6. BEZIER CURVE
How to Draw a Bézier Curve
Bézier curves are widely used in graphics and animations for creating smooth curves that can be scaled indefinitely. They are defined by points called control points. Here’s a step-by-step guide on how to draw a Bézier curve:
Step 1: Understand the Control Points
A Bézier curve requires at least two control points:
- Linear Bézier (1st degree): 2 control points (P0, P1)
- Quadratic Bézier (2nd degree): 3 control points (P0, P1, P2)
- Cubic Bézier (3rd degree): 4 control points (P0, P1, P2, P3)
Step 2: Define Control Points
- Choose your points on a coordinate system. For example:
- P0 at (x0, y0)
- P1 at (x1, y1)
- P2 at (x2, y2) for quadratic.
- P3 at (x3, y3) for cubic.
Step 3: Calculate the Bézier Curve
Using the Bézier curve formula, you can calculate the position of points on the curve. For a cubic Bézier curve, it can be represented as:
$$
B(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t) t^2 P_2 + t^3 P_3 \quad \text{for } t \in [0, 1]
$$
Step 4: Draw the Curve
- Choose values for t (from 0 to 1 in increments, e.g., 0, 0.1, 0.2,…, 1).
- Calculate B(t) for each value of t using the formula.
- Plot the points calculated from B(t) on a graph.
Step 5: Connect the Points
After plotting the points, connect them smoothly to form the Bézier curve.
Applications of Bézier Curves
Bézier curves have numerous applications across various fields, including:
- Graphic Design: Used to create smooth, scalable shapes and paths in vector graphics.
- Animation: To define motion paths for objects, allowing for fluid movement transitions.
- Font Design: Used to design characters and glyphs in scalable vector formats.
- Computer-Aided Design (CAD): Employed in architecture and engineering for curves in design.
- Game Development: For creating paths and trajectories for moving objects.
The combination of control points offers flexibility and precision, making Bézier curves a foundational component in many digital design and animation applications
SCIENCE LABS 