Perhaps I can help with the markings on the new face:
Originally Posted by max6166
One full revolution around the dial corresponds to 20 (measured) kilograms. We can then express the measurement reported on the dial as a function of the angle between the needle and the axis (we'll use the zero point as the axis):
Now we already know that if a measurement is reported, the actual weight will be
Solving for ,
So what does this mean? It means that if we want to know where the marking for a weight of should go on the dial, we substitute it into the above equation to get the angle.
Now, suppose we want to make a marking for a particular weight (for example, you will probably want markings for 1, 2, 3, ... kilograms). Call the weight . For simplicity, let us allow the marking to be a single point that lies some given distance away from the origin (the center of the dial). Then we will have something that looks like this:
Looking at this triangle, we know from simple trigonometry that the coordinates of our marking will be
| x M
| / |
y | t / |
|θ / |
| / |
Letting vary between two positive values will give us a line segment. This line segment can be used as a marking for that particular weight. For example, if you were constructing the new face to have markings 5 cm from the center of the dial (let's say), you wanted each marking to be 1 cm in length, and you wanted to make markings at each whole number weight in kilograms, you could simply graph the parametric equations
for each positive integer between 0 and 20 or so. The process is similar for other measurements.
Below is an example that I made using the plotting capabilities of the open source program Maxima, by plotting 220 lines to give 22 whole value markings each with 10 subdivisions (I used different thicknesses to make the markings properly readable; the numbers were added separately in Paint Shop Pro).