Calculate The Bezier Curve Given The Following Points

Bezier curves are mathematical curves used in computer graphics and animation to model smooth paths. This calculator helps you plot a Bezier curve given control points.

Introduction

A Bezier curve is a parametric curve frequently used in computer graphics to define smooth curves. It's defined by a set of control points, where the shape of the curve is determined by the positions of these points.

Bezier curves are widely used in vector graphics editors, animation software, and CAD systems. They provide a flexible way to create complex shapes with smooth transitions.

How to Use This Calculator

Enter the coordinates of your control points in the input fields.
Click the "Calculate" button to generate the Bezier curve.
View the resulting curve in the chart below the calculator.
Use the "Reset" button to clear all inputs and start over.

For best results, use between 2 and 10 control points. The calculator will automatically adjust the curve based on the number of points you provide.

Formula

The parametric form of a Bezier curve of degree n is given by:

B(t) = Σ (from i=0 to n) [ C(n,i) * (1-t)^(n-i) * t^i * P_i ]

Where:

B(t) is the point on the curve at parameter t
C(n,i) is the binomial coefficient
t is the parameter (0 ≤ t ≤ 1)
P_i are the control points

For a quadratic Bezier curve (n=2), the formula simplifies to:

B(t) = (1-t)²P₀ + 2(1-t)tP₁ + t²P₂

Example Calculation

Let's calculate a quadratic Bezier curve with these control points:

Point	X Coordinate	Y Coordinate
P₀	0	0
P₁	1	2
P₂	2	0

Using the quadratic formula:

B(t) = (1-t)²(0,0) + 2(1-t)t(1,2) + t²(2,0)

For t=0.5:

B(0.5) = (0.25)(0,0) + 0.5(1,2) + 0.25(2,0) = (0.5,1)

FAQ

What are control points in a Bezier curve?

Control points are the points that define the shape of the Bezier curve. The curve will pass through the first and last control points but not necessarily through the intermediate ones.

How many control points can I use?

You can use between 2 and 10 control points. The calculator will automatically adjust the curve based on the number of points you provide.

What is the difference between a Bezier curve and a B-spline?

Bezier curves are defined by control points and have a fixed number of control points, while B-splines are defined by a set of control points and a knot vector, allowing for more complex shapes and local control.