|
Butterfly Curve (transcendental)
The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989. __TOC__ Equation The curve is given by the following parametric equations: :x = \sin t \!\left(e^ - 2\cos 4t - \sin^5\!\Big(\Big)\right) :y = \cos t \!\left(e^ - 2\cos 4t - \sin^5\!\Big(\Big)\right) :0 \le t \le 12\pi or by the following polar equation: :r = e^ - 2\cos 4\theta + \sin^5\left(\frac\right) The term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye. Developments In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent. See also https://books.google.com/books?id=AsYaCgAAQBAJ&dq=OSCAR+RAMIREZ+POLAR+EQUATION&pg=PA732 * Butterfly curve (algebraic) In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation :x^6 + y^6 = x^2. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Transcendental Curve
Butterflies are insects in the macrolepidopteran clade Rhopalocera from the order Lepidoptera, which also includes moths. Adult butterflies have large, often brightly coloured wings, and conspicuous, fluttering flight. The group comprises the large superfamily Papilionoidea, which contains at least one former group, the skippers (formerly the superfamily "Hesperioidea"), and the most recent analyses suggest it also contains the moth-butterflies (formerly the superfamily "Hedyloidea"). Butterfly fossils date to the Paleocene, about 56 million years ago. Butterflies have a four-stage life cycle, as like most insects they undergo complete metamorphosis. Winged adults lay eggs on the food plant on which their larvae, known as caterpillars, will feed. The caterpillars grow, sometimes very rapidly, and when fully developed, pupate in a chrysalis. When metamorphosis is complete, the pupal skin splits, the adult insect climbs out, and after its wings have expanded and dried, it flie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Definition Formally, an analytic function ''f''(''z'') of one real or complex variable ''z'' is transcendental if it is algebraically independent of that variable. This can be extended to functions of several variables. History The transcendental functions sine and cosine were trigonometric tables, tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote: A revolutionary understanding of these circular functions occurred in the 17th century and was expli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Southern Mississippi
The University of Southern Mississippi (Southern Miss or USM) is a public research university with its main campus located in Hattiesburg, Mississippi. It is accredited by the Southern Association of Colleges and Schools to award bachelor's, master's, specialist, and doctoral degrees. The university is classified among "R1: Doctoral Universities – Very high research activity". Founded on March 30, 1910, the university is a dual campus institution, with its main campus located in Hattiesburg and its other large campus – Gulf Park – located in Long Beach. It has five additional teaching and research sites, including the John C. Stennis Space Center and the Gulf Coast Research Laboratory (GCRL). Originally called the Mississippi Southerners, the Southern Miss athletic teams became the Golden Eagles in 1972. The school's colors, black and gold, were selected by a student body vote shortly after the school was founded. While mascots, names, customs, and the campus ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Animated Construction Of Butterfly Curve
Animation is a method by which still figures are manipulated to appear as moving images. In traditional animation, images are drawn or painted by hand on transparent celluloid sheets to be photographed and exhibited on film. Today, most animations are made with computer-generated imagery (CGI). Computer animation can be very detailed 3D animation, while 2D computer animation (which may have the look of traditional animation) can be used for stylistic reasons, low bandwidth, or faster real-time renderings. Other common animation methods apply a stop motion technique to two- and three-dimensional objects like paper cutouts, puppets, or clay figures. A cartoon is an animated film, usually a short film, featuring an exaggerated visual style. The style takes inspiration from comic strips, often featuring anthropomorphic animals, superheroes, or the adventures of human protagonists. Especially with animals that form a natural predator/prey relationship (e.g. cats and mice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
_male_in_flight.jpg "Click for more on -> Butterfly Transcendental Curve")
