Refraction at a Spherical Surface

Snell's law bends a ray at a flat window and leaves it going straight again — offset, but never gathered. Curve that window into a slice of a sphere and something new happens: rays from a single point, spreading out, are each bent by a slightly different amount, and if the curvature is right they all cross again at one place. A curved boundary between two transparent media does not merely deflect light — it forms an image. That is the whole of lens optics in one sentence, and this article is where it begins, at a single curved surface. This is the atom of the lens. A magnifying glass, a spectacle lens, a camera objective, the cornea of your own eye — every one of them is two spherical surfaces back to back, and you cannot understand the pair until you can do one. The governing relation for paraxial rays is

\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}

three distances and two refractive indices, and it is the bridge from "a ray bends at a boundary" to "a boundary makes a picture". Notice at once what is not here: no focal length yet, no lens, no thin-glass approximation — just one interface between a medium of index $n_1$ and a medium of index $n_2$ , with a radius of curvature $R$ . The signs are the New Cartesian signs already used for mirrors in The Mirror Formula — distances against the incident light are negative, $R$ counted from the surface to its centre of curvature — but the equation itself is new, and the next two articles build the lens-maker's formula and the thin-lens formula directly on top of it. Get this one surface right and lenses become bookkeeping.

Key takeaways

A single spherical surface between media $n_1$ and $n_2$ obeys $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ for paraxial rays — the first relation in this book that forms an image at a curved boundary.
The right-hand side $\dfrac{n_2 - n_1}{R}$ is the power of the surface: a bigger index jump or a tighter curve bends light harder.
New Cartesian signs (as in the mirror articles): real object $u < 0$ ; $R > 0$ when the centre of curvature is on the outgoing-light side; the image distance $v$ is measured into the second medium.
Worked anchor: an object 60 cm in air, in front of a convex glass surface ( $n_1 = 1$ , $n_2 = 1.5$ , $R = +20$ cm), gives $v = +180$ cm — a real image formed inside the glass.
Flatten the surface ( $R \to \infty$ ) and the equation collapses to $n_2/v = n_1/u$ — the apparent-depth result, so a pool looks $3/4$ as deep.
Key idea: one curved refracting surface locates an image through $n_2/v - n_1/u = (n_2-n_1)/R$ ; stack two of them and you have a lens, which is the entire next chapter.

The core insight

Refraction at a single spherical surface between indices $n_1$ and $n_2$ is governed, for paraxial rays, by $n_2/v - n_1/u = (n_2-n_1)/R$ in New Cartesian signs; it is the building block from which the lens-maker's formula and the thin-lens equation are assembled, surface by surface.

Where this lives today

The single refracting surface is not a textbook abstraction — it is the front of your eye. The cornea, the transparent dome over the iris, is a convex surface separating air ( $n \approx 1$ ) from the aqueous humour behind it ( $n \approx 1.34$ ), and because the index jump there is large, the cornea does about two-thirds of the eye's total focusing; the lens inside only fine-tunes it. Ophthalmologists model corneal reshaping in LASIK surgery with exactly equation (1), changing $R$ by micrometres to shift where the image lands on the retina. The same one-surface physics designs the plano-convex front element of a camera or projector, sizes the curved window of a fibre-optic lens that couples light into a strand, and governs the immersion objectives of high-end microscopes, where a drop of oil is chosen to make $n_1$ nearly equal to $n_2$ and so tame the surface's aberrations — the aplanatic trick Abbe invented. Every one of these is the atom this article isolates; the instruments of the coming chapters are simply many atoms in a row.

The physical picture

Everything in this chapter rests on one shift of viewpoint. Snell's law, from two articles ago, is a rule about one ray crossing one boundary: it tells you the new direction and nothing else. But an object is a point that sends out a fan of rays in every direction, and imaging is the question of whether that fan is put back together. At a flat surface it never is — parallel-in stays parallel-out, a diverging fan stays diverging (only shifted), and you see the object "through" the glass, apparently displaced but never focused. At a curved surface the fan is treated unequally: a ray striking near the axis meets the surface almost head-on and bends little, while a ray striking higher up meets a more slanted piece of surface and bends more. Curvature is precisely the machine that bends the outer rays harder than the inner ones — which is exactly what it takes to fold a diverging fan back to a point. We keep only paraxial rays — those that stay close to the axis and make small angles with it — for the same reason the mirror chapter did: only then do all the rays from a point cross at one image point, and only then are the angles small enough that $\sin\theta$ and $\tan\theta$ may both be replaced by the angle itself. Rays far from the axis focus slightly short of the paraxial image; that blurring is spherical aberration, a real defect of real lenses that we set aside here and that belongs to a later, university-level treatment.

Figure

Refraction at a convex spherical surface: an object O in medium n1 sends a paraxial ray that strikes the surface, bends toward the normal on entering the denser medium n2, and meets the axis again at a real image I. The pole is P, the centre of curvature is C, the object distance is u, the image distance is v and the radius is R.

The working equation

Two or three lines of geometry (done in full in the derivation below) turn Snell's law at a curved surface into a single relation between the object distance, the image distance, the two indices and the radius:

\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \qquad (1)

where: $u$ is the object distance and $v$ the image distance, both measured from the pole $P$ where the axis pierces the surface; $n_1$ is the index of the medium the light starts in and $n_2$ the index it ends in; and $R$ is the radius of curvature of the surface. Read the left side as "the second medium claims the image, the first medium owned the object", and the right side as a single number built from the surface alone. The image forms in the second medium, so $v$ is genuinely a distance into the glass — a fact students routinely forget, and one the worked example below makes vivid. The right-hand side deserves a name. The quantity

P_{\text{surf}} = \frac{n_2 - n_1}{R} \qquad (2)

is the power of the single surface — how strongly this one interface converges light. It grows with the index jump $n_2 - n_1$ (a bigger optical mismatch bends harder) and with the curvature $1/R$ (a tighter surface bends harder). This is the quantity that simply adds when you stack two surfaces to make a lens, and that addition is the entire content of the next article.

The signs are the mirror's signs — but the equation is not

The convention here is exactly the New Cartesian convention set out in The Mirror Formula and Spherical Mirrors: Concave and Convex in the previous chapter: measure everything from the pole, take the incident-light direction as positive, so distances in front (where a real object sits) are negative and distances beyond the surface positive; $R$ is positive when the centre of curvature $C$ lies on the outgoing-light side of the surface. A real object therefore has $u < 0$ , just as for a mirror. But do not confuse the conventions, which agree, with the formulas, which differ. The mirror formula is $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ with a plus and magnification $m = -v/u$ . Refraction carries the two indices and a minus: $\dfrac{n_2}{v} - \dfrac{n_1}{u}$ . Reflection sends light back into the same medium; refraction passes it into a new one, and the index bookkeeping is the visible sign of that difference. Keep the shared sign convention, but never write the mirror's equation for a refracting surface.

Derivation ARefraction at a spherical surface, from Snell and two triangles

The goal is to prove $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ for a paraxial ray crossing a single convex surface, using nothing beyond Snell's law in its small-angle form and the exterior-angle theorem for triangles. We work throughout with plain positive lengths and attach the New Cartesian signs only in the last line — exactly the discipline the mirror derivation used — because that keeps the geometry honest and the signs bookkeeping.

Step 1. Draw the ray. An object $O$ sits on the axis in medium $n_1$ ; a paraxial ray leaves it, strikes the surface at a point $N$ a small height $h$ above the axis, and refracts into medium $n_2$ , crossing the axis at the image $I$ . The normal at $N$ is the radius $NC$ , since every radius of a sphere is perpendicular to its surface, so the angle of incidence $i$ and refraction $r$ are measured from $NC$ . Let the incident ray make angle $\alpha$ with the axis, the refracted ray make angle $\beta$ with it, and the radius $NC$ make angle $\gamma$ with it.

Step 2. Get $i$ and $r$ from the geometry. In triangle $ONC$ , the angle $i$ at $N$ is an exterior angle, so it equals the sum of the two remote interior angles: $i = \alpha + \gamma$ . In triangle $INC$ , the angle $\gamma$ at $C$ is the exterior angle, so $\gamma = r + \beta$ , giving $r = \gamma - \beta$ . No approximation yet — this is pure triangle geometry.

Step 3. Apply Snell's law in the small-angle form. For paraxial rays every angle is small, so $\sin\theta \approx \theta$ , and $n_1 \sin i = n_2 \sin r$ becomes $n_1\, i = n_2\, r$ . Substitute Step 2: $n_1(\alpha + \gamma) = n_2(\gamma - \beta)$ , which rearranges to $n_1\alpha + n_2\beta = (n_2 - n_1)\gamma$ .

Step 4. Turn the angles into distances. For small angles each angle is its height-over-distance: $\alpha \approx h/OP$ , $\beta \approx h/PI$ , $\gamma \approx h/PC$ , where $OP$ , $PI$ and $PC$ are the plain lengths from the pole. Substitute and cancel the common $h$ : $\dfrac{n_1}{OP} + \dfrac{n_2}{PI} = \dfrac{n_2 - n_1}{PC}$ .

Step 5. Attach New Cartesian signs. The object is in front, so $OP = -u$ ; the image is beyond, so $PI = +v$ ; the centre of curvature is on the outgoing side, so $PC = +R$ . Then $\dfrac{n_1}{-u} + \dfrac{n_2}{v} = \dfrac{n_2 - n_1}{R}$ , i.e. $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ . A quick test: $n_1 = 1$ , $n_2 = 1.5$ , $u = -60$ , $R = +20$ gives $v = +180$ cm, the real image of the worked example. Verified: the anchor $v = +180$ cm and $v = -180$ cm, the two focal lengths $f_2 = 60$ cm and $f_1 = 40$ cm, and the surface power $0.025\ \mathrm{cm^{-1}}$ are recomputed in verify_hs2-refraction-spherical-surface.py.

Go deeper · Derivation B · 1 minThe flat-surface limit is apparent depth

Derivation BThe flat-surface limit is apparent depth

The goal is to show that a flat boundary is just the $R \to \infty$ special case of equation (1), and that it reproduces the apparent-depth result — a coin under water looking raised — that the glass-slab article obtained straight from Snell. This is the check that the new machinery contains the old, and it fixes forever which index goes on top.

Step 1. Flatten the surface. A plane boundary is a sphere of infinite radius, $R \to \infty$ , so the right-hand side of equation (1) vanishes: $\dfrac{n_2}{v} - \dfrac{n_1}{u} = 0$ , that is $\dfrac{n_2}{v} = \dfrac{n_1}{u}$ , or $v = \dfrac{n_2}{n_1}\,u$ . A flat surface forms no real image — it only relocates the object along the axis.

Step 2. Set up the coin. A coin lies at real depth $d$ under water and is viewed from directly above in air. The light travels water-to-air, so $n_1 = n_{\text{water}} = 4/3$ (the medium the light starts in), $n_2 = 1$ (air). The object is in front: $u = -d$ .

Step 3. Substitute: $v = \dfrac{n_2}{n_1}\,u = \dfrac{1}{4/3}\times(-d) = -\dfrac{3}{4}d$ . The image sits at depth $\tfrac{3}{4}d$ , on the same side as the coin — virtual, raised. So the apparent depth is $d/n = \tfrac{3}{4}d$ , and for $d = 4\ \mathrm{cm}$ the coin appears $3.0\ \mathrm{cm}$ down, exactly the figure the glass-slab article found from Snell alone.

Step 4. Read the lesson. The index that sits on top in "apparent $=$ real $/n$ " is the index of the medium the light must leave — the object's medium — which is why a pool ( $n = 4/3$ ) looks $3/4$ as deep and a diamond's flaws ( $n = 2.42$ ) look less than half as deep. Equation (1) contains this as its flattest special case, which is a good sign that the curved cases are trustworthy too. Verified: the flat-surface relation $v = (n_2/n_1)u$ and the coin result $4\ \mathrm{cm} \to 3.0\ \mathrm{cm}$ are recomputed in verify_hs2-refraction-spherical-surface.py.

Figure

The flat-surface limit R going to infinity collapses the surface equation to apparent depth: a coin at real depth d under water sends rays that bend away from the normal at the flat surface, and an eye above sees a raised virtual image at depth d over n, so a 4 cm deep coin in water appears only 3 cm down.

Limiting cases

Case	Result	Meaning
$R \to \infty$ (flat surface)	$n_2/v = n_1/u$	apparent depth: a pool looks $d/n$ deep, no real image forms
$u \to -\infty$ (distant object)	$v = n_2 R/(n_2 - n_1) = f_2$	the second focal point — parallel light converges inside the glass
$v \to \infty$ (parallel emergent rays)	$u = -n_1 R/(n_2 - n_1) = -f_1$	the first focal point — the two focal lengths differ, $f_1/f_2 = n_1/n_2$
$n_2 = n_1$ (no index jump)	$1/v = 1/u$ , $v = u$	matched media: light passes as if the surface were not there
$R > 0$ , real object, $n_2 > n_1$	$v > 0$ for a distant object	a convex denser surface converges — the raw ingredient of a lens

Common trapIt is not the mirror formula, and v lives in the second medium

Two mistakes account for almost every lost mark here. The first is writing the mirror relation for a refracting surface — reaching for $1/v + 1/u = 1/f$ out of habit. There is no single $f$ at one surface, there is no plus sign, and the two indices must appear: the correct object is $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ , with $n_1$ the medium light leaves and $n_2$ the medium it enters. Swap the two indices and every answer inverts. The second mistake is forgetting where the image is: because the light ends up in medium $n_2$ , the image distance $v$ is measured into that second medium, and a positive $v$ means a real image sitting inside the glass, not in the air where the object was. If a worked answer comes out as a real image floating back on the object's side, you have almost certainly mislabelled which medium is $n_1$ and which is $n_2$ . Write the two indices down, with an arrow showing the direction of travel, before you touch the algebra.

Worked examples

Board level — a real image formed inside glass (CBSE and NEET pattern)

One end of a long glass rod ( $n_2 = 1.5$ ) is ground into a convex spherical surface of radius 20 cm. A small object is placed in air ( $n_1 = 1$ ) on the axis, 60 cm in front of this surface. Locate the image.

Step 1. Fix the signs first. Real object in front: $u = -60\ \mathrm{cm}$ . The surface bulges toward the object, and its centre of curvature $C$ lies inside the glass — on the outgoing side — so $R = +20\ \mathrm{cm}$ . Light goes air-to-glass, so $n_1 = 1$ , $n_2 = 1.5$ .

Step 2. Substitute into equation (1): $\dfrac{1.5}{v} - \dfrac{1}{-60} = \dfrac{1.5 - 1}{20}$ , that is $\dfrac{1.5}{v} + \dfrac{1}{60} = \dfrac{0.5}{20} = 0.025\ \mathrm{cm^{-1}}$ .

Step 3. Solve: $\dfrac{1.5}{v} = 0.025 - \dfrac{1}{60} = 0.025 - 0.016\overline{6} = 0.008\overline{3}\ \mathrm{cm^{-1}}$ , so $v = \dfrac{1.5}{0.008\overline{3}} = \boxed{+180\ \mathrm{cm}}$ .

Step 4. Read the sign. $v > 0$ means the image is on the outgoing side — a real image, 180 cm inside the glass rod. This is the single most surprising thing about a curved surface: air-to-glass refraction alone, with no second surface, can throw a real image, and it lands deep in the second medium. Move the object closer, to $u = -30\ \mathrm{cm}$ , and $\dfrac{1.5}{v} = 0.025 - \dfrac{1}{30} = -0.008\overline{3}$ , giving $v = -180\ \mathrm{cm}$ — now a virtual image, back on the object's side, because the object has moved inside the surface's first focal point.

Competitive level — the two focal points of one surface (JEE pattern)

For the same surface ( $n_1 = 1$ , $n_2 = 1.5$ , $R = +20$ cm), find (a) where a very distant object is imaged, and (b) where an object must sit so that the refracted rays leave parallel. Comment on the two answers.

Step 1. A distant object means $u \to -\infty$ , so $n_1/u \to 0$ and equation (1) becomes $\dfrac{n_2}{v} = \dfrac{n_2 - n_1}{R}$ . Hence $v = \dfrac{n_2 R}{n_2 - n_1} = \dfrac{1.5 \times 20}{0.5} = \boxed{+60\ \mathrm{cm}}$ . This is the second focal length $f_2$ : parallel light from far away converges 60 cm inside the glass.

Step 2. For the refracted rays to leave parallel, the image is at infinity, $v \to \infty$ , so $n_2/v \to 0$ and equation (1) gives $-\dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ , hence $u = -\dfrac{n_1 R}{n_2 - n_1} = -\dfrac{1 \times 20}{0.5} = \boxed{-40\ \mathrm{cm}}$ . This is the first focal length $f_1 = 40\ \mathrm{cm}$ : an object 40 cm in front produces a parallel beam in the glass.

Step 3. Compare. The two focal lengths are unequal — $f_1 = 40\ \mathrm{cm}$ in air, $f_2 = 60\ \mathrm{cm}$ in glass — and their ratio is $f_1/f_2 = 40/60 = n_1/n_2 = 1/1.5$ . A single refracting surface, unlike a thin lens in one medium, has different focal lengths on its two sides, precisely because the media are different. When the two surfaces of a lens sit in the same medium (air on both sides), this asymmetry cancels and the familiar single focal length returns — which is why the next article can speak of one $f$ . Timing note: about 90 seconds. The whole competitive skill here is recognising that "distant object" kills the $u$ term and "parallel emergent rays" kills the $v$ term — each limit strips equation (1) down to a one-line answer. Spot which term vanishes and the algebra is trivial.

Exam traps

Reaching for the mirror or lens formula. A single refracting surface is neither a mirror nor a lens. Use $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ every time; do not drop the indices, do not flip the minus to a plus.

Getting $n_1$ and $n_2$ backwards. $n_1$ is the medium the light starts in, $n_2$ the one it ends in — decided by the direction of travel, not by which glass is "the important one". Air-to-glass has $n_1 = 1$ , $n_2 = 1.5$ ; glass-to-air has them the other way. An arrow on your diagram prevents the slip.

The sign of $R$ . $R > 0$ when the centre of curvature lies on the outgoing-light side. A convex surface facing the incoming light has $R > 0$ ; a concave one has $R < 0$ . Read $R$ 's sign off the position of $C$ , exactly as for a mirror.

Forgetting the image is inside the second medium. A positive $v$ places the image at distance $v$ into $n_2$ . In a glass rod the real image forms deep inside the glass, and that is correct, not an error to "fix".

Board scope. Single-surface refraction is examined by CBSE, ICSE, JEE Main, JEE Advanced and NEET. It is not on AP Physics 2, IB, IGCSE 0625 or any A-Level — those courses jump straight to the thin-lens equation and never derive it from one surface. Treat this article as the Indian-syllabus bridge that explains where the lens formula comes from; if you sit only one of those other boards, it is enrichment, and a beautiful piece of enrichment at that.

Origins · 1 minErnst Abbe and how this was discovered

Ernst Abbe

1840–1905 · Jena

Ernst Abbe was the theorist who turned lens-making from craft into physics. The son of a spinning-mill worker, he rose through scholarships to a professorship at Jena, where the instrument-maker Carl Zeiss hired him in 1866 to work out, from first principles, why microscopes behaved as they did. His answer reached far beyond this page — his 1873 theory of image formation showed that a microscope's resolution is set by diffraction, not by ever-finer grinding — but its foundation is exactly the physics of refraction at a spherical surface. His sine condition and the aplanatic points of a refracting sphere are the rules for building an objective surface by surface, so the paraxial image survives even for wide-angle rays. The single-surface formula this article derives is the first line of that theory; Abbe wrote the rest.

History

150

— Ptolemy tabulates air-to-water refraction angles, the first quantitative refraction data, though with no law behind them.

984

— Ibn Sahl, in Baghdad, records a manuscript geometry equivalent to the sine law of refraction for curved burning surfaces.

1621

— Willebrord Snellius finds the sine law of refraction empirically, the tool this derivation rests on, though he never publishes it.

1637

— René Descartes prints the sine law in La Dioptrique and treats refraction with coordinate geometry.

1873

— Ernst Abbe publishes his theory of image formation, built on refraction at spherical surfaces and the diffraction limit of the microscope.

Going further

Texts. NCERT Physics Class 12, Ch. 9 ("Ray Optics and Optical Instruments"), §9.5 "Refraction at Spherical Surfaces" · H. C. Verma, Concepts of Physics Vol. 1, "Geometrical Optics", refraction at spherical surfaces · Irodov, Problems in General Physics, geometrical optics problems.

Provenance. Standard paraxial optics. Recompute: air-to-glass $n_1 = 1$ , $n_2 = 1.5$ , $R = +20$ cm — $u = -60 \to v = +180$ cm (real, in glass); $u = -30 \to v = -180$ cm (virtual); distant object $\to v = f_2 = n_2 R/(n_2-n_1) = 60$ cm; parallel emergent $\to u = -f_1 = -n_1 R/(n_2-n_1) = -40$ cm, with $f_1/f_2 = 1/1.5$ ; surface power $(n_2-n_1)/R = 0.025\ \mathrm{cm^{-1}}$ . Flat limit $R \to \infty$ : $v = (n_2/n_1)u$ , giving the coin $4\ \mathrm{cm} \to 3.0\ \mathrm{cm}$ in water. A concave surface $R = -20$ , $u = -60 \to v = -36$ cm (virtual).

Prerequisites. Refraction of Light & Snell's Law — the boundary law and $n = c/v$ this page bends around a sphere. Leads to. The Lens Maker's Formula, which applies this surface equation twice to build a lens; and The Thin Lens Formula, the workhorse image equation that follows.

Go deeper (UG). The full treatment — general (non-paraxial) surfaces, the sign-consistent matrix method, aberrations and Abbe's sine condition — is developed in Refraction at a Spherical Surface in the university optics track. It is enrichment only; every result on this page follows from Snell's law and two triangles.

Why is there no focal length f in this equation, when mirrors and lenses have one?

Because a single surface has two focal lengths, not one, and they are unequal. The distance at which a distant object images (the second focal length, $f_2 = n_2 R/(n_2-n_1)$ ) differs from the object distance that produces a parallel beam (the first focal length, $f_1 = n_1 R/(n_2-n_1)$ ), and their ratio is $f_1/f_2 = n_1/n_2$ . This happens because the media on the two sides are different, so a ray "sees" a different medium coming in than going out. A thin lens sits with the same medium (usually air) on both sides, which forces the two focal lengths to be equal, and only then can you speak of a single $f$ . That is exactly the simplification the lens-maker's formula exploits.

How can a real image form when there is only one surface and no lens?

It surprises everyone, but it is genuine. Refraction at a convex surface into a denser medium bends every ray toward the axis, and for a paraxial fan that convergence is enough to bring the rays back to a point on the far side — a real image, catchable in principle on a screen placed inside the second medium. In the worked example that image sits 180 cm inside the glass rod. You do not need two surfaces to focus; you need one curved surface with an index jump across it. A lens uses two surfaces mainly so that the image can be thrown back out into the air where you can use it, and so that its focal length no longer depends on how deep the second medium runs.

What does it mean that the image is "inside the glass"?

Exactly what it says: if the glass were long enough and you could look along it, the rays would cross and form a sharp point at that depth within the glass. In the worked example the real image is 180 cm into the rod. If the rod ends before 180 cm, the light hits the far surface first and refracts again — and that second refraction is precisely how a lens works. So "image inside the glass" is not a paradox; it is the intermediate image that becomes the object for the lens's back surface, as the lens-maker's derivation will use directly.

How do I decide the sign of R?

Use the same rule as for mirrors: $R$ is positive when the centre of curvature $C$ is on the outgoing side of the surface — the side the light is heading into after refraction — and negative when $C$ is on the incoming side. For a convex surface bulging toward the incoming light, $C$ is beyond the surface, so $R > 0$ ; for a concave surface, $C$ is back on the object's side, so $R < 0$ . Draw the surface, mark $C$ , ask which side the refracted light goes, and read the sign off. Never assign $R$ 's sign from the words "convex" or "concave" alone without checking which way the light travels.

Why must the rays be paraxial? What breaks if they are not?

The whole derivation replaced $\sin\theta$ and $\tan\theta$ by the angle itself, which is only accurate for small angles — rays close to the axis. If a ray strikes the surface far from the axis, those approximations fail, and it crosses the axis slightly short of the paraxial image point. Different ray heights then focus at different places, and instead of a sharp point you get a small blur. This defect is called spherical aberration, and it is why real camera lenses use several elements and aspheric surfaces to correct it. At the school level we work only with paraxial rays, where a single sharp image exists and equation (1) is exact.

Is this the same as Snell's law, or something new?

It is Snell's law — applied at a curved boundary and combined with a little triangle geometry. Snell's law alone ( $n_1 \sin\theta_1 = n_2 \sin\theta_2$ ) tells you how one ray bends; it says nothing about images. The new content here is geometric: because the surface is curved, rays at different heights meet normals pointing in different directions, and when you track a whole paraxial fan through Snell's law the fan reconverges to an image. So there is no new law of physics on this page — only Snell's law put to work on a sphere, which is what makes an image possible.