Bezier Curve
A Bezier curve is a parametric curve used in computer graphics, animation, and engineering for modeling smooth curves. It was named after Pierre Bézier, who developed the curve for his work at Renault in the 1960s to design automobile bodies. Here are key aspects of the Bezier curve:
- Definition: A Bezier curve is defined by a set of control points. The number of control points determines the degree of the curve. For instance, a cubic Bezier curve is defined by four points: two endpoints and two control points.
- Mathematical Formulation: The curve can be expressed mathematically using the Bernstein polynomial:
\[
B(t) = \sum_{i=0}^n \binom{n}{i}(1-t)^{n-i} t^i P_i
\]
where \( P_i \) are the control points, \( n \) is the degree of the polynomial, and \( t \) is a parameter in the range [0,1].
- Properties:
- The curve starts at the first control point and ends at the last control point.
- The curve lies within the convex hull of its control points.
- It is tangent to the line segment from the first point to the second at the start, and from the penultimate point to the last at the end.
- Applications:
- Computer Aided Design (CAD): Used for designing smooth curves in engineering and industrial design.
- Font Design: TrueType fonts use cubic Bezier curves for glyph outlines.
- Animation and Motion Paths: For smooth transitions and paths in animations.
- Graphic Design: For creating curved paths in vector graphics software like Adobe Illustrator.
- History:
- Pierre Bézier developed the curve independently of Paul de Casteljau, who was working on similar problems at Citroen. Bézier's work became more widely known due to his publication and the adoption by the CAD industry.
- Initially, Bézier curves were used in automotive design, but their applications expanded as computer graphics technology developed.
- Types:
- Linear Bezier curve (degree 1): Straight line between two points.
- Quadratic Bezier curve (degree 2): Defined by three control points.
- Cubic Bezier curve (degree 3): Defined by four control points, widely used in graphics.
- Higher-order Bezier curves are less common but can be used for more complex shapes.
- De Casteljau's Algorithm: This recursive algorithm is used to evaluate points on a Bezier curve by successively interpolating between control points.
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