Thin-lens (geometrical optics) applicability of the lens formula: Under what conditions is the standard lens equation 1/f = 1/u + 1/v considered valid for engineering surveying optics and simple optical instruments?

Introduction / Context:
The thin-lens equation, written as 1/f = 1/u + 1/v, is used in geometrical optics to relate object distance u, image distance v, and focal length f. Surveying instruments (levels, theodolites, and telescopes) rely on this relation when forming images on cross-hair diaphragms. Knowing when this formula applies avoids errors in instrument design, adjustment, and basic optics questions in civil engineering exams.

Given Data / Assumptions:

• Lenses are idealized as thin, so thickness is negligible in first-order calculations.
• Distances are measured along the principal axis (paraxial approximation).
• Required accuracy is at the geometrical-optics level; wave effects and thick-lens corrections are not demanded.

Concept / Approach:
The thin-lens model assumes the principal planes coincide (negligible thickness) and rays make small angles with the axis (paraxial rays). Under these conditions, the simple reciprocal relation predicts image location and magnification accurately enough for most educational and field-calibration purposes. Off-axis points, high-aperture systems, or thick compound lenses need more advanced models (principal planes separated, nodal points, or matrix optics).

Step-by-Step Solution:

Verification / Alternative check:
For a simple telescope objective forming a sharp image on the diaphragm, predicted image distances closely match actual values for small apertures and modest fields, validating thin-lens use in most surveying optics questions.

Why Other Options Are Wrong:

• Each individual statement (thin assumption, axial distances, geometrical accuracy) is correct but incomplete alone; all together define the applicability domain.

Common Pitfalls:
Applying the formula to thick lenses without principal plane corrections; using skew-ray distances; expecting diffraction-limited predictions from a purely geometrical model.

Final Answer:
All of the above