The Bézier curve representation is one that is utilized most frequently in computer graphics and geometric modeling. The curve is defined geometrically, which means that the parameters have geometric meaning - they are just points in three-dimensional space. It was developed by two competing European engineers in the late 1960s to attempt to draw automotive components.
In these notes, we develop the quadratic Bézier curve. This curve can be developed through a divide-and-conquer approach whose basic operation is the generation of midpoints on the curve. However, this time we develop the curve by calculating points other than midpoints - resulting in a useful parameterization for the curve.
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Development of the Quadratic Bézier Curve
Given three control points and we develop a divide procedure that is based upon a parameter , which is a number between 0 and (the illustrations utilize the value ). This proceeds as follows:
This is a similar procedure to the divide-and-conquer method in that geometric means are used to define points on the curve. Each time a new point is calculated, the control points are subdivided into two sets, each of which may be use to generate new subcurves. The method is identical to the divide-and-conquer method in the case .
Developing the Equation of the Curve
There is a different way of looking at this procedure - because there is a parameter involved. Each one of the points , , and is really a function of the parameter - and can be equated with since it is a point on the curve that corresponds to the parameter value . In this way, becomes a functional representation of the Bézier curve.
Writing down the algebra, we see that
Properties of the Quadratic Curve
The quadratic Bézier curve has the following properties, which can be easily verified.
Since the blending functions are non-negative and add to one, is an affine combination of the points , , and . Thus must lie in the convex hull of the control points for all . The convex hull of a triangle is the triangle itself.
If the points are colinear, then the convex hull is a straight line, and the curve must lie within the convex hull.
Clearly is the first point calculated by the divide and conquer method.
Lets show that is exactly the point obtained by performing the divide-and-conquer method, on the control points , and which were generated in the first step of the divide-and-conquer method. If we call this point , then by the divide-and-conquer method
Summarizing the Development of the Curve
We now have two methods by which we can generate points on the curve. The first of which is geometrically based - points are found on the curve by selecting successive points on line segments. The other is an analytic formula, which expresses the curve in functional notation.
The quadratic curve serves as a good example for discussing the development of the Bézier curve, but really only generates parabolas. This eliminates the curve for many applications where smooth curves with inflection points are necessary. The problem can be addressed by performing exactly the same steps as above, but utilizing the procedure on four control points - resulting in the cubic Bézier curve.