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Math Formula for Bezier Curve

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Math Formula for Bezier Curve - Formula Quest Mania

Math Formula for Bézier Curve: Explanation and Examples

The Bézier curve is a fundamental concept in computer graphics, animation, and design. Named after Pierre Bézier, who used it to model car surfaces, this mathematical curve is widely used in applications like vector graphic design, font creation, and animation paths. This article explains the math formula for Bézier curves, its types, and practical examples.

What is a Bézier Curve?

A Bézier curve is a parametric curve defined by a set of control points. The curve is constructed mathematically using Bernstein polynomials, with the shape determined by the positions of the control points. Bézier curves are highly versatile, making them ideal for creating smooth and scalable shapes in graphic design and computational geometry.

The Math Formula for Bézier Curves

The Bézier curve of degree n is defined as:

B(t) = Σ (n choose i) * (1 - t)n-i * ti * Pi for i = 0 to n

Where:

  • t: The parameter, ranging from 0 to 1.
  • n: The degree of the curve (one less than the number of control points).
  • Pi: The control points (P0, P1, ..., Pn).
  • (n choose i): The binomial coefficient, calculated as n! / (i! * (n - i)!).

This formula generates the curve by blending the control points using the weights defined by Bernstein polynomials. As t varies from 0 to 1, the curve is traced from the starting point (P0) to the endpoint (Pn).

Types of Bézier Curves

1. Linear Bézier Curve

A linear Bézier curve is the simplest form, defined by two control points. It is a straight line between the points.

Formula: B(t) = (1 - t) * P0 + t * P1

Example: If P0 = (0, 0) and P1 = (1, 1), the curve is a straight line from (0, 0) to (1, 1).

2. Quadratic Bézier Curve

A quadratic Bézier curve is defined by three control points and forms a parabolic shape. The formula is:

B(t) = (1 - t)2 * P0 + 2 * (1 - t) * t * P1 + t2 * P2

Example: With control points P0 = (0, 0), P1 = (0.5, 1), and P2 = (1, 0), the curve forms an arch from (0, 0) to (1, 0) passing through (0.5, 1).

3. Cubic Bézier Curve

A cubic Bézier curve is defined by four control points and is commonly used in graphic design tools like Adobe Illustrator. The formula is:

B(t) = (1 - t)3 * P0 + 3 * (1 - t)2 * t * P1 + 3 * (1 - t) * t2 * P2 + t3 * P3

Example: For control points P0 = (0, 0), P1 = (0.33, 1), P2 = (0.66, 1), and P3 = (1, 0), the curve creates a smooth S-shape.

Practical Examples of Bézier Curves

1. Bézier Curves in Graphic Design

Graphic designers use Bézier curves to create smooth and scalable shapes. For example, a quadratic Bézier curve can be used to design a curved logo or a rounded corner in an icon.

2. Animation Paths

In animation, Bézier curves define smooth motion paths for objects. By manipulating control points, animators can create natural movements.

3. Font Design

Most modern fonts use Bézier curves to define the shapes of letters and symbols. This ensures scalability without losing quality, making fonts clear at any size.

Step-by-Step Example: Drawing a Quadratic Bézier Curve

Let's create a quadratic Bézier curve using three control points:

  1. Control Points: P0 = (0, 0), P1 = (0.5, 1), and P2 = (1, 0).
  2. Formula: B(t) = (1 - t)2 * P0 + 2 * (1 - t) * t * P1 + t2 * P2.
  3. Calculate for t: Let t = 0.5:
    • B(0.5) = (1 - 0.5)2 * P0 + 2 * (1 - 0.5) * 0.5 * P1 + 0.52 * P2.
    • B(0.5) = 0.25 * (0, 0) + 0.5 * (0.5, 1) + 0.25 * (1, 0).
    • B(0.5) = (0, 0) + (0.25, 0.5) + (0.25, 0).
    • B(0.5) = (0.5, 0.5).

The point (0.5, 0.5) lies on the Bézier curve, illustrating how the formula determines positions.

Applications of Bézier Curves

  • Vector Graphics: Used in tools like Adobe Illustrator and Inkscape for creating scalable graphics.
  • Motion Planning: Bézier curves define paths in robotics and video games.
  • Data Interpolation: Used in computational geometry for smoothing data points.

Conclusion

Bézier curves are essential tools in mathematics and computer graphics, enabling the creation of smooth, precise curves and shapes. Whether designing logos, animating objects, or planning robotic paths, understanding the math formula for Bézier curves and their applications is invaluable for professionals in various fields. By mastering Bézier curves, you unlock endless possibilities for creativity and innovation.

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