What is a Bézier curve?

Bézier curves serve as a primary tool to model smooth and scalable curves. These curves find their applications in diverse fields such as Computer-aided design (CAD), Computer-aided manufacturing (CAM), and graphic design.

Definition

A Bézier curve is a parametric curve used in two-dimensional graphic applications. The curve is defined by a set of at least two control points. The path followed by the bezier curve starts at the first control point and ends at the last control point, being controlled by the positions of intermediate control points.

Constructing Bézier curves: De Casteljau's algorithm

De Casteljau's algorithm is a recursive method that constructs a Bézier curve using linear interpolation between the two control points.

To construct a Bézier curve with any number of control points:

• Divide the control polygon into segments by computing the control points between the consecutive (two) control points.

• Interpolate new points by finding control points between the previously calculated control points.

• Continue this process until a single point remains; it lies on the Bézier curve.

Bézier curves: algebraic formulas

Given below are the formulas to find the control points between two control points.

• To find the control point between the two points $b_0$ and $b_1$:

Here,

$b_0^1$is the new control point between control points $b_0$ and $b_1$and $t$ is the parameter that determines the position of the control point between the two points. It ranges from 0 to 1.

• To interpolate a new point ($b_1^1$) between the two points $b_1$ and $b_2$:

• To interpolate a new point ($b_0^2$) between the two added control points $b_0^1$ and $b_1^1$:

• We can substitute the values of $b_0^1$ and $b_1^1$and rewrite the above equation as:

Note: You can use the above equations to find other points on the Bézier curve, but the process becomes increasingly complex as more points are added, making it difficult to show the complete construction here.

Demonstration: quadratic Bézier

The demonstration shows how to connect three points smoothly using quadratic Bézier curves.

Note: You can visit this link for a nice Bézier curve animation. You can modify the control points and see how the changes effect the curve.

Properties of Bézier curves

Bézier curves have several important properties, some of which are explained below.

• Affine invariance: Bézier curves are invariant under affine transformationsA linear mapping method that preserves points, straight lines, and planes., such as translation, rotation, scaling, and skewing. It means that any transformations applied to the control points are reflected in the curve.

• Convex hull property: The Bézier curve lies within the convex hullA minimum convex set enclosing a sample of points. of its control points. It means that it stays close to the shape formed by its control points.

• Variation diminishing property: The property states that Bézier curves are smoother than their control polygons. When a line intersects the curve, it will have either fewer or the same number of intersections with the curve compared to the control polygon.

• Subdivision property: Bézier curves can be subdivided at any point into two separate Bézier curves without changing the overall shape of the curve.

Conclusion

Bézier curves are smooth curves used in computer graphics. They are defined by control points that determine their shape and smoothness. Such curves offer a simple and powerful way to create attractive curves for various applications.

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