The Butterfly Curve
The Butterfly Curve
复制标题
蝴蝶曲线
DOI：
10.1080/00029890.1989.11972217
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发表时间：
1989
影响因子：
0.5
通讯作者：
T. H. Fay
中科院分区：
文献类型：
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作者：
T. H. Fay
我们中的许多人被平面曲线的绘制所吸引，因为它们内在的美表现在对称性、树叶和叶瓣、渐近行为、不对称或观察者眼中的任何东西上。经典的玫瑰曲线之类的似乎不会激起多少学生的热情，但在计算机的帮助下绘制更复杂的曲线似乎会激发更多的兴趣，并经常带来惊喜。在电脑屏幕上看到一条曲线被画出来的动力有一些特殊之处，这使得整个过程更加有趣。这本笔记的目的是要指出，在获得教学目标的同时，绘制曲线是相当愉快和有趣的。从教学上讲，设置自变量的界限、绘制步长、屏幕窗口大小等重要问题都需要在“按下运行按钮”之前解决。在这里，关于大小、周期、对称性、域等的观察对学生来说具有新的相关性：最小化运行时间。此外，绘制形状未知的曲线会让人兴奋不已。例如，由极方程p=a+b cos(No)和p2=a+b cos(No)描述的曲线比由p=a cos(No)描述的曲线有趣得多(见[1])。当a b，a=b，n是偶数，n是奇数等等时，学生们经常会对他们的行为感到惊讶。其他有趣的花瓣曲线，学生可以预测花瓣的数量，通过用n和m奇数绘制方程p=(4cos(No)+cos(Mo))/cos(O)得到；如果n或m中的一个是偶数，曲线有一条渐近线和相当不同的外观(见[2])。例如，有趣的曲线由(n，m)E{(5，3)，(3，5)(1，9)，(3，2)}产生。也许在作者发现的所有曲线中，最有趣和最美丽的是附图中显示的“蝴蝶”。蝴蝶的方程为p=ecos(O)2cos(40)+sin5(0/12)。上一学期，
Many of us are attracted to the drawing of plane curves because of their inherent beauty expressed in symmetry, leaves and lobes, asymptotic behavior, asymmetry, or whatever is in the eye of the beholder. The classical rose curves and the like do not seem to provoke much student enthusiasm, but sketching more complicated curves, with the aid of a computer, seems to spark more interest and often brings surprises. There is something special about the dynamics of seeing a curve being drawn on the computer screen that makes the whole process more enjoyable. It is the purpose of this note to point out that there is considerable pleasure and fun to be had with curve plotting while simultaneously obtaining teaching objectives. Pedagogically, the important questions of setting the bounds on the independent variable, the plotting step size, the screen window size, all need to be addressed prior to "hitting the RUN button." Here observations about size, period, symmetry, domain, and the like, take on new relevance for the student: minimize run time. Moreover, there is an excitement generated by plotting curves whose shape is not a priori known. For example, the curves described by the polar equations p = a + b cos( nO) and p2 = a + b cos(nO ) are much more interesting than those described by p = a cos(nO) (see [1]). Students are often surprised by their behavior when a b, a = b, n is even, n is odd, etc. Other interesting petal curves, for which students can predict the number of petals, are obtained from plotting the equations p = (4 cos(nO) + cos(mO))/cos(O) with both n and m odd; if one of n or m is even, the curve has an asymptote and quite a different appearance (see [2]). For example, amusing curves are produced with (n, m) E {(5, 3), (3, 5) (1, 9), (3, 2)}. Perhaps the most interesting and beautiful of all the curves that the author has discovered is that of the "butterfly" displayed in the accompanying figure. The equation of the butterfly is p = ecos(O) 2 cos(40) + sin5(0/12). The last term,