Drawing Path by Bezier Curves

Overview

Bezier curves are a popular choice for drawing smooth curves in various fields, including computer-aided design (CAD), computer graphics, and animation. In this article, we will explore how to draw a path using Bezier curves, focusing on the connection between curves.

Understanding Bezier Curves

A Bezier curve is defined by a set of control points, which are used to create a smooth curve that passes through these points. The order of the control points determines the shape of the curve. A Bezier curve can be classified into several types, including:

Quadratic Bezier Curve: Defined by two control points
Cubic Bezier Curve: Defined by three control points

In this article, we will focus on cubic Bezier curves.

Step 1: Defining the Control Points

The control points of a cubic Bezier curve are defined as follows:

The first point is called the start point and has weight 1.
The second and third points are called control points, each having weight 1/2.
The fourth point is called the end point, which has weight 0.

Step 2: Calculating the Curve

To calculate the curve, we use a technique called de Casteljau’s algorithm. This algorithm involves recursively calculating the point at which each segment of the curve passes through.

Here are the steps:

Start with the initial control points and weight values.
For each point along the curve, calculate the point using de Casteljau’s algorithm.
Repeat step 2 until you reach the desired number of points.

Step 3: Implementing Bezier Curves in Code

To implement Bezier curves in code, we can use a simple recursive function that calculates each segment of the curve:

def bezier_curve(points):
    # Initialize an empty list to store the curve points
    curve = []

    def de_casteljau(x, t, points):
        # If there's only one point left, return its coordinates
        if len(points) == 1:
            return points[0]

        x1 = de_casteljau(x, t, points[:3])
        x2 = de_casteljau(x, t, [points[0], points[1]])

        # Return the interpolated point
        return (x1 * (1-t) + x2*t), (x2 * (1-t) + x1*t)

    # Calculate each segment of the curve using de Casteljau's algorithm
    for i in range(0, len(points)-1):
        t = i / (len(points)-1)
        x, y = de_casteljau(0, t, points)
        curve.append((x, y))

    return curve

# Define the control points
points = [(100, 200), (300, 400), (500, 600), (700, 800)]

# Calculate and print the Bezier curve points
curve_points = bezier_curve(points)

print(curve_points)

Connecting Curves Smoothly

To connect curves smoothly, we can use a combination of techniques:

Catmull-Rom spline: A smooth curve that passes through two control points.
B-Spline: A generalization of Catmull-Rom splines.

Here’s how to implement them in code:

import numpy as np

def catmull_rom_spline(points, num_segments):
    # Calculate the intermediate points using Catmull-Rom formula
    curve_points = []
    for i in range(num_segments):
        t1 = 2 * i / (num_segments - 1) + i * 2 / num_segments
        t2 = 2 * (i+1) / (num_segments - 1)
        
        p0, p1, p2, p3 = points[i], points[i+1], points[i+2], points[i+3]
        x, y = catmull_rom(p0[0], p0[1], p1[0], p1[1], t1), catmull_rom(p0[0], p0[1], p1[0], p1[1], t1)
        
        curve_points.append((x, y))
    
    return curve_points

def b_spline(points, num_segments):
    # Calculate the intermediate points using B-Spline formula
    curve_points = []
    for i in range(num_segments):
        t1 = 2 * i / (num_segments - 1) + i * 2 / num_segments
        t2 = 2 * (i+1) / (num_segments - 1)
        
        p0, p1, p2, p3 = points[i], points[i+1], points[i+2], points[i+3]
        x, y = b_spline(p0[0], p0[1], p1[0], p1[1], t1), b_spline(p0[0], p0[1], p1[0], p1[1], t1)
        
        curve_points.append((x, y))
    
    return curve_points

def catmull_rom(x1, y1, x2, y2, t):
    # Calculate the Catmull-Rom interpolated point
    t1 = 0.5 - t/2 + t**2 / 2
    t2 = t * (t-2) + 4*t*3 + 6*t**2
    
    return x1*(1-t)+x2*t + (1-2*t)*t*t + 3*t**2 - 2*t*t**2

def b_spline(x1, y1, x2, y2, t):
    # Calculate the B-Spline interpolated point
    t1 = t * (3-t)
    
    return x1*(2*t**3-3*t**2+t)+x2*(t**3-2*t**2+t)

# Define the control points
points = [(100, 200), (300, 400), (500, 600), (700, 800)]

# Calculate and print the Catmull-Rom spline curve points
catmull_rom_curve_points = catmull_rom_spline(points, num_segments=10)

# Calculate and print the B-Spline curve points
b_spline_curve_points = b_spline_spline(points, num_segments=10)

The code above calculates the Catmull-Rom spline curve points using a simple recursive function. The same can be applied to calculate the B-Spline curve points.

Conclusion

Bezier curves are a powerful tool for drawing smooth curves in various fields. By understanding how Bezier curves work and implementing them correctly, you can create beautiful and realistic graphics.

In this article, we have covered the basics of Bezier curves and provided examples of how to implement them using code. We also discussed techniques for connecting curves smoothly, including Catmull-Rom splines and B-Splines.

By mastering Bezier curves and related techniques, you’ll be well-equipped to tackle a wide range of graphical challenges in your projects.

Last modified on 2024-08-21