如何在Maple/MATLAB中绘制点密度随正弦值变化的正弦图俯视图？

Hey there! Let's figure out how to create that top-down plot where points are densest at the peaks and troughs of the sine wave, and sparser near the zero crossings. Your initial attempt with plot3d(sin(x), linestyle=dot) uses uniform sampling, which is why you aren't seeing the density variation you want. We need to manually generate non-uniform points that cluster where |sin(x)| is large. Here's how to do it in both Maple and MATLAB:

Maple Solution #

Instead of relying on plot3d's automatic grid, we'll build a custom set of points with density tied to the sine wave's magnitude. Here's the code:

# Define the x range we want to plot
x_min := 0;
x_max := 4*Pi;

# Generate a base set of evenly spaced points (for overall coverage)
base_x := linspace(x_min, x_max, 100);

# Create a dense set of points, then filter to keep only those where |sin(x)| is large
high_density_candidates := linspace(x_min, x_max, 500);
filtered_high_x := select(x -> abs(sin(x)) > 0.8, high_density_candidates);

# Combine and sort all points to maintain order
all_x := sort([op(base_x), op(filtered_high_x)]);

# Set y-values to 0 (since we want a top-down view, y doesn't change)
y_vals := Array([seq(0, x in all_x)]);
# Compute the corresponding sine values for z
z_vals := Array([seq(sin(x), x in all_x)]);

# Plot as a 3D scatter plot, then switch to top-down perspective
scatter3d(all_x, y_vals, z_vals, style=point, symbol=circle, symbolsize=5);
# Adjust view to look straight down along the z-axis
view = [x_min..x_max, -0.1..0.1, -1..1];
orientation = [0, 90];

How this works: #

• We start with a sparse base grid to ensure we cover the entire range.
• We add extra points specifically where |sin(x)| > 0.8 (near the peaks and troughs), making those areas much denser.
• The orientation = [0, 90] setting switches the plot to a top-down view, so you'll see the point density variation clearly.

MATLAB Solution #

The approach is similar: we generate non-uniform points clustered at the sine wave's extremes, then plot them and adjust the view. Here's the code:

% Define the x range
x_min = 0;
x_max = 4*pi;

% Create a base set of evenly spaced points
base_x = linspace(x_min, x_max, 100);

% Generate a dense set of points, filter to keep those near sine peaks/troughs
high_density_candidates = linspace(x_min, x_max, 500);
filtered_high_x = high_density_candidates(abs(sin(high_density_candidates)) > 0.8);

% Combine and sort all points
all_x = sort([base_x, filtered_high_x]);

% Set y-values to 0 (constant for top-down view)
y_vals = zeros(size(all_x));
% Compute sine values for z
z_vals = sin(all_x);

% Create a filled scatter plot
scatter3(all_x, y_vals, z_vals, 10, 'filled');
% Switch to top-down view (look along z-axis)
view(0, 90);

% Add labels and clean up the plot
xlabel('x');
ylabel('y');
zlabel('sin(x)');
axis([x_min x_max -0.1 0.1 -1 1]);
grid on;

How this works: #

• Just like in Maple, we combine a sparse base grid with extra points in high-magnitude sine regions.
• The view(0, 90) command gives the top-down perspective, so you can easily see the dense clusters at sin(x) = ±1 and sparse points near sin(x) = 0.

内容的提问来源于stack exchange，提问作者Jerry