Drag the planet around its epicycle. Watch the deviation rise, cross zero, go negative, return. The smooth, symmetric pattern is the structure Āryabhaṭa noticed in the residuals — and encoded as a sine table.
The gray dot on the dashed circle is the mean planet — it moves at perfectly uniform speed, easy to compute from time alone. The orange dot is the true planet, riding a smaller circle (the epicycle) centered on the mean. The green arrow shows the deviation — the jya — between the two.
One thing worth noticing: at θ = 0° and θ = 180°, the orange dot sits in line with the gray dot as seen from Earth — same longitude, just a different distance. The green arrow vanishes because what it measures is the perpendicular displacement, the part that actually changes the planet's angular position in the sky. The equation tracks longitude, not distance. The epicycle moves the planet both radially and laterally; only the lateral part shows up as a correction.
What makes the argument click: pause the animation and drag the epicycle slider alone. The true planet traces a loop around the mean position, and the deviation value oscillates — rising, crossing zero, going negative, returning. That smooth, symmetric pattern is exactly what Āryabhaṭa noticed in the residuals after subtracting uniform motion from observation. It wasn't noise; it was structure.
The equation in the panel formalizes it: true = mean ± k · jya(θ). The sine table encodes the jya values so that correction becomes a lookup, not a fresh geometric calculation. The table is a compressed map of deviation — and once you see it that way, it stops being a list of mysterious numbers.
The visualization uses a single epicycle and an exaggerated ratio (k = 0.25) so that the loop is clearly visible. In the historical model, Āryabhaṭa's epicycle ratios for the five planets ranged from about 0.038 for the Sun's apparent motion to about 0.20 for Mars. The geometry is identical; only the magnitude differs.