Find new control point when endpoint change in cubic bezier curve

Finding New Control Points for Cubic Bezier Curves

Cubic Bezier curves are widely used in computer graphics and design for their smooth and flexible shapes. A cubic Bezier curve is defined by four control points: two endpoints and two control points.

Understanding the Problem

When the endpoint of a cubic Bezier curve changes, we often need to adjust the control points to maintain the curve’s shape and smoothness. This article explores techniques for finding the new control points when an endpoint is modified.

Methods for Finding New Control Points

There are various methods to calculate new control points based on the endpoint changes. Here are two common approaches:

• Keeping Existing Shape: This approach aims to preserve the original shape of the curve as much as possible while moving the endpoint.
• Matching Tangents: This method focuses on maintaining the tangent directions at the endpoints, ensuring a smooth transition between the curve segments.

Keeping Existing Shape

In this method, we aim to shift the control points in a way that minimizes the change in the overall curve shape. One technique involves using the following steps:

• Find the Direction Vector: Calculate the vector from the old endpoint to the new endpoint.
• Shift Control Points: Shift both control points closest to the changed endpoint along the direction vector. The amount of shift is proportional to the distance between the old and new endpoints and the distance between the control point and the endpoint.

Matching Tangents

This method ensures a smooth curve by matching the tangent directions at the endpoints. The procedure can be summarized as follows:

• Calculate Tangent Directions: Determine the tangent vectors at the old endpoint and the new endpoint.
• Find New Control Points: Adjust the control points to ensure that the tangent vectors at the endpoints remain consistent. This involves solving a system of equations based on the tangent directions.

Code Example

Here’s a Python example demonstrating how to find a new control point while maintaining the shape of a cubic Bezier curve using the “Keeping Existing Shape” approach:

import numpy as np

def find_new_control_point(old_endpoint, new_endpoint, control_point):
  """Finds a new control point to maintain the curve shape.

  Args:
    old_endpoint: The original endpoint of the curve.
    new_endpoint: The new endpoint of the curve.
    control_point: The control point closest to the endpoint.

  Returns:
    The new control point.
  """
  direction_vector = new_endpoint - old_endpoint
  shift_factor = np.linalg.norm(control_point - old_endpoint) / np.linalg.norm(old_endpoint - new_endpoint)
  new_control_point = control_point + shift_factor * direction_vector
  return new_control_point

Comparison of Methods

Conclusion

Finding new control points for a cubic Bezier curve when the endpoint changes requires balancing the need to maintain the shape, smoothness, and tangent properties. The method chosen will depend on the specific requirements of the application. Both the “Keeping Existing Shape” and “Matching Tangents” approaches provide useful tools for adjusting Bezier curves effectively.