Polar and Vector Plots

🎯 Specialty Plots: Polar and Vector Plots

The Compass and Wind Story

Imagine you’re a ship captain. You have two special tools:

1. A compass that shows directions in a circle (like a pizza!)
2. A wind map that shows arrows pointing where the wind blows

Matplotlib gives us these same superpowers for data! Let’s discover them together.

🧭 Polar Plot Basics

What is a Polar Plot?

Think of a circular dartboard. Instead of going left-right (X) and up-down (Y), we use:

• Angle (theta/θ): Which direction to point (like clock hands)
• Distance ®: How far from the center

import matplotlib.pyplot as plt
import numpy as np

# Create angles (0 to 360 degrees)
theta = np.linspace(0, 2*np.pi, 100)

# Distance from center
r = np.cos(4 * theta)

# Make a polar plot
fig, ax = plt.subplots(
    subplot_kw={'projection': 'polar'}
)
ax.plot(theta, r)
plt.show()

The Magic Key: projection='polar'

This tiny phrase transforms a regular plot into a circular one!

graph TD
    A["Regular Plot"] -->|Add projection='polar'| B["Circular Plot!"]
    B --> C["Angle = Direction"]
    B --> D["Distance = From Center"]

Simple Example: Drawing a Circle

theta = np.linspace(0, 2*np.pi, 50)
r = np.ones(50)  # All same distance

fig, ax = plt.subplots(
    subplot_kw={'projection': 'polar'}
)
ax.plot(theta, r)
plt.show()

Result: A perfect circle! Why? Same distance (r=1) at every angle.

📊 Polar Bar Chart

What is it?

Imagine pizza slices of different sizes arranged in a circle. Each slice shows a value!

Real-life uses:

• Wind rose (weather)
• Daily schedule
• Compass directions

Basic Code

import matplotlib.pyplot as plt
import numpy as np

# 8 directions (like a compass)
N = 8
theta = np.linspace(
    0, 2*np.pi, N, endpoint=False
)

# Values for each direction
values = [3, 4, 2, 5, 4, 3, 2, 4]

# Bar width
width = 2*np.pi / N

fig, ax = plt.subplots(
    subplot_kw={'projection': 'polar'}
)
ax.bar(theta, values, width=width)
plt.show()

Understanding the Parts

Make it Colorful!

colors = plt.cm.viridis(
    np.linspace(0, 1, N)
)
ax.bar(
    theta, values,
    width=width, color=colors
)

graph TD
    A["Polar Bar Chart"] --> B["theta = Bar Position"]
    A --> C["height = Bar Size"]
    A --> D["width = Bar Thickness"]
    A --> E["color = Make it Pretty!"]

🏹 Quiver Plot

What is a Quiver?

A quiver holds arrows. A quiver plot shows many arrows on a grid!

Think of it like: Tiny flags showing wind direction at different spots.

The Arrow Recipe

Each arrow needs:

• X, Y: Where to put it
• U, V: Which way it points and how long

import matplotlib.pyplot as plt
import numpy as np

# Grid of positions
X, Y = np.meshgrid(
    np.arange(0, 5, 1),
    np.arange(0, 5, 1)
)

# Arrow directions
U = np.ones_like(X)  # Right
V = np.ones_like(Y)  # Up

plt.quiver(X, Y, U, V)
plt.title('Arrows pointing up-right')
plt.show()

Understanding U and V

Make Arrows Change Based on Position

X, Y = np.meshgrid(
    np.linspace(-2, 2, 10),
    np.linspace(-2, 2, 10)
)

# Arrows point AWAY from center
U = X
V = Y

plt.quiver(X, Y, U, V)
plt.title('Exploding outward!')
plt.show()

Magic: When U=X and V=Y, arrows point away from (0,0)!

graph TD
    A["Quiver Plot"] --> B["X,Y = Arrow Positions"]
    A --> C["U,V = Arrow Directions"]
    B --> D["meshgrid creates grid"]
    C --> E["Bigger U,V = Longer arrows"]

🌊 Stream Plot

What is a Stream Plot?

Instead of separate arrows, stream lines flow like rivers!

Think of it like: Dropping leaves in water and watching where they float.

Basic Stream Plot

import matplotlib.pyplot as plt
import numpy as np

# Create grid
X, Y = np.meshgrid(
    np.linspace(-3, 3, 30),
    np.linspace(-3, 3, 30)
)

# Flow direction
U = -Y  # Left when above center
V = X   # Up when right of center

plt.streamplot(X, Y, U, V)
plt.title('Circular Flow!')
plt.show()

What Makes Streams Special?

Add Color to Show Speed

speed = np.sqrt(U**2 + V**2)

plt.streamplot(
    X, Y, U, V,
    color=speed,
    cmap='coolwarm'
)
plt.colorbar(label='Speed')
plt.show()

Result: Blue = slow, Red = fast!

Control Stream Density

# Fewer lines
plt.streamplot(
    X, Y, U, V, density=0.5
)

# More lines
plt.streamplot(
    X, Y, U, V, density=2
)

graph TD
    A["Stream Plot"] --> B["X,Y = Grid Points"]
    A --> C["U,V = Flow Direction"]
    A --> D["density = How many lines"]
    A --> E["color = Speed or value"]
    C --> F["Lines connect smoothly"]

🎨 Quick Reference

Polar Plot Basics

fig, ax = plt.subplots(
    subplot_kw={'projection': 'polar'}
)
ax.plot(theta, r)

Polar Bar Chart

ax.bar(theta, values, width=width)

Quiver Plot

plt.quiver(X, Y, U, V)

Stream Plot

plt.streamplot(X, Y, U, V)

🌟 The Big Picture

graph TD
    A["Specialty Plots"] --> B["Polar Plots"]
    A --> C["Vector Plots"]
    B --> D["Polar Line: Circular data"]
    B --> E["Polar Bar: Direction values"]
    C --> F["Quiver: Arrow field"]
    C --> G["Stream: Flow lines"]

You did it! You now understand:

• ✅ Polar plots for circular data
• ✅ Polar bars for direction-based values
• ✅ Quiver plots for arrow fields
• ✅ Stream plots for flow visualization

These tools help you show wind patterns, circular schedules, magnetic fields, and so much more!