Figure 1: Figure 1: Daylength by latitude and day in the year for ( lat_step = 5, lon_step = 1, year_step = 10)

Is there a better way of generating the grid?

While it is possible to smooth the contours by increasing the number of points in the grid at the expense of computation and rendering time, something struck me: the resolution around the poles is higher than at the Equator.

The number of points in each parallel is constant irrespective of the length of the parallel, being the Equator the largest parallel and the closer to the Poles, the shorter.

In the first post of this series the construction method of the grid is analogous to sampling the angles \(\theta\) [0,\(\pi\)] and \(\phi\) [0,\(2\pi\)] at equal distances as per the step parameters, lat_step and lon_step respectively. This results in a grid of a number of points equal to latitude points \(\times\) longitude points.

The figure 2 shows the points as a function of the angles \(\theta\) and \(\phi\) and then plotted on the unit sphere. While the points are equidistributed in the \(\phi\)–\(\theta\) plane, that is not the case on the unit sphere.

 1m <- list(
 2  l = 0,
 3  r = 0,
 4  b = 0,
 5  t = 0,
 6  pad = 4
 7)
 8
 9lat_step <- 5
10lat_pts <- seq(from =0, to = pi, by = pi/(180/lat_step))
11num_lat_pts <- length(lat_pts)
12lon_step <- 1
13lon_pts <- seq(from = 0, to = 2*pi, by = pi/24)
14num_lon_pts <- length(lon_pts)
15dfgridr <- data.frame(x = rep(lon_pts, num_lat_pts), y = rep(lat_pts, each=num_lon_pts))
16fig_rect <- plot_ly(width = 400)
17fig_rect <- fig_rect %>% add_trace(data = dfgridr, x = ~x , y = ~y ,type ='scatter', mode='markers', marker = list(size = 3))
18
19scene = list(camera = list(eye = list(x = 0, y = 2, z = 0))) 
20fig_sph <- plot_ly(width = 800)
21fig_sph <- fig_sph %>% add_trace(data = coor_lat_lon, x = coor_lat_lon[,'x'], y = coor_lat_lon[,'y'], z = coor_lat_lon[,'z'], type ='scatter3d', mode='markers', marker = list(size = 2))
22
23fig <- subplot(fig_rect, fig_sph, widths = c(0.4,0.5)) %>%
24  layout(title = 'Points distribution', scene = list(camera = list(eye = list(x = 0, y = 2, z = 0)), domain = list(x = c(0.5, 1), y = c(0,1))), showlegend = F) %>%
25  config(displayModeBar = FALSE, scrollZoom = FALSE)  %>%  
26  layout(xaxis = list(title = 'φ [0,2π]',
27                      zeroline = F,
28                      showgrid = F
29                     ),
30         yaxis = list(title = 'θ [0,π]',
31                      zeroline = F,
32                      showgrid = F))
33fig