Axes Method - plot (3D Lines)¶

The ax.plot method on 3D axes creates line plots in three-dimensional space. This is equivalent to ax.plot3D --- both methods are interchangeable. A parametric 3D curve is a trajectory through space: as the parameter \(t\) advances, the point \((x(t), y(t), z(t))\) traces a path.

Basic Usage¶

1. Simple 3D Line¶

```python import matplotlib.pyplot as plt import numpy as np

t = np.linspace(0, 2 * np.pi, 100) x = np.cos(t) y = np.sin(t) z = t

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(x, y, z) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel('z') plt.show() ```

Interpreting the helix: x = cos(t) and y = sin(t) produce circular motion in the xy-plane, while z = t adds linear progression upward. The combined curve is a helix --- rotating motion over time. This decomposition (circular + linear) is the key to reading parametric curves: each component tells you what happens along one axis.

2. plot vs plot3D¶

Both methods produce identical results:

```python fig, axes = plt.subplots(1, 2, figsize=(12, 5), subplot_kw={'projection': '3d'})

axes[0].plot(x, y, z, 'blue') axes[0].set_title('ax.plot')

axes[1].plot3D(x, y, z, 'blue') axes[1].set_title('ax.plot3D')

plt.tight_layout() plt.show() ```

3. With Color Argument¶

python fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(x, y, z, 'red') plt.show()

Line Styling¶

1. Colors¶

```python t = np.linspace(0, 4 * np.pi, 200)

fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

colors = ['blue', 'green', 'red']

for ax, color in zip(axes, colors): ax.plot(np.cos(t), np.sin(t), t, color=color, linewidth=2) ax.set_title(f"color='{color}'")

plt.tight_layout() plt.show() ```

2. Line Styles¶

```python fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

styles = ['-', '--', ':'] titles = ['Solid', 'Dashed', 'Dotted']

for ax, style, title in zip(axes, styles, titles): ax.plot(np.cos(t), np.sin(t), t, linestyle=style, linewidth=2) ax.set_title(title)

plt.tight_layout() plt.show() ```

3. Line Width¶

```python fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

widths = [0.5, 2, 4]

for ax, lw in zip(axes, widths): ax.plot(np.cos(t), np.sin(t), t, linewidth=lw) ax.set_title(f'linewidth={lw}')

plt.tight_layout() plt.show() ```

4. Alpha Transparency¶

```python fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

for offset in range(5): ax.plot(np.cos(t), np.sin(t) + offset, t, alpha=0.3 + offset * 0.15)

ax.set_title('Varying Alpha') plt.show() ```

Multiple Lines¶

1. Multiple Helices¶

```python t = np.linspace(0, 4 * np.pi, 200)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

ax.plot(np.cos(t), np.sin(t), t, 'b-', label='Helix 1') ax.plot(np.cos(t + np.pi), np.sin(t + np.pi), t, 'r-', label='Helix 2')

ax.legend() ax.set_title('Double Helix') plt.show() ```

2. Parallel Lines¶

```python fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

for i, color in enumerate(['blue', 'green', 'red', 'purple', 'orange']): ax.plot(np.cos(t), np.sin(t), t + i * 5, color=color, label=f'z offset = {i*5}')

ax.legend() plt.show() ```

3. Radial Lines¶

```python fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

for angle in np.linspace(0, 2 * np.pi, 8, endpoint=False): x = np.cos(t + angle) y = np.sin(t + angle) ax.plot(x, y, t, alpha=0.7)

ax.set_title('Radial Helix Array') plt.show() ```

Parametric Curves¶

1. Helix¶

```python t = np.linspace(0, 6 * np.pi, 500)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(np.cos(t), np.sin(t), t, 'b-', linewidth=1.5) ax.set_title('Helix: \((\\cos t, \\sin t, t)\)') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel('z') plt.show() ```

2. Conical Helix¶

```python t = np.linspace(0, 6 * np.pi, 500)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(t * np.cos(t) / 10, t * np.sin(t) / 10, t, 'g-', linewidth=1.5) ax.set_title('Conical Helix') plt.show() ```

3. Trefoil Knot¶

```python t = np.linspace(0, 2 * np.pi, 1000) x = np.sin(t) + 2 * np.sin(2 * t) y = np.cos(t) - 2 * np.cos(2 * t) z = -np.sin(3 * t)

fig, ax = plt.subplots(figsize=(8, 8), subplot_kw={'projection': '3d'}) ax.plot(x, y, z, 'purple', linewidth=2) ax.set_title('Trefoil Knot') plt.show() ```

4. Lissajous 3D¶

```python t = np.linspace(0, 2 * np.pi, 1000) x = np.sin(3 * t) y = np.sin(4 * t) z = np.sin(5 * t)

fig, ax = plt.subplots(figsize=(8, 8), subplot_kw={'projection': '3d'}) ax.plot(x, y, z, 'darkorange', linewidth=1.5) ax.set_title('3D Lissajous Curve') plt.show() ```

5. Parametric Curve Gallery¶

```python t = np.linspace(0, 4 * np.pi, 500)

fig, axes = plt.subplots(2, 2, figsize=(12, 10), subplot_kw={'projection': '3d'})

Helix¶

axes[0, 0].plot(np.cos(t), np.sin(t), t, 'blue') axes[0, 0].set_title('Helix')

Conical Helix¶

axes[0, 1].plot(t * np.cos(t) / 10, t * np.sin(t) / 10, t, 'green') axes[0, 1].set_title('Conical Helix')

Lissajous¶

t2 = np.linspace(0, 2 * np.pi, 1000) axes[1, 0].plot(np.sin(3t2), np.sin(4t2), np.sin(5*t2), 'orange') axes[1, 0].set_title('3D Lissajous')

Trefoil Knot¶

axes[1, 1].plot( np.sin(t2) + 2np.sin(2t2), np.cos(t2) - 2np.cos(2t2), -np.sin(3*t2), 'purple' ) axes[1, 1].set_title('Trefoil Knot')

plt.tight_layout() plt.show() ```

Lines on Planes¶

1. XY Plane (z = constant)¶

```python t = np.linspace(0, 2 * np.pi, 100)

fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(np.cos(t), np.sin(t), np.zeros_like(t), 'b-', linewidth=2) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel('z') ax.set_title('Circle on XY Plane') plt.show() ```

2. Multiple Horizontal Slices¶

```python fig, ax = plt.subplots(subplot_kw={'projection': '3d'})

for z_level in np.linspace(0, 10, 6): ax.plot(np.cos(t), np.sin(t), np.ones_like(t) * z_level, alpha=0.7)

ax.set_title('Circles at Different Heights') plt.show() ```

3. XZ and YZ Planes¶

```python fig, axes = plt.subplots(1, 3, figsize=(15, 5), subplot_kw={'projection': '3d'})

t = np.linspace(0, 2 * np.pi, 100)

XY plane¶

axes[0].plot(np.cos(t), np.sin(t), np.zeros_like(t), 'b-', linewidth=2) axes[0].set_title('XY Plane (z=0)')

XZ plane¶

axes[1].plot(np.cos(t), np.zeros_like(t), np.sin(t), 'g-', linewidth=2) axes[1].set_title('XZ Plane (y=0)')

YZ plane¶

axes[2].plot(np.zeros_like(t), np.cos(t), np.sin(t), 'r-', linewidth=2) axes[2].set_title('YZ Plane (x=0)')

plt.tight_layout() plt.show() ```

View Control¶

1. view_init Parameters¶

```python t = np.linspace(0, 4 * np.pi, 200) x, y, z = np.cos(t), np.sin(t), t

fig, axes = plt.subplots(2, 2, figsize=(12, 10), subplot_kw={'projection': '3d'})

views = [(30, -60), (90, 0), (0, 0), (30, 45)] titles = ['Default', 'Top', 'Side', 'Isometric']

for ax, (elev, azim), title in zip(axes.flat, views, titles): ax.plot(x, y, z, 'blue', linewidth=1.5) ax.view_init(elev=elev, azim=azim) ax.set_title(f'{title}\n(elev={elev}°, azim={azim}°)')

plt.tight_layout() plt.show() ```

2. Rotating Views¶

```python fig, axes = plt.subplots(2, 3, figsize=(15, 10), subplot_kw={'projection': '3d'})

azimuths = [0, 60, 120, 180, 240, 300]

for ax, azim in zip(axes.flat, azimuths): ax.plot(x, y, z, 'blue', linewidth=1.5) ax.view_init(elev=30, azim=azim) ax.set_title(f'azim={azim}°')

plt.tight_layout() plt.show() ```

3D Axes Customization¶

1. Labels and Title¶

python fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(x, y, z, 'blue', linewidth=2) ax.set_xlabel('X axis', fontsize=12) ax.set_ylabel('Y axis', fontsize=12) ax.set_zlabel('Z axis', fontsize=12) ax.set_title('3D Helix', fontsize=14) plt.show()

2. Custom Ticks¶

python fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(x, y, z, 'blue', linewidth=2) ax.set_xticks([-1, 0, 1]) ax.set_yticks([-1, 0, 1]) ax.set_zticks([0, 5, 10]) plt.show()

3. Pane Colors¶

```python fig, ax = plt.subplots(subplot_kw={'projection': '3d'}) ax.plot(x, y, z, 'blue', linewidth=2)

White panes¶

ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0))

ax.set_title('White Background Panes') plt.show() ```

4. Grid Control¶

```python fig, axes = plt.subplots(1, 2, figsize=(12, 5), subplot_kw={'projection': '3d'})

axes[0].plot(x, y, z, 'blue', linewidth=2) axes[0].grid(True) axes[0].set_title('Grid On')

axes[1].plot(x, y, z, 'blue', linewidth=2) axes[1].grid(False) axes[1].set_title('Grid Off')

plt.tight_layout() plt.show() ```

Combined 2D and 3D¶

1. Mixed Subplots¶

```python t = np.linspace(0, 4 * np.pi, 200)

fig = plt.figure(figsize=(14, 5))

2D projection¶

ax1 = fig.add_subplot(1, 3, 1) ax1.plot(np.cos(t), np.sin(t), 'b-') ax1.set_title('XY Projection') ax1.set_aspect('equal') ax1.grid(alpha=0.3)

Another 2D view¶

ax2 = fig.add_subplot(1, 3, 2) ax2.plot(t, np.cos(t), 'r-', label='x(t)') ax2.plot(t, np.sin(t), 'g-', label='y(t)') ax2.set_title('Components vs Time') ax2.legend() ax2.grid(alpha=0.3)

3D view¶

ax3 = fig.add_subplot(1, 3, 3, projection='3d') ax3.plot(np.cos(t), np.sin(t), t, 'purple', linewidth=1.5) ax3.set_title('3D Helix')

plt.tight_layout() plt.show() ```

Practical Applications¶

1. Space Curve with Projections¶

```python t = np.linspace(0, 4 * np.pi, 300) x = np.cos(t) y = np.sin(t) z = t

fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'})

Main curve¶

ax.plot(x, y, z, 'blue', linewidth=2, label='Space curve')

Projections¶

ax.plot(x, y, np.zeros_like(z), 'r--', alpha=0.5, label='XY projection') ax.plot(x, np.ones_like(y) * y.max() * 1.2, z, 'g--', alpha=0.5, label='XZ projection') ax.plot(np.ones_like(x) * x.max() * 1.2, y, z, 'm--', alpha=0.5, label='YZ projection')

ax.legend() ax.set_title('Space Curve with Projections') plt.show() ```

2. Multiple Time Series¶

```python np.random.seed(42) t = np.linspace(0, 10, 200)

fig, ax = plt.subplots(figsize=(12, 8), subplot_kw={'projection': '3d'})

n_series = 5 colors = plt.cm.viridis(np.linspace(0, 1, n_series))

for i, color in enumerate(colors): y = np.sin(t + i * 0.5) + np.random.randn(len(t)) * 0.1 ax.plot(t, np.ones_like(t) * i, y, color=color, linewidth=1.5)

ax.set_xlabel('Time') ax.set_ylabel('Series') ax.set_zlabel('Value') ax.set_title('Multiple Time Series') ax.view_init(20, -50) plt.show() ```

3. Publication-Quality Figure¶

```python t = np.linspace(0, 4 * np.pi, 400) x = np.cos(t) y = np.sin(t) z = t / (2 * np.pi)

fig, ax = plt.subplots(figsize=(10, 8), subplot_kw={'projection': '3d'})

ax.plot(x, y, z, 'steelblue', linewidth=2)

ax.set_xlabel('\(x = \\cos(t)\)', fontsize=12, labelpad=10) ax.set_ylabel('\(y = \\sin(t)\)', fontsize=12, labelpad=10) ax.set_zlabel('\(z = t/2\\pi\)', fontsize=12, labelpad=10) ax.set_title('Helix: \((\\cos t, \\sin t, t/2\\pi)\)', fontsize=14, fontweight='bold')

ax.view_init(25, -60) ax.tick_params(labelsize=10)

ax.w_xaxis.set_pane_color((0.98, 0.98, 0.98, 1.0)) ax.w_yaxis.set_pane_color((0.98, 0.98, 0.98, 1.0)) ax.w_zaxis.set_pane_color((0.98, 0.98, 0.98, 1.0))

plt.tight_layout() plt.show() ```

Summary¶

Exercises¶

Exercise 1. Plot two intertwined helices in 3D. The first helix uses x = cos(t), y = sin(t), z = t, and the second uses x = cos(t + pi), y = sin(t + pi), z = t for t in \([0, 6\pi]\). Color them differently, add a legend, and label all axes.

import matplotlib.pyplot as plt
import numpy as np

t = np.linspace(0, 6 * np.pi, 1000)

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')

ax.plot(np.cos(t), np.sin(t), t, label='Helix 1', color='blue')
ax.plot(np.cos(t + np.pi), np.sin(t + np.pi), t, label='Helix 2', color='red')

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('Intertwined Helices')
ax.legend()
plt.show()

Exercise 2. Create a 3D Lissajous curve with x = sin(3t), y = sin(4t), z = sin(5t) for t in \([0, 2\pi]\). Color the line by t using a colormap (use LineCollection or plot segments with varying color). Add a colorbar showing the parameter t.

import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d.art3d import Line3DCollection

t = np.linspace(0, 2 * np.pi, 1000)
x = np.sin(3 * t)
y = np.sin(4 * t)
z = np.sin(5 * t)

points = np.array([x, y, z]).T.reshape(-1, 1, 3)
segments = np.concatenate([points[:-1], points[1:]], axis=1)

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')

norm = plt.Normalize(t.min(), t.max())
lc = Line3DCollection(segments, cmap='viridis', norm=norm)
lc.set_array(t[:-1])
ax.add_collection3d(lc)

ax.set_xlim(-1, 1)
ax.set_ylim(-1, 1)
ax.set_zlim(-1, 1)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('3D Lissajous Curve')

sm = plt.cm.ScalarMappable(cmap='viridis', norm=norm)
plt.colorbar(sm, ax=ax, label='t')
plt.show()

Exercise 3. Plot the trefoil knot in 3D using parametric equations x = sin(t) + 2*sin(2t), y = cos(t) - 2*cos(2t), z = -sin(3t) for t in \([0, 2\pi]\). Use linewidth=3 and set a viewing angle of elev=30, azim=60. Add a title and axis labels.

import matplotlib.pyplot as plt
import numpy as np

t = np.linspace(0, 2 * np.pi, 1000)
x = np.sin(t) + 2 * np.sin(2 * t)
y = np.cos(t) - 2 * np.cos(2 * t)
z = -np.sin(3 * t)

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')
ax.plot(x, y, z, linewidth=3, color='purple')
ax.view_init(elev=30, azim=60)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('Trefoil Knot')
plt.show()