NBPhO 2026

Difficulty hard

Prerequisites

• Thin-lens imaging equation $1/u + 1/v = 1/f$
• The chief ray through the lens centre is undeviated
• Tangent lines from an external point to a circle and the angle-bisector property
• Similar triangles and ruler-and-compass length constructions
• Polar parametrisation of plane curves

Learning objectives

• Recognise that $P, L, P'$ collinear forces the image curve $P'$ to share its angular extent at $P$ with the lens-centre circle
• Decompose the lens equation along an oblique chord: $|PL|\cos\alpha = u$, $|LP'|\cos\alpha = v$
• Manipulate $1/u + 1/v = 1/f$ into the chord form $1/|PL| + 1/|LP'| = \cos\alpha/f$
• Construct an auxiliary point whose locus is straight by isolating a quantity independent of the moving parameter
• Use a ruler-and-compass harmonic-mean (similar-triangles) construction in a physics problem
• Cross-check a multi-step construction by exploiting redundant pairings (both branches of the curve, both source points)

Watch out for

• Confusing the tangent point on the curve with the tangent point on the lens circle. The two tangent points lie on the same tangent line, but at different distances from $P$ (the curve point is further out by a factor $\mu = u/(u-f) > 1$). Only the tangent line — not the contact points — is shared.
• Mistaking interior tangent rays of the figure-eight curve $Q'$ for outermost tangents. Only the rays bounding the angular extent of the curve as seen from $Q$ are tangent to the lens circle; rays touching internal loops are not.
• Treating $|PL|$ and $|LP'|$ as $u$ and $v$ directly. They are Euclidean lengths along the chord, not axial distances; the lens equation in chord form picks up a $\cos\alpha$ factor.
• Choosing the wrong intersection of line $PP'$ with the lens circle. In fact both intersections give the same auxiliary point $M$, so the choice is irrelevant — but the construction is fragile if one assumes only one matters.
• Skipping the consistency checks. Four tangent lines from two source points yield four perpendicular distances to the centre that must agree; the lines of $M$'s built from $P$ and from $Q$ must come out parallel. These are free safety nets — use them.