Gaussian Bump
z = exp(−((x²+y²)/4)) · cos(t)
Open in the app The dials and keys named below are the app's.
What it draws
One bell over the middle of the sheet, breathing. The height is e raised to −(x² + y²)/4, times cos t. The formula is stored as it was pasted from a textbook, in LaTeX, and converted on the way into the field; typing e^{−(x² + y²)/4}·cos t gives exactly the same sheet.
Since x² + y² is r², the bump depends on distance from the origin alone and its contours are circles. It is 1 at the origin and half that at r = 2√(ln 2) ≈ 1.67, so the width across the bell at half its height is 3.33. As a bell curve its standard deviation is √2 ≈ 1.41. cos t takes 2π ≈ 6.28 seconds at Speed 1 to swing the bump up, flat at t = π/2, into a pit of the same shape, and back.
The bell
The exponent splits: e^{−(x² + y²)/4} = e^{−x²/4}·e^{−y²/4}. So the bump is round, and it is also a product of one bell in x with one bell in y. Every straight cut across it, in any direction, comes out the same bell shape, scaled down. No other profile does both, which is the argument James Clerk Maxwell used in 1860 to derive the distribution of molecular velocities from symmetry alone.
The same shape turns up whenever many small independent effects are added, as the normal distribution, and whenever heat spreads. With a unit of heat placed at the origin, the temperature after one second is this bell divided by 4π, which is the volume under it. What the clock does here is not that spreading: the width never changes, only the sign and the height.
Try
- Top in the deck: circular contours, evenly shaded, with no ridge anywhere.
- Widen it, e^{−(x² + y²)/16}·cos t: the standard deviation doubles to 2.83 and the bell runs off the edge of the plot.
- Subtract a wider bell from a taller one, 2·e^{−(x² + y²)/4} − e^{−(x² + y²)/16}: a peak in a circular moat, the shape used as a wavelet and as a model of a retinal receptive field.
- Put a ripple under the bell, e^{−(x² + y²)/4}·cos(t − r): rings run outward and the bell becomes their envelope.
- Lower Span of x, y to 2 to fill the plot with the peak alone.