matlab 3d plot

Step 1: Setting up the Canvas

figure(‘Position’, [100, 100, 800, 600]);

This line initializes a new graphics window. The ‘Position’ vector defines the window’s location (100 pixels from the left and bottom of your screen) and its size (800×600 pixels).

Step 2: Creating the Coordinate Grid

[X, Y] = meshgrid(-2:0.1:2, -2:0.1:2);

In MATLAB, we cannot simply plot a continuous function; we must evaluate it at discrete points. meshgrid creates a 2D grid of coordinates from -2 to 2. The 0.1 step size determines the “resolution”—smaller values result in a smoother surface but require more memory.

Step 3: Calculating the Z-Values (The Math)

Z = X .* exp(-X.^2 – Y.^2) + 0.5 * sin(2*X) .* cos(3*Y);

This is the heart of the visualization. We are calculating the height (Z) for every (X, Y) pair.
The Formula: $Z = X \cdot e^{-(X^2 + Y^2)} + 0.5 \cdot \sin(2X) \cdot \cos(3Y)$
Note the use of the dot operator (.* and .^). This tells MATLAB to perform element-wise multiplication, ensuring the operation is applied to every point in the grid individually.

Step 4: Rendering the Surface

surf(X, Y, Z, ‘EdgeColor’, ‘none’, ‘FaceAlpha’, 0.9);

The surf command creates the 3D colored surface. We set ‘EdgeColor’ to ‘none’ to remove the black wireframe lines for a cleaner look, and ‘FaceAlpha’ to 0.9 to make the surface slightly translucent.

Step 5: Color and Legend

colormap(jet);

c = colorbar(‘EastOutside’);

We apply the jet colormap (ranging from blue for low values to red for high values). The colorbar adds a vertical scale on the right side of the plot to translate colors into numerical values.

Step 6: Lighting and Materiality

light(‘Position’, [1, 1, 5]);

lighting gouraud;

material shiny;

To make the plot look realistic, we add a virtual light source. lighting gouraud calculates the color across the surfaces smoothly, and material shiny adds specular highlights (reflections) to the peaks.

Step 1: Defining the Spiral Path

t = 0:0.01:8*pi;
x = exp(-0.05*t) .* sin(t);
y = exp(-0.05*t) .* cos(t);
z = t/4;

We define our spiral using a parameter t. The equations for x and y include a decaying exponential $e^{-0.05t}$, which causes the spiral to “shrink” inward over time, while z grows linearly to give the spiral height.

Step 2: Calculating Tangent Vectors

dx = gradient(x); dy = gradient(y); dz = gradient(z);

To orient a ribbon, we first need to know which direction the curve is “pointing” at any given moment. The gradient function calculates the derivative of our coordinates, giving us the Tangent Vector at every point along the path.

Step 3: Creating the Binormal Vector

binormal_x = dy * up(3) – dz * up(2);
binormal_y = dz * up(1) – dx * up(3);
binormal_z = dx * up(2) – dy * up(1);

A ribbon needs width. To find the “sideways” direction relative to our curve, we calculate the Cross Product of our tangent vector and an “up” vector [0, 0, 1]. This resulting Binormal Vector tells MATLAB exactly which way to stretch the surface to create the ribbon’s flat face.

Step 4: Generating the Ribbon Surface

[S, T_idx] = meshgrid(s, 1:num_points);

We use a loop to iterate through every point on our spiral. For each point, we project outward along the Binormal vector to create a set of coordinates (X_ribbon, Y_ribbon, Z_ribbon). This effectively “sweeps” a 2D line along our 3D path, transforming a 1D line into a 2D surface.

Step 5: Multi-Layered Plotting

surf(X_ribbon, Y_ribbon, Z_ribbon, ‘EdgeColor’, ‘none’, ‘FaceAlpha’, 0.8);
scatter3(x(1:200:end), y(1:200:end), z(1:200:end), 40, ‘filled’, ‘MarkerFaceColor’, ‘r’, ‘MarkerEdgeColor’, ‘k’);
plot3(x, y, z, ‘k-‘, ‘LineWidth’, 1);

This plot is actually composed of three layers:

• Surf: Draws the main colored ribbon using the hsv colormap.
• Scatter3: Adds red markers at specific intervals to highlight discrete data points.
• Plot3: Draws a black “center line” through the middle of the ribbon for better definition.

Step 1: Reproducibility and Initialization

figure('Position', [100, 100, 850, 650]);
n = 500;
rand('seed', 42);

We define n = 500 points per cluster. Setting the seed ensures that every time you run this code, the “random” points are generated in the exact same positions—essential for consistent research results.

Step 2: Generating the Spherical Cluster

theta1 = 2*pi*rand(n,1);
phi1 = pi*rand(n,1);
x1 = 2*sin(phi1).*cos(theta1) + randn(n,1)*0.1;
y1 = 2*sin(phi1).*sin(theta1) + randn(n,1)*0.1;
z1 = 2*cos(phi1) + randn(n,1)*0.1;

To create a sphere, we use spherical coordinates transformed into Cartesian coordinates (X, Y, Z).
The Math: x = r · sin(φ) · cos(θ).
We add a small amount of Gaussian noise (randn * 0.1) so the points don’t sit perfectly on the surface, giving it a realistic “cloud” feel.

Step 3: Creating Linear Cluster

x2 = randn(n,1)*1.5 + 4;
y2 = randn(n,1)*0.5 + 4;
z2 = 0.5*x2 + randn(n,1)*0.3;

For the Linear Cluster, we create points that follow a linear relationship. The x-values are centered around 4 with more spread, the y-values are more tightly clustered, and z is linearly related to x with some noise. This creates a stretched, line-like cluster in 3D space.

Step 4: Creating Ring Cluster

theta3 = 2*pi*rand(n,1);
x3 = 3*cos(theta3) + randn(n,1)*0.2;
y3 = 3*sin(theta3) + randn(n,1)*0.2;
z3 = randn(n,1)*0.5 - 2;

For the Ring Cluster, we use polar coordinates (sine and cosine) to create points in a circular pattern. The x and y coordinates form a ring with radius 3, while the z-coordinate is clustered around -2 with some vertical spread, creating a flattened ring shape.

Step 5: Advanced 3D Scattering

scatter3(x1, y1, z1, 50, 'r', 'MarkerFaceColor', 'r', 'MarkerEdgeColor', 'k');
hold on;
scatter3(x2, y2, z2, 50, 'b', '^', 'MarkerFaceColor', 'b', 'MarkerEdgeColor', 'k');
scatter3(x3, y3, z3, 50, 'g', 's', 'MarkerFaceColor', 'g', 'MarkerEdgeColor', 'k');

The scatter3 function is used to plot individual points. To make the clusters distinct, we use different properties:

• Red Circles: Spherical cluster.
• Blue Triangles ('^'): Linear cluster.
• Green Squares ('s'): Ring cluster.
• MarkerEdgeColor: We add a black (‘k’) edge to each point to make them stand out against the white background.
• Size (50): Controls the marker size for better visibility.

Step 6: Transparency and Proportions

h = findobj(gca, 'Type', 'patch');
for i = 1:length(h)
alpha(h(i), 0.7);
end

We use alpha to make the points 70% opaque. This allows you to see “through” the clusters to see points hidden in the middle.

Step 7: Labels and Final Touches

title('3D Scatter Plot of Synthetic Clusters', 'FontSize', 14, 'FontWeight', 'bold');
xlabel('Feature X'); ylabel('Feature Y'); zlabel('Feature Z');
grid on; view(45, 20);
legend('Spherical Cluster', 'Linear Cluster', 'Ring Cluster', 'Location', 'Best');
axis equal;

Finally, we add labels, a grid for better spatial orientation, set a nice viewing angle, create a legend, and use axis equal to ensure proper proportions—this prevents the sphere from looking like an oval by making all axes use the same scale.