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A Product–Delay algorithm for graphic designAsok K. SenDepartment of Mathematical Sciences, Indiana University, 402 N. Blackford Street, Indianapolis, IN 46202, USAAvailable online 9 February Product–Delay algorithm is presented for creating graphic designs on a computer. In this algorithm two functions u(t) and v(t) are multiplied yielding a function x(t). Another function y(t) is formed by delaying or advancing x(t) by a fixed amount of time t. These functions are evaluated over a suitable time interval and the results are plotted in the x–y plane. For appropriate choices of the functions and parameters, the x–y displays exhibit interesting geometric patterns. In this paper the algorithm is illustrated with a pair of sine and square waves. It is shown that a wide variety of graphic designs can be created with these simple waveforms. By virtue of its simplicity this algorithm can be programmed easily and quickly using general purpose software such as Maple, Matlab or Mathematica. It can be executed on standard platforms such as IBM PC compatibles, Macintosh computers or workstations. Some results in polar coordinates are also Keywords: graphic design; product-delay algorithm; computer art1. IntroductionIt has long been recognized that mathematics and computers can be used in a complementary fashion for the creation of visual arts. Several mathematical methods have been developed over the years for generating graphic designs on the computer. The methods range from simple iteration schemes such as fractal generation techniques to solving systems of differential equations of nonlinear dynamics and chaos[1, 2]. In this paper we will present a different methodology for creating graphic designs using computers. This methodology is based on the idea of Lissajous figures and is referred to as the Product–Delay algorithm. We will illustrate its application to graphic design with a pair of sine and square waves. It will be seen that with these simple waveforms the Product–Delay algorithm can produce a multitude of interesting geometric formulation is based on multiplying two functions u(t) and v(t), yielding a new function x(t). It is convenient to think of the variable t as time. The function x(t) is delayed or advanced by a fixed amount of time. The resulting function y(t) is plotted against x(t) in a phase-plane like representation. For suitable choices of the functions and parameters, the x–y displays exhibit a rich variety of geometric patterns. We will appropriately call this procedure the Product–Delay algorithm. The concept of a time delay is often used in nonlinear signal processing applications including the analysis of chaotic dynamics. The purpose is to construct a phase portrait from the measurement of the time series of a single system variable. In these applications the original time series is delayed by a certain amount of time. The original time series and the delayed series are plotted to construct a phase portrait of the system dynamics[3]. The notion of a discrete time delay has also been used recently in a different context[4].All the computations reported in this paper are performed using the software Maple V (Release 4) on a SUN Ultrasparc workstation. They can also be carried out with other general purpose software such as Matlab or Mathematica, and on different platforms such as IBM PC compatibles, Macintosh computers and other . Pattern formation with sine and square wavesTo begin with our development we consider a simple combination of sine and square waves. In particular, we takeu(t)=a1+sin(2 πf1t), (1)v(t)=a2+sin(2 πf2t)/|sin(2 πf2t)|. (2)Here the vertical bars denote the absolute value of the function inside them. The function u(t) is a sine wave of amplitude unity, frequency f1 and offset a1, whereas the function v(t) represents a square wave of amplitude unity, frequency f2 and offset a2. We now form the productx(t)=u(t)v(t), (3)and select a delay/advance time δ to construct the functiony(t)=x(t+δ).. Effect of a nonzero offsetHere we investigate the effect of a nonzero offset in the sine or the square wave. First consider the sine wave u(t) as before with no offset (a1=0) but the square wave with an offset a2=−. In other words, we use u(t) as given by Eq. 5, but Eq. 6 is replaced withv(t)=−(2π.3000t)/|sin(2π.3000t)|. (8)These functions together with those given by Eq. 3 and Eq. 7 are evaluated over the time interval [0, ] at 500 numpoints. The result is the pattern depicted in Fig. 2. From a comparison of Fig. 1 and Fig. 2 it is apparent that Fig. 2 possesses a similar structure as Fig. 1 around its center but the presence of an offset in the square wave has led to significant deformation in Fig. 2 from the outer periphery inward. As the offset a2 is increased through negative values, the pattern continues to lose its central structure and in the limiting case a2=−1, ., when the magnitude of the negative offset equals the amplitude of the square wave, the inner structure around the center is completely lost and the twelve limbs intersect in a single point at the center. We now consider an offset whose magnitude is larger than the amplitude of the square wave. Fig. 3 portrays the scenario for a2=−2. Observe the emergence of a clear region around the center. If the offset is increased through large negative values, the central clear region tends to increase in size and the resulting pattern tends to approach a circle with no internal structure when the magnitude of the offset is very large. Note that all these patterns are rotationally Full Size version of this image (5K)Fig. 2. Same parameter values are used as in Fig. 1 except with an offset of − in the square waveDisplay Full Size version of this image (5K)Fig. 3. Same parameter values are used as in Fig. 1 except with an offset of − in the square waveNext we examine the effect of introducing a nonzero offset in the sine wave. Consider, for example, a1=−1 with a2=0. These choices lead to a bundle-like pattern with bilateral symmetry as shown in Fig. 4. If the offset a1 is increased through negative values, the bundle continues to become narrower, and for large negative values of a1, the pattern approaches a line of slope unity. Other interesting patterns can be created by introducing a nonzero offset in both the sine and square Full Size version of this image (5K)5. Concluding remarksWe have presented a few prototypical examples of graphic patterns that can be created with the Product–Delay algorithm using sine and square waves. Clearly it is possible to create numerous other patterns with these waveforms by appropriately selecting the various parameter values. An even richer variety of patterns can be produced by using other types of functions. In a future paper we will explore the effect of amplitude and frequency modulated waves and other complex waveforms on pattern generation

毕业论文开题报告 论文题目: 学生姓名: 学 号: 专 业: 指导教师: 年 月 日 开题报告填写要求 1.开题报告作为毕业设计(论文)答辩委员会对学生答辩资格审查的依据材料之一.此报告应在指导教师指导下,由学生在毕业设计(论文)工作前期内完成,经指导教师签署意见审查后生效. 2.开题报告内容必须用黑墨水笔工整书写,按成教处统一设计的电子文档标准格式打印,禁止打印在其它纸上后剪贴,完成后应及时交给指导教师签署意见. 3.学生查阅资料的参考文献应在3篇及以上(不包括辞典,手册),开题报告的字数要在1000字以上. 4.有关年月日等日期的填写,应当按照国标GB/T 7408—94《数据元和交换格式,信息交换,日期和时间表示法》规定的要求,一律用阿拉伯数字书写.如"2004年9月26日"或"2004-09-26". 毕 业 论 文 开 题 报 告 1.本课题的研究意义 中国互联网经过10年的持续发展。目前在普及应用上正步入崭新的多元化应用阶段。有关数据显示，中国宽带用户、网络国际出口带宽、上网方式和途径、网络应用服务更趋多样化。人们对互联网的使用广度、信用度、依赖度正在逐步提高。随着网络提供的功能和服务的进一步完善，网络应用化、生活化服务正逐步成熟。互联网的影响正逐步渗透到人们生产、生活、工作、学习的各个角落。中国互联网整体呈现较快的增长态势。但中国地区之间互联网发展水平、普及水平还存在明显的差距，呈现"东快、西慢，城快、村慢"的特点，因此，加大对于互联网应用和发展的研究力度，借鉴国外互联网应用的成功范例引入和普及互联网应用的先进经验是当务之急。 2.本课题的重点和难点 第一，从全国人口来看，互联网普及率还很低，仅有，与世界平均水平约14%还有较大差距。因此要普及互联网，让更多人来使用互联网是任重道远的事情。 第二，网上信息资源还不够丰富，质量比较好的、能反映我国优秀文化的、对广大网民有真正用处的信息还不够多。根据国信办的调查，截至2004年底，我国共有亿中文网页，比2003年底差不多翻了一番，但是仅占全世界网页数量(300多亿)的2%，比例很低。因此我们需要网上有更多丰富的内容，特别是健康的、有质量的、有针对性的内容。 第三，目前，互联网产业虽然在电子政务、电子商务方面进行了不少探索，也取得了一些成果，但是从整体上看网络应用水平和实效(即网民的用户体验)还比较初级。在技术驱动下产生的包括网络游戏、电子商务、无线宽带、VoIP、P2P等新的应用还没有形成成熟的盈利模式。 第四，新技术发展遭遇机遇和挑战。当前国外互联网新技术层出不穷，一直处于互联网发展的领先地位，而我国的自主创新能力比较薄弱，因此需要更加努力，迎头赶上。 第五，网络安全和网络文明面临严峻挑战。网络文明要靠政府法制、行业自律、网民的自觉来维护，而最关键的应该是网民素质的提高。就像交通管理一样，有交通法规的限制，也有警察的监管，但是最关键的还是司机素质的提高，否则交通事故还是无法避免的。同时，提供内容、服务的企业也应当承担其责任，实施行业自律。 3.论文提纲 我国互联网在若干领域的应用 1.互联网在政府中的应用 2.互联网在企业中的应用 3.互联网在消费群体中的应用 我国互联网应用前景 1.互联网将加速融入我们的生活 2.互联网经济逐渐产生效益 3.宽带网络建设打通互联网应用瓶颈 4.互联网成为国民经济新的增长点 毕 业 论 文 开 题 报 告 指导教师意见: (对本课题的深度,广度及工作量的意见) 指导教师: (亲笔签名) 年 月 日 系部审查意见: 系部负责人: (亲笔签名) 年 月 日 参考资料：