What are Bézier Curves?
Bézier curves are parametric curves widely used in computer graphics, animation, CAD/CAM software, and font design. They were developed by Pierre Bézier at Renault in the 1960s for car body design.
Key Properties
- Convex Hull: The curve always lies within the convex hull of its control points.
- Endpoint Interpolation: The curve passes through the first and last control points.
- Tangent Properties: The curve is tangent to the control polygon at the endpoints.
- Affine Invariance: The curve transforms correctly under affine transformations.
Applications
- Vector Graphics: SVG paths, TrueType fonts, Adobe Illustrator
- Animation: Motion paths, easing functions, keyframe interpolation
- CAD/CAM: Industrial design, automotive body shapes
- Game Development: Camera movements, particle paths, UI animations
de Casteljau Algorithm
The de Casteljau algorithm is a recursive method for evaluating Bézier curves. At each level, it performs linear interpolation between adjacent points. This geometric construction makes the algorithm numerically stable and provides insight into the curve's structure.
Watch the construction lines animation to see how intermediate points are computed: linearly interpolate between control points at each level until convergence to a single point on the curve.
How to Interact
- Drag Control Points: Click and drag any control point (P₀, P₁, P₂, etc.) to reshape the curve.
- Adjust Parameter t: Use the slider or press Play to animate and see the construction.
- Change Curve Type: Switch between Linear, Quadratic, Cubic, and Higher Order curves.
- Toggle Construction: Show/hide the de Casteljau algorithm construction lines.