Inputs
The user's inputs produce the following values:
- \(g_s =\) stitch gauge, the number of stitches per inch/cm (horizontal),
- \(g_r =\) row gauge, the number of rows per inch/cm (vertical),
- \(d =\) desired sphere diameter.
Stitch Counts
The total number of rows needed, \(n = \frac{\pi d}{2} \cdot g_r\), rounded to the nearest whole number. We can then calculate the adjusted diameter due to rounding, \(d' = n \cdot \frac{2}{\pi g_r}\).
We can now calculate a set of indexed variables for each row:
- \(\theta_i\), the angle of elevation/depression from the center of the sphere to row \(i\): \(\theta_i = \pi (\frac{i}{n} - \frac{1}{2n})\),
- \(r_i\), the radius of row \(i\): \(r_i = \frac{d}{2} \sin \theta_i\),
- \(c_i\), the circumference of row \(i\): \(c_i = 2 \pi r_i\),
- \(\hat s_i\), the number of stitches in row \(i\): \(s_i = c_i \cdot g_s\); \(\hat s_i = s_i\) rounded to the nearest whole number.
Or, mashing it all up to give \(s_i\) in terms of the original variables,
$$s_i = \sin(\frac{2i-1}{dg_r}) \cdot \pi d \cdot g_s.$$
Because this produces a decimal value, \(s_i\) is rounded to the nearest whole number to get \(\hat s_i\), the functional number of stitches for row \(i\).
Error Calculation
Now, let's calculate the accumulated error (relative to \(d'\), not \(d\)). I chose to use \(r^2\), as many people are familiar with this metric. Let's start by laying out the variables that will be used for these calculations:
- \(c_i\), the calculated circumference of row \(i\), as above,
- \(\hat c_i\), the real circumference of row \(i\): \(\hat c_i = \frac{\hat s_i}{g_s}\),
- \(\bar c\), the average of all of the circumferences, \(\frac{\sum_{i=1}^n \hat c_i}{n}\).
The formula for \(r^2\) is
$$r^2 = \frac{ESS}{TSS}.$$
where ESS is the explained sum of squares and TSS is the total sum of squares (see
the Wiki article on ESS for more info).
$$r^2 = \frac{ESS}{TSS} = \frac{\sum_{i=1}^n (\hat c_i\ - \bar c)^2}{\sum_{i=1}^n (c_i\ - \bar c)^2}.$$
This \(r^2\) is given as a decimal value between 0 and 1; the value is converted to a percentage before being displayed.