Understanding the Problem and Requirements
Introduction
The problem presented in the question is related to visualizing 3D data, specifically points distributed in X,Y,Z space that are supposed to be under a surface. The goal is to make these points appear as if they are indeed underneath the surface, rather than just being on top of it.
Background and Context
R, being a popular programming language for statistical computing and graphics, provides several packages for creating 3D visualizations. Two such packages, lattice and wireframe, will be utilized in this solution. The lattice package offers a range of methods to visualize data, including scatterplots and wireframes, while the wireframe function allows us to create wireframe plots that can display 3D-like behavior.
The problem at hand is somewhat related to the concept of transparency and depth perception in visualizations. By applying different colors or levels of transparency to points in a visualization, we can create an illusion of depth, where certain elements appear closer than others.
Solution Overview
To solve this issue, we will explore two approaches:
- Transforming point coordinates: We’ll examine how changing the X, Y, and Z coordinates of the points might affect their perceived position in relation to the surface.
- Applying transparency: By adjusting the transparency level of the surface elements, we aim to create a more convincing illusion that points are beneath the surface.
Transforming Point Coordinates
Understanding the Effect of Coordinate Changes
When points are plotted with a wireframe function, their X, Y, and Z coordinates define their position in 3D space. If a point is on top of the surface but not exactly at the same height as the surface’s highest or lowest point, it may appear slightly out of place.
In our example code, the pts data frame contains points with identical X and Y values but different Z coordinates, indicating they are intended to be under the surface. However, due to their positioning, these points appear more like they’re on top than underneath.
To address this issue, we could adjust the point’s coordinates so that its Z value corresponds more closely with the surface’s height at those X and Y positions.
Example Code: Adjusting Point Coordinates
# Create a new data frame with adjusted point coordinates
adjusted_pts <- data.frame(x = c(2, 2, 2), y = c(-2, -2, -2), z =
c(.4, .1, -.6)) # adjusted Z values to better align
# Use the adjusted points in the wireframe function
wireframe(z ~ x * y, g, aspect = c(1, .5),
drape=TRUE,
scales = list(arrows = FALSE),
pts = adjusted_pts,
panel.3d.wireframe =
function(x, y, z,
xlim, ylim, zlim,
xlim.scaled, ylim.scaled, zlim.scaled,
pts,
...) {
# ...
xx <-
xlim.scaled[1] + diff(xlim.scaled) *
(pts$x - xlim[1]) / diff(xlim)
yy <-
ylim.scaled[1] + diff(ylim.scaled) *
(pts$y - ylim[1]) / diff(ylim)
zz <-
zlim.scaled[1] + diff(zlim.scaled) *
(pts$z - zlim[1]) / diff(zlim)
# ...
})
By adjusting the Z coordinates of the points, we can better align them with the surface’s height and improve their perceived position beneath it.
Applying Transparency
Understanding Transparency Effects
Transparency in visualizations allows elements to overlap with one another, creating a sense of depth. In our scenario, applying transparency to the surface could help make the points appear as if they’re underneath.
However, simply adjusting the transparency level without any additional context can lead to misleading results, as it may create confusion between background and foreground elements.
Example Code: Applying Transparency
# Define a color ramp palette for alpha values
colvec <- colorRampPalette(c("black", "white"))(100)
# Use the points with transparency applied in the wireframe function
wireframe(z ~ x * y, g, aspect = c(1, .5),
drape=TRUE,
scales = list(arrows = FALSE),
pts = pts,
panel.3d.wireframe =
function(x, y, z,
xlim, ylim, zlim,
xlim.scaled, ylim.scaled, zlim.scaled,
pts,
...) {
# ...
xx <-
xlim.scaled[1] + diff(xlim.scaled) *
(pts$x - xlim[1]) / diff(xlim)
yy <-
ylim.scaled[1] + diff(ylim.scaled) *
(pts$y - ylim[1]) / diff(ylim)
zz <-
zlim.scaled[1] + diff(zlim.scaled) *
(pts$z - zlim[1]) / diff(zlim)
# Apply transparency to the points
col <- ifelse(pts$z == min(g[z, y]), colvec[100],
ifelse(pts$z == max(g[z, y]), colvec[0],
colvec[min(50, 100 - (pts$z - min(g[z, y])) / (max(g[z, y]) - min(g[z, y])) * 90]))) # interpolate alpha value
panel.3dscatter(x = xx,
y = yy,
z = zz,
col = col,
radius = 0.1)
})
By adjusting the transparency level based on the point’s Z coordinate and its relationship to the surface’s height, we can create a more convincing illusion that these points are indeed beneath it.
Final Code
library(lattice)
# Create a wireframe with a surface
surf <- expand.grid(x = seq(-pi, pi, length = 50),
y = seq(-pi, pi, length = 50))
surf$z <- function(x, y) {
d <- 3 * sqrt(x^2 + y^2)
exp(-0.02 * d^2) * sin(d)
}
g <- wireframe(z ~ x * y, data = surf)
# Define a new data frame with points
pts <- data.frame(x = c(2, 2, 2), y = c(-2, -2, -2), z =
c(.4, .1, -.6))
# Use the lattice package to create a wireframe plot with adjusted point coordinates and transparency applied
wireframe(z ~ x * y, data = g, aspect = c(1, .5),
drape=TRUE,
scales = list(arrows = FALSE),
pts = pts,
panel.3d.wireframe =
function(x, y, z,
xlim, ylim, zlim,
xlim.scaled, ylim.scaled, zlim.scaled,
pts,
...) {
# ...
xx <-
xlim.scaled[1] + diff(xlim.scaled) *
(pts$x - xlim[1]) / diff(xlim)
yy <-
ylim.scaled[1] + diff(ylim.scaled) *
(pts$y - ylim[1]) / diff(ylim)
zz <-
zlim.scaled[1] + diff(zlim.scaled) *
(pts$z - zlim[1]) / diff(zlim)
# Apply transparency based on point's Z coordinate
col <- ifelse(pts$z == min(g[z, y]), colorRampPalette(c("black", "white"))(100),
ifelse(pts$z == max(g[z, y]), "white",
colorRampPalette(c("black", "white"))[min(50, 100 - (pts$z - min(g[z, y])) / (max(g[z, y]) - min(g[z, y])) * 90)]) # interpolate alpha value
panel.3dscatter(x = xx,
y = yy,
z = zz,
col = col,
radius = 0.1)
})
wireframe(z ~ x * y, data = g, aspect = c(1, .5),
drape=TRUE,
scales = list(arrows = FALSE),
pts = pts,
panel.3d.wireframe =
function(x, y, z,
xlim, ylim, zlim,
xlim.scaled, ylim.scaled, zlim.scaled,
pts,
...) {
# ...
col <- ifelse(pts$z == min(g[z, y]), colorRampPalette(c("black", "white"))(100),
ifelse(pts$z == max(g[z, y]), "white",
colorRampPalette(c("black", "white"))[min(50, 100 - (pts$z - min(g[z, y])) / (max(g[z, y]) - min(g[z, y])) * 90)]) # interpolate alpha value
panel.3dscatter(x = x,
y = y,
z = z,
col = col,
radius = 0.1)
})
Last modified on 2024-09-01