Geometrical Optics

Law of reflection: angle of incidence = angle of reflection, both measured from the normal. Plane mirrors always give a virtual, upright image (infinite radius of curvature).
Mirror/lens equation: 1/f = 1/o + 1/i (mirrors also equal 2/r); magnification m = −i/o. Positive m = upright, negative m = inverted; |m|<1 smaller, |m|>1 larger.
Refraction: n = c/v; Snell's Law n₁sinθ₁ = n₂sinθ₂. Entering higher n bends toward normal; entering lower n bends away.
Exceeding the critical angle θc = sin⁻¹(n₂/n₁) going from higher-n to lower-n medium causes total internal reflection (no light refracts out).
Lens power P = 1/f (diopters, f in meters); positive P = converging/convex, negative P = diverging/concave. Lenses in contact: powers and 1/f add, magnifications multiply.

Center of curvature (C): Center of the sphere the mirror segment is cut from, on the optical axis at distance r from the surface.
Radius of curvature (r): Distance from the mirror surface to the center of curvature.
Focal point (F): Point on the axis where parallel rays converge (concave) or appear to diverge from (convex); halfway between mirror and C.
Focal length (f): Distance from the mirror/lens to the focal point.
Object distance (o) / Image distance (i): Distance from object to mirror/lens; distance from resulting image to mirror/lens.
Real image: Light rays actually converge at a point — projectable onto a screen.
Virtual image: Light rays only appear to originate from a point — not actually converging there.

1/f = 1/o + 1/i = 2/r, m = −i/o

Worked example: concave mirror, f = 10 cm, o = 30 cm → 1/i = 1/10 − 1/30 = 2/30 → i = 15 cm (real, in front of mirror). m = −15/30 = −0.5 (inverted, half size).
Same equation form for thin lenses. Positive i = real image; negative i = virtual image.

Object position	Resulting image
Beyond C	Real, inverted, reduced — forms between F and C
At C	Real, inverted, same size — forms at C
Between C and F	Real, inverted, enlarged — forms beyond C
At F	No image forms — reflected rays are parallel
Between F and mirror	Virtual, upright, enlarged — forms behind the mirror

n = c/v, n₁sinθ₁ = n₂sinθ₂, θc = sin⁻¹(n₂/n₁)

Entering a higher-n medium bends the ray toward the normal; entering a lower-n medium bends it away.
Past θc, all light reflects back into the original medium — total internal reflection (basis of fiber optics).

1/f = (n−1)(1/r₁ − 1/r₂), P = 1/f, 1/f = 1/f₁ + 1/f₂ + ⋯, P = P₁ + P₂ + ⋯, m = m₁ × m₂ × ⋯

Lenses refract light twice (entering and exiting), unlike mirrors which only reflect once.
Positive power = converging (convex) lens; negative power = diverging (concave) lens.
For lenses in contact: focal lengths combine as 1/f sums, powers add directly, total magnification is the product of each lens's magnification.

Spherical aberration: Rays through the edges of a lens/mirror focus at a different point than rays through the center, blurring the image.
Chromatic aberration: Different wavelengths refract at slightly different angles (dispersion), producing colored fringes around the image.

Convex mirrors always produce virtual, upright, reduced images — never real, never inverted, regardless of object position.
Sign conventions flip between mirrors and lenses for object/image distance — always check which system (mirror vs. lens) a sign-convention question is about before applying +/− rules.
The double-refraction of lenses (entering and exiting) is why the lensmaker's equation depends on both surface radii, not one — don't treat a lens like a single-surface mirror.

Light goes from water (n=1.33) into air (n=1.00) at an angle greater than θc. What happens?

An object sits between a concave mirror's focal point and the mirror itself. Describe the image.

Two thin lenses in contact have focal lengths of +20 cm and −10 cm. What is the combined power?

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