What are Bezier curves?

Question

Nathen Lamb · Accepted Answer

Step 1
A curve is an infinitely large set of points. Each point has two neighbors except endpoints.
A Bezier curve is a parametric curve used in computer graphics and related fields to produce smooth curves and it uses Burnstein polynomial as a basis.
Given

n + 1

points

(P 0, \dots, P_{n})

called the control points, the Bezier curve defined by these points is defined as:

P (t) = \sum B_{i_{i = 0}}^{n^{n}} (t) . P_{i}, t \in [0, 1]

where B(t) is the Burnstein polynomial and:

B_{i}^{n} (t) = (\begin{matrix} n \\ i \end{matrix}) t^{i} {(1 - t)}^{n - i}, (\begin{matrix} n \\ i \end{matrix}) = \frac{n!}{i! (n - i)!}

Step 2
Bezier curve follows the following properties:
They generally follow the shape of the control polygon, which consists of the segments joining the control points.
They always pass through the first and last control points.
They are contained in the convex hull of their defining control points.
The degree of the polynomial defining the curve segment is one less that the number of defining polygon point. Therefore, for 4 control points, the degree of the polynomial is 3, i.e. cubic polynomial.
A Bezier curve generally follows the shape of the defining polygon.
The direction of the tangent vector at the end points is same as that of the vector determined by first and last segments.
The convex hull property for a Bezier curve ensures that the polynomial smoothly follows the control points.
No straight line intersects a Bezier curve more times than it intersects its control polygon.
They are invariant under an affine transformation.
Bezier curves exhibit global control means moving a control point alters the shape of the whole curve.
A given Bezier curve can be subdivided at a point

t = t_{0}

into two Bezier segments which join together at the point corresponding to the parameter value

t = t_{0}

.

What are Bezier curves?

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