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How to use
Pick a preset or move any slider. Top: a silvered bauble with an object that moves back and forth; the blue image stays small and upright inside the ball. Bottom: sunlight crossing a clear glass ball and heating a sheet of paper. Try: (1) raise n to 2 and watch the focus slide onto the rear surface; (2) open the aperture to 1.0 and see the focus smear; (3) untick auto-focus and drag the paper through the focus to see the spot size and temperature change; (4) move the object close to the mirror to enlarge the image. Drag to orbit, wheel to zoom. Space pauses, R resets.
The table, wall, Christmas tree with ornaments, gifts, bench rails, stands, snowman and paper card were modelled and baked in Blender; Three.js r160 renders the live mirror reflection (cube camera), the glass ball and the exactly traced rays from the same model as the 2D lab.
The science
Two balls, two kinds of optics. A silvered Christmas bauble is a convex spherical mirror. A clear glass ball is a ball lens. Lengths are in centimetres; the sphere has radius R and its centre is on the optical axis.
1. Silvered bauble (convex mirror). A spherical mirror has focal length f = R/2, negative for a convex surface, so the focus is a virtual point behind the surface. The mirror equation and magnification are
1/do + 1/di = 1/f = −2/R, m = −di/do
di is always negative (virtual image behind the surface) and 0 < m < 1: you see a small, upright, wide-angle picture of the room. For do >> R the image sits near R/2 behind the surface and its height shrinks like R/(2do). The drawn rays are traced exactly with the law of reflection, r = d − 2(d·N)N, and the dashed blue lines show where the reflected rays seem to come from.
2. Glass ball (ball lens). Each ray is refracted twice with Snell's law n1 sinθ1 = n2 sinθ2. In the paraxial limit the two surfaces combine into
EFL = n R / (2 (n − 1)), BFD = EFL − R
EFL is the effective focal length (from the centre) and BFD the back focal distance (from the rear surface). For n = 2 the focus lies exactly on the rear surface; for n = 1.5 it is 0.5 R behind it. Rays far from the axis bend too much (spherical aberration): the longitudinal error grows as h², so the focus is a blur, not a point. The simulator traces rays exactly, finds the plane of least blur, and adds the size of the Sun's disc image, EFL × 0.0093 rad.
3. Why a glass ball is a fire hazard. Sunlight of about 1000 W/m² entering an aperture of diameter D = 2 a R is squeezed into a spot of diameter d. The concentration is C = T (D/d)², where T is the transmission after two Fresnel reflections, r = ((n−1)/(n+1))² per surface. A small bauble can give C of a few hundred suns. The paper graph integrates a lumped heat balance, Cp dT/dt = α C I0 − 2h (T − T0) − 2εσ(T&sup4; − T0&sup4;) with α = 0.7, h = 25 W/m²K, ε = 0.9 and a heat capacity of 107 J/m²K (80 g/m² paper). Paper can ignite from about 230 C (the line on the graph). Do not leave glass balls, water bottles or magnifiers in direct sunlight near anything flammable.
Checks. (1) the exact trace for h → 0 reproduces the paraxial BFD; (2) n = 2 puts the focus on the back surface; (3) the spherical aberration ratio LSA(2h)/LSA(h) is 4; (4) two exactly reflected rays meet at the mirror-equation image distance; (5) the same two rays give the magnification −di/do; (6) Fresnel reflectance plus transmittance equals one (transmittance from the amplitude coefficients) and (7) the Fresnel formula reduces to ((n−1)/(n+1))² at normal incidence.
Simplifications. Meridional (2D) rays only, monochromatic light (no dispersion), no internal reflections beyond the transmission factor, a perfect mirror, a point-like paper patch with no lateral heat conduction or moisture (so the time to 233 C is an optimistic lower bound), and the object is a flat arrow on the axis.