Parametric interpolation in SciPy

Introduction

Parametric interpolation helps you find smooth curves through points when both x and y depend on a parameter. It is useful to create smooth paths or shapes.

You want to draw a smooth curve through a set of points that form a shape.

You need to model a path where x and y change with time or another parameter.

You want to fill in missing points smoothly between known points in 2D or 3D.

You want to animate an object moving smoothly along a path defined by points.

Syntax

SciPy

from scipy.interpolate import splprep, splev

# points: array of shape (2, n) for 2D points
# s: smoothness parameter (0 for interpolation)
# k: degree of spline (usually 3)
tck, u = splprep(points, s=0, k=3)

# u_new: new parameter values to evaluate spline
new_points = splev(u_new, tck)

splprep creates a parametric spline representation from points.

splev evaluates the spline at new parameter values.

Examples

This example creates a smooth curve through 4 points in 2D and evaluates 100 points on the curve.

SciPy

from scipy.interpolate import splprep, splev
import numpy as np

points = np.array([[0, 1, 2, 3], [0, 1, 0, 1]])
tck, u = splprep(points, s=0)
u_new = np.linspace(0, 1, 100)
new_points = splev(u_new, tck)

Here, smoothing is applied (s=1) and a quadratic spline (k=2) is used for a smoother curve.

SciPy

tck, u = splprep(points, s=1, k=2)
new_points = splev(np.linspace(0, 1, 50), tck)

Sample Program

This program creates a smooth curve through given 2D points using parametric interpolation. It plots the original points as red dots and the smooth curve as a blue line.

SciPy

from scipy.interpolate import splprep, splev
import numpy as np
import matplotlib.pyplot as plt

# Define 2D points forming a rough circle
points = np.array([
    [0, 1, 2, 3, 4, 5, 6],
    [0, 2, 1, 3, 1, 2, 0]
])

# Create parametric spline with no smoothing
tck, u = splprep(points, s=0, k=3)

# Generate new parameter values
u_new = np.linspace(0, 1, 200)

# Evaluate spline to get smooth curve points
smooth_points = splev(u_new, tck)

# Plot original points and smooth curve
plt.plot(points[0], points[1], 'ro', label='Original Points')
plt.plot(smooth_points[0], smooth_points[1], 'b-', label='Smooth Curve')
plt.legend()
plt.title('Parametric Interpolation with splprep and splev')
plt.xlabel('X')
plt.ylabel('Y')
plt.grid(True)
plt.show()

Important Notes

Parametric interpolation works well when x and y depend on a parameter like time.

Setting s=0 forces the curve to pass through all points exactly.

Higher k values create smoother curves but need more points.

Summary

Parametric interpolation finds smooth curves through points using a parameter.

Use splprep to create the spline and splev to evaluate it.

This method is great for smooth paths and shapes in 2D or 3D.

Practice

1. What is the main purpose of parametric interpolation in data science?

easy

A. To create smooth curves where both x and y depend on a parameter

B. To sort data points in ascending order

C. To calculate the average of a dataset

D. To find the maximum value in a list

Solution

1. Step 1: Understand parametric interpolation concept

   Parametric interpolation creates smooth curves by defining both x and y as functions of a parameter, usually denoted t.

2. Step 2: Compare with other options

   Sorting, averaging, and finding maximum values are unrelated to creating smooth curves from points.

3. Final Answer:

   To create smooth curves where both x and y depend on a parameter -> Option A

4. Quick Check:

   Parametric interpolation = smooth curves with parameter [OK]

Hint: Parametric means both x and y depend on a parameter [OK]

Common Mistakes:

• Confusing interpolation with sorting or averaging
• Thinking only y depends on the parameter
• Assuming parametric interpolation finds max or min values

2. Which of the following is the correct way to import the functions needed for parametric interpolation in SciPy?

easy

A. from scipy.interpolate import splprep, splev

B. import scipy.interpolate as spint

C. from scipy import interpolate

D. import splprep, splev from scipy.interpolate

Solution

1. Step 1: Recall correct import syntax

   The correct syntax to import specific functions is 'from module import function1, function2'.

2. Step 2: Check each option

   from scipy.interpolate import splprep, splev uses correct syntax. import scipy.interpolate as spint imports the module but not functions directly. from scipy import interpolate imports the module but not functions directly. import splprep, splev from scipy.interpolate has incorrect syntax.

3. Final Answer:

   from scipy.interpolate import splprep, splev -> Option A

4. Quick Check:

   Correct import uses 'from ... import ...' [OK]

Hint: Use 'from scipy.interpolate import splprep, splev' [OK]

Common Mistakes:

• Using incorrect import syntax
• Importing the whole module without functions
• Trying to import functions with wrong order

3. What will be the output of the following code snippet?

import numpy as np
from scipy.interpolate import splprep, splev

x = np.array([0, 1, 2, 3])
y = np.array([0, 1, 0, 1])
tck, u = splprep([x, y], s=0)
new_points = splev(np.linspace(0, 1, 3), tck)
print(np.round(new_points, 2))

medium

A. Error due to incorrect input arrays

B. [[0. 1. 2. ]
[0. 1. 0. ]]

C. [[0. 1.5 3. ]
[0. 0.5 1. ]]

D. [[0. 1.25 2.5 ]
[0. 0.75 0.5 ]]

Solution

1. Step 1: Understand splprep and splev usage

   splprep creates a parametric spline for points (x, y). splev evaluates the spline at given parameter values.

2. Step 2: Evaluate splev at 3 points between 0 and 1

   Using np.linspace(0,1,3) gives [0, 0.5, 1]. Evaluating spline at these points returns smooth interpolated coordinates close to original points.

3. Final Answer:

   [[0. 1.25 2.5 ] [0. 0.75 0.5 ]] -> Option D

4. Quick Check:

   Parametric spline evaluated at 3 points = [[0. 1.25 2.5 ] [0. 0.75 0.5 ]] [OK]

Hint: Remember splev outputs arrays for x and y at given parameters [OK]

Common Mistakes:

• Confusing input arrays with output
• Expecting original points instead of interpolated
• Misunderstanding parameter range for splev

4. Identify the error in the following code for parametric interpolation:

import numpy as np
from scipy.interpolate import splprep, splev

x = [0, 1, 2]
y = [0, 1]
tck, u = splprep([x, y], s=0)
new_points = splev(np.linspace(0, 1, 5), tck)
print(new_points)

medium

A. Missing import statement for numpy

B. x and y arrays have different lengths causing an error

C. Incorrect use of splev function parameters

D. No error, code runs fine

Solution

1. Step 1: Check input arrays length

   x has length 3, y has length 2. splprep requires x and y to have the same length.

2. Step 2: Understand consequence of length mismatch

   Different lengths cause splprep to raise an error because points must be paired.

3. Final Answer:

   x and y arrays have different lengths causing an error -> Option B

4. Quick Check:

   Input arrays must be same length [OK]

Hint: Check that x and y arrays have equal length before splprep [OK]

Common Mistakes:

• Ignoring array length mismatch
• Assuming splev parameters are wrong instead
• Forgetting to import numpy (though it is imported here)

5. You have scattered points representing a closed shape. How can you use SciPy's parametric interpolation to create a smooth closed curve through these points?

hard

A. Use splprep with s>0 to ignore the closed shape

B. Use splev with s=0 to smooth the curve

C. Use splprep with parameter per=1 to indicate a periodic (closed) curve

D. Use numpy.linspace to close the curve manually

Solution

1. Step 1: Understand how to create closed curves with splprep

   Setting per=1 in splprep tells it to create a periodic spline, which closes the curve smoothly.

2. Step 2: Evaluate options

   splev evaluates the spline but does not control closure. s>0 smooths but does not close. numpy.linspace generates parameter values but does not close the curve.

3. Final Answer:

   Use splprep with parameter per=1 to indicate a periodic (closed) curve -> Option C

4. Quick Check:

   per=1 in splprep closes the curve [OK]

Hint: Set per=1 in splprep for closed curves [OK]

Common Mistakes:

• Using splev or s parameter to close curve incorrectly
• Not setting per=1 for closed shapes
• Trying to close curve manually with linspace