Secondary Axes¶

Secondary axes display an alternative scale or transformation of the same data, useful for showing different units or transformations.

secondary_xaxis / secondary_yaxis¶

Unlike twinx()/twiny() which create independent axes for different data, secondary axes transform the same data to a different scale.

Basic Usage¶

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

fig, ax = plt.subplots()

x = np.linspace(0, 100, 100) y = np.sin(x * np.pi / 50)

ax.plot(x, y) ax.set_xlabel('Distance (km)')

Secondary axis showing same data in miles¶

def km_to_miles(x): return x * 0.621371

def miles_to_km(x): return x / 0.621371

secax = ax.secondary_xaxis('top', functions=(km_to_miles, miles_to_km)) secax.set_xlabel('Distance (miles)')

plt.show() ```

Temperature Conversion¶

```python fig, ax = plt.subplots()

temp_c = np.linspace(-40, 100, 100) y = np.exp(-temp_c / 50)

ax.plot(temp_c, y) ax.set_xlabel('Temperature (°C)') ax.set_ylabel('Value')

Secondary axis in Fahrenheit¶

def c_to_f(x): return x * 9/5 + 32

def f_to_c(x): return (x - 32) * 5/9

secax = ax.secondary_xaxis('top', functions=(c_to_f, f_to_c)) secax.set_xlabel('Temperature (°F)')

plt.show() ```

Secondary Y-Axis¶

```python fig, ax = plt.subplots()

x = np.linspace(0, 10, 100) y = x ** 2

ax.plot(x, y) ax.set_ylabel('Energy (Joules)')

Secondary axis showing same energy in calories¶

def j_to_cal(x): return x / 4.184

def cal_to_j(x): return x * 4.184

secax = ax.secondary_yaxis('right', functions=(j_to_cal, cal_to_j)) secax.set_ylabel('Energy (calories)')

plt.show() ```

Logarithmic Secondary Axis¶

```python fig, ax = plt.subplots()

x = np.linspace(1, 100, 100) ax.plot(x, x ** 2) ax.set_xlabel('Frequency (Hz)') ax.set_yscale('log')

Secondary axis in decades¶

def hz_to_decades(x): return np.log10(x)

def decades_to_hz(x): return 10 ** x

secax = ax.secondary_xaxis('top', functions=(hz_to_decades, decades_to_hz)) secax.set_xlabel('Frequency (decades)')

plt.show() ```

Comparison: Secondary vs Twin Axes¶

When to Use Each¶

Secondary Axes:

• Same measurement in different units (km ↔ miles)
• Same scale with transformation (linear ↔ log)
• Wavelength and frequency

Twin Axes:

• Price and volume on same chart
• Temperature and humidity
• Different physical quantities

Practical Examples¶

1. Wavelength and Frequency¶

```python fig, ax = plt.subplots()

wavelength = np.linspace(400, 700, 100) # nm intensity = np.exp(-((wavelength - 550) ** 2) / 5000)

ax.plot(wavelength, intensity) ax.set_xlabel('Wavelength (nm)') ax.set_ylabel('Intensity')

Frequency = c / wavelength¶

c = 3e8 # m/s

def wavelength_to_freq(wl): return c / (wl * 1e-9) / 1e12 # THz

def freq_to_wavelength(f): return c / (f * 1e12) / 1e-9 # nm

secax = ax.secondary_xaxis('top', functions=(wavelength_to_freq, freq_to_wavelength)) secax.set_xlabel('Frequency (THz)')

plt.show() ```

2. Pressure Units¶

```python fig, ax = plt.subplots()

altitude = np.linspace(0, 50000, 100) # meters pressure = 101325 * np.exp(-altitude / 8500) # Pa

ax.plot(altitude / 1000, pressure / 1000) ax.set_xlabel('Altitude (km)') ax.set_ylabel('Pressure (kPa)')

Secondary in atmospheres¶

def kpa_to_atm(x): return x / 101.325

def atm_to_kpa(x): return x * 101.325

secax = ax.secondary_yaxis('right', functions=(kpa_to_atm, atm_to_kpa)) secax.set_ylabel('Pressure (atm)')

plt.show() ```

3. Date and Day Number¶

```python import matplotlib.pyplot as plt import matplotlib.dates as mdates import numpy as np from datetime import datetime, timedelta

fig, ax = plt.subplots()

Data for one year¶

start = datetime(2024, 1, 1) days = np.arange(0, 365) dates = [start + timedelta(days=int(d)) for d in days] values = np.sin(days * 2 * np.pi / 365)

ax.plot(days, values) ax.set_xlabel('Day of Year') ax.set_ylabel('Value')

plt.show() ```

Key Parameters¶

Common Pitfalls¶

1. Inverse Function Required¶

```python

Both forward AND inverse functions needed¶

def forward(x): return x * 2

def inverse(x): return x / 2

CORRECT¶

secax = ax.secondary_xaxis('top', functions=(forward, inverse))

WRONG: Missing inverse¶

secax = ax.secondary_xaxis('top', functions=(forward,))¶

```

2. Non-Monotonic Transforms¶

Secondary axes work best with monotonic transformations. Non-monotonic transforms may produce unexpected results.

3. Log Scale Interactions¶

When using log scales, ensure transformations handle the log space correctly.

Exercises¶

Exercise 1. Plot temperature data in Celsius for 12 months (e.g., [5, 7, 12, 18, 23, 28, 31, 30, 25, 18, 11, 6]) and add a secondary y-axis that shows the values in Fahrenheit using the conversion \(F = C \times 9/5 + 32\). Use secondary_yaxis with forward and inverse functions.

import matplotlib.pyplot as plt
import numpy as np

months = np.arange(1, 13)
temps_c = [5, 7, 12, 18, 23, 28, 31, 30, 25, 18, 11, 6]

fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(months, temps_c, 'o-', color='tab:blue')
ax.set_xlabel('Month')
ax.set_ylabel('Temperature (°C)', color='tab:blue')
ax.set_xticks(months)

secax = ax.secondary_yaxis('right',
                            functions=(lambda c: c * 9/5 + 32,
                                       lambda f: (f - 32) * 5/9))
secax.set_ylabel('Temperature (°F)', color='tab:red')
ax.set_title('Monthly Temperature with Celsius and Fahrenheit')
plt.show()

Exercise 2. Plot the function \(y = e^x\) for x in \([0, 5]\) with a primary x-axis in linear scale. Add a secondary x-axis on top that shows the natural log of the x values. Use secondary_xaxis with functions=(np.log, np.exp).

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(0.1, 5, 200)
y = np.exp(x)

fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(x, y, color='navy')
ax.set_xlabel('x (linear)')
ax.set_ylabel(r'$e^x$')

secax = ax.secondary_xaxis('top', functions=(np.log, np.exp))
secax.set_xlabel('ln(x)')
ax.set_title(r'$y = e^x$ with Secondary Log Axis')
plt.show()

Exercise 3. Create a plot showing distance in kilometers over time in hours. Add a secondary y-axis that converts kilometers to miles (1 km = 0.621371 miles) and a secondary x-axis that converts hours to minutes. Use sample data time = [0, 1, 2, 3, 4, 5] and distance = [0, 10, 25, 45, 60, 80].

import matplotlib.pyplot as plt

time_h = [0, 1, 2, 3, 4, 5]
distance_km = [0, 10, 25, 45, 60, 80]

fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(time_h, distance_km, 'o-', color='steelblue', linewidth=2)
ax.set_xlabel('Time (hours)')
ax.set_ylabel('Distance (km)')

secax_y = ax.secondary_yaxis('right',
                              functions=(lambda km: km * 0.621371,
                                         lambda mi: mi / 0.621371))
secax_y.set_ylabel('Distance (miles)')

secax_x = ax.secondary_xaxis('top',
                              functions=(lambda h: h * 60,
                                         lambda m: m / 60))
secax_x.set_xlabel('Time (minutes)')

ax.set_title('Distance over Time with Unit Conversions')
plt.show()

Exercise 4. Create a plot of temperature in Celsius over 24 hours. Add a secondary y-axis showing Fahrenheit using the conversion F = C * 9/5 + 32. Verify that the two axes stay synchronized by checking that 0°C aligns with 32°F.

import matplotlib.pyplot as plt
import numpy as np

hours = np.arange(24)
temp_c = 10 + 8 * np.sin((hours - 6) * np.pi / 12)

fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(hours, temp_c, 'b-o', markersize=4)
ax.set_xlabel('Hour of Day')
ax.set_ylabel('Temperature (°C)')

secax = ax.secondary_yaxis('right',
                            functions=(lambda c: c * 9/5 + 32,
                                      lambda f: (f - 32) * 5/9))
secax.set_ylabel('Temperature (°F)')
ax.set_title('Daily Temperature with Dual Units')
ax.axhline(0, color='gray', linestyle=':', alpha=0.5, label='0°C = 32°F')
ax.legend()
plt.show()

Exercise 5. Explain the difference between secondary_yaxis and twinx. When should you use each? Give a concrete example where using twinx for a unit conversion would be incorrect.

# WRONG — manually plotting the same data twice
fig, ax1 = plt.subplots()
ax1.plot(time, temp_c, 'b-')
ax1.set_ylabel('°C')
ax2 = ax1.twinx()
ax2.plot(time, temp_c * 9/5 + 32, 'r-')  # Same data, duplicated!
ax2.set_ylabel('°F')