Bezier Curves/Surfaces



Bezier Curves

What they are and how they are used in today’s society

Introduction:

For years, engineers and architects relied solely on compasses and straight edges to develop and depict curves and surfaces. While the curves produced through the use of compasses and straight edges could be accurately reproduced, it was difficult to create complex curves using these tools. The other option in depicting a curve or surface was to free-hand the drawing. As one can imagine depictions produced in this way were extremely difficult to reproduce and required extremely complex mathematical analysis to quantify. The difficulty of reproducing such curves and surfaces began to become more and more of a problem as companies began to mass produce specifically shaped molds of plastics and metals. Realizing that this was a problem and hoping to make surfaces far easier to reproduce, Pierre Bezier developed what are known today as Bezier surfaces. These surfaces proved to be far more convenient to use in practical applications than did their free hand and compass and straight edge predecessors.

These curves/surfaces were first used by Bezier himself in 1972 to design automobiles for the Renault car company. These curves/surfaces are widely used today in the design and implementation of buildings and play a large role in many computer programs such as CAD, Adobe Illustrator and Corel Draw. They are also used to digitize letter forms (design new fonts).

What are they?

The method by which Bezier developed these curves is relatively simple. He began by defining a curve contained within a cube. This curve was given by a parametric equation equal to y=x2. (r(t)=) NOTE:This curve becomes three dimensional if control points are in three dimensions.. By shifting the cube in which the curve is contained, naturally the curve changes shape. In other words the shape of the resulting parallelepiped determines the shape of the parametric curve. Bezier used the four vertices to describe the shift of the parallelepiped and therefore the shape of the resulting curve. Because the curve is always within this imaginary parallelepiped, it can be said that the curve is contained completely within the “convex hull” of its control points. (The control points are the vertices that define the parallelepiped.)

Below is a Bezier curve showing control points as well as the line segments connecting those control points. Notice that the curve connects the two endpoints but is not in contact with the other two control points. All Bezier curves connect the two end points. They may or may not, however, be in contact with any of the other control points. While it may not be obvious, it is also interesting to note that the slope of the curve at the initial endpoint is equal to the slope of the line connecting the first two control points. The slope of the curve at the final endpoint is also equal to the slope of the lines connecting the final two control points. This characteristic is true for all Bezier curves of all degrees and in all dimensions.

[pic]

When converting this curve in to three dimensions it is important to note that our curve is a cubic. This is important because a quadratic Bezier curve may have control points in three dimensions but will still only be two dimensional. This is so because all three control points (We know there will only be three control points see below) will lie on a single plane. Points on a plane need only two dimensions to be expressed accurately. Because you may need three dimensions to accurately describe a collection of four points, (Bezier cubics have four control points) any Bezier curve of degree three or higher may need to be expressed in three dimensions. Below is a cubic curve, similar to the one shown above, that has been converted to three dimensions. The line segments connecting the control points have also been shown. Notice that the curve lies completely within the convex hull of the control points. Again, the slope of the curve at the first end point is equal to the slope of the line segment connecting the first two control points. Likewise for the other endpoint and the line segment connecting it with the third control point.

[pic]

How are they made?

Now that a very general idea has been given as to how Bezier curves were developed and what they look like, let us now examine the mathematical functions that give rise to these unique curves.

A Bezier curve can be expressed parametrically as:

r(t)=

Here the points (x0,y0,z0), (x1,y1,z1)etc. are the control points and n is the degree of the curve. (0 ................
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