is more precise. However, the essence of the visible horizon is about which rays just graze the Earth surface from the observer point, not about their relation to the meridian. Another pitfall is confusing visible horizon with other horizon types that use different reference planes. To answer such questions correctly, focus on the geometry: the visible horizon is the circle where the Earth is touched by the cone of visual rays that originate at the observer and are tangent to the surface.

Final Answer:
Therefore, the visible horizon is defined as the circle of contact of the Earth and the cone of visual rays that pass through the point of observation and just graze the Earth surface.