This tool does not try to predict the exact outcome of a session on the mirror. There are too many unknowns and too much physics for the model: the pitch’s real hardness that day, the room’s temperature changing from one hour to the next, how the lap was pressed, the hands’ pressure and rhythm, the cerium, the wetness… It is there to explain the influence of the main parameters. You see how each one changes the mirror’s figure (the stroke, mirror on top or tool on top, the weight, the lap’s facets, the pitch and its temperature…), and why the classic advice works.
Reading the charts
The Glass removed chart, at the top, shows where the glass went and, starting from a sphere, what would have to be removed to head towards a perfect paraboloid, or a hyperboloid if you keep going; starting from a paraboloid, what would have to be removed to return to the sphere.
The figure as worked chart shows the mirror against its best sphere, or its best paraboloid, as a Foucault test would. Against its best sphere, the paraboloid is low at the centre and at the edge, and high around the 70 % zone.
The model
The first principle is that glass wears in proportion to the local pressure and to the speed of the tool over it: Preston’s law. The model takes a definite movement, such as a typical W (which you can change), then assumes the mirror is turned a little between strokes, going round the barrel many times, which gives the mean wear as a function of the distance to the centre. The duration of the work is not modelled: you choose the total glass removed, and the tool computes the estimated curve of glass removed and the resulting figure from the chosen starting point.
What is taken into account
Preston’s law alone is far too simple for what happens between the glass and the pitch. The model includes the stroke’s kinematics, the weight and balance of the piece on top, the pitch’s softness and flow along with the uneven pressure across the facets, the mirror’s edge that can sink into the pitch, and the feedback of the mirror’s own shape.
The pitch
Each Gugolz grade is taken at the middle of the temperature range its maker gives (13 to 18 °C for the 55, 18 to 24 °C for the 64, 24 to 32 °C for the 73). Its viscosity there is about 10⁸ Pa·s and it is divided by ten each time the temperature rises by 6 °C, so the room’s temperature matters as much as the pitch’s number. Two quantities follow, shown under the grade. The softness says how far the pitch gives under pressure over one stroke; a high zone of the glass is pressed harder, the less so the softer the pitch. The flow says how fast it takes the mirror’s shape, relative to the glass removed.
The feedback
Between two rigid pieces, the mutual fitting always brings the surfaces back towards a sphere: in grinding, the one on top hollows and the one below bulges, but both stay spherical. What the pitch adds is that it adjusts as the work goes. It sinks and flows, which lets a stroke dig the centre more than the edge, and so leave the sphere to parabolise. Hence the importance of hardness and temperature: a hard lap imposes its shape, a soft one follows the mirror’s. A stroke’s effect also depends on the starting point (you can choose the sphere or the paraboloid), because the pitch acts more on the glass that stands above the sphere, and pulls the mirror towards the stroke’s own balance figure.
Machines
The machines are those of the classic families: Draper’s with one or two eccentrics (the offset one too), Hindle’s, which holds the top disc by its edge so the mirror can be on top, Elgin’s or Waineo’s swinging arm, and Zeiss’s two cranks. The turntable turns the body below; the one on top, carried by its centre, spins freely, dragged by the friction: its spin is computed at each instant, where the friction’s torque about its pin vanishes. A machine strokes faster or slower than a hand, and the pitch flows with time, not with the glass worn: its flow is scaled by the machine’s mean speed against a hand’s, a third’s stroke each second.