A DIVIDE AND CONQUER METHOD FOR CURVE DRAWING
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1 ECS 78 Course Notes A DIVIDE AND CONQUER METHOD FOR CURVE DRAWING Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview In the late 960s, two European engineers independently developed a mathematical curve formulation which was extremely useful for modeling and design and also easily adaptable to use on a computer system. The primary feature of this method was that the controlling parameters of the curve were simply points in three-dimensional space, and each of these points had an influence on the curve. This curve, commonly called the Bézier curve, is the representation that is most frequently used in computer graphics and geometric modeling. Here we present a method to generate a Bézier curve through a simple divide-and-conquer, or subdivision method. This will not give us a rigorous definition of the curve, but will serve as motivation as it follows the general construction procedure for the curve. Plus, it s cool!!! The Subdivision Procedure Our curve is defined by using three control points, P, and. [We ll start with three points, but the procedure will generally work with curves defined by n points.] Normally, these points can be arbitrarily placed in three-dimensional space, but for illustration purposes, we will limit ourselves to two-dimensions, and will use the points given in the illustration below. You will get the idea how things work.
2 P We normally call the points, P and control points, as they define and influence the curve. The curve is going to pass through the points and and will lie within the triangle P. P is a control point that serves as a handle or a influence on the curve. It is normally off the curve, only lying on the curve in very specific cases. Our divide and conquer procedure will split or subdivide the curve into two segments, each of which is again specified by three control points. With this procedure, we can recursively generate many small segments of the curve, which can be eventually approximated by straight lines when the curve is to be drawn. The procedure is quite simple, the most complicated mathematics being the calculation of midpoints of the lines connecting control points. The Basic Subdivision Procedure The procedure to subdivide the curve into two segments can be described as follows: First, let be the midpoint of the segment P, P then, let be the midpoint of segment P,
3 P and finally let P () be the midpoint of the segment P(). P P () We define P () to be a point on the curve. Simple procedure. Right??? Let s see what we have done. First, the notation is rather unusual, but we will use this over and over as we go along in the course, and I wanted to introduce it here. Just let me say get used to it! When you see notation of this type, you should anticipate that a P (j) i will appear soon, and it certainly will. We have generated another point on the curve P (). We already had two points on the curve, and, and now we have a third. We have effectively subdivided the curve into two segments. The first segment has control { } points,, P() (note that the beginning and end points are both on the curve, and the middle point is influencing the curve.), and the second segment has control points { } P (), P(),. 3
4 P Point on the curve P () New control points New control points So, our simple procedure has generated an additional point on the curve and two new sets of three control points. Now, we will work on each new piece to generate additional points on the curve. See what s coming??? Continuing the Subdivision Performing the procedure again, we use the control points {,, P() }, relabeling them for convenience as, P, and respectively, and again apply our procedure to generate a point on the curve. P First let be the midpoint of the segment P. 4
5 P then let be the midpoint of segment P. P and finally let P () as the midpoint of the segment P(). P P () We now define P () to be a point on the curve. So, we have produced another point on the curve, and created two new sets of control points. If we consider the control points P (), P(), and, generated in the first subdivision, and relabel them as, P, and respectively, we can again apply the subdivision procedure 5
6 P P P P as the midpoint of the segment P. as the midpoint of segment P. P () as the midpoint of the segment P(). P () 6
7 And again, P () is on the curve. The Subdivision Algorithm We now have five points on the curve,, P, and our three new generated points, and have generated four subcurves that we can use to generate additional points. You should now see how to proceed. At each step the process creates both a point on the curve and two new sets of control points, recursively subdividing the curve segments into two new curve segments, each of which can be handled in the same way. P The curve drawn by this method is a quadratic Bézier curve. Summary This is a somewhat unique method to define a curve, and probably not previously seen by many students. It is a geometric method, as it uses only the midpoint formula as it s fundamental tool. It uses the basic computer science paradigm of (sub)divide and conquer to calculate points on the curve. The curve can be drawn using computer graphics by calculating a somewhat-dense set of points, and connecting them with straight lines. All contents copyright (c) Computer Science Department, University of California, Davis All rights reserved. 7
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