1 / 9

Spirals

Spirals. By Jonathan Dolan, Daniel Cavanagh, and Ryan osak. Types of Spirals. The Archimedean spiral: r = a+b * Θ The Euler spiral, Cornu spiral or clothoid Fermat's spiral: r = Θ 1/2 The hyperbolic spiral: r = a/ Θ The lituus : r = Θ -1/2

kristian-vidar
Télécharger la présentation

Spirals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Spirals By Jonathan Dolan, Daniel Cavanagh, and Ryan osak

  2. Types of Spirals • The Archimedean spiral: r = a+b*Θ • The Euler spiral, Cornu spiral or clothoid • Fermat's spiral: r = Θ1/2 • The hyperbolic spiral: r = a/Θ • The lituus: r = Θ-1/2 • The logarithmic spiral: r=a*ebΘ approximations of this are found in nature • The Fibonacci spiral and golden spiral • The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles

  3. Standard Graph • All spiral graphs continue on forever in both directions or until max Θ and min Θ is met • All graphs have a “loop” given the correct Θ min and max values • More on this later! • r = Θ, pictured right, is the “standard” spiral graph • It starts at 0 and moves counterclockwise

  4. Multiply Θ by a Number • Θ • When x<1, the graph becomes more compressed • When x>1, the graph becomes stretched

  5. Invert over BOTH axes • r = -Θ • Not only flipped over x-axis, also flipped over y-axis

  6. Loops always goes through (0,0) • Starts at X on x-axis • Loops to point (2π-X,2 π) • Top is r = Θ - 3 • Starts at -3, spirals counterclockwise past y-axis • Middle is r = Θ– 6 • Starts at -6, spirals counterclockwise past y-axis • Bottom is r = Θ – 7 • Starts at -7, spirals counterclockwise past y-axis

  7. Graph to the right is r=+3 • Graph starts at point (0,X) and never overlaps itself • When min is 0 and coefficient is positive • When max is 0 and and coefficient is negative • When there are no restrictions the spiral will hit (0,0), overlap itself, and is symmetrical • These are the exact same graphs, one with restrictions between 0 and 4π and the other with no restrictions

  8. Applications of spiral graphs • Most well known is the Fibonacci Sequence • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... • Each number is found by adding the previous 2 numbers • When graphed, it makes a nice spiral! • Related to the Golden Ratio, which is used extensively to create aesthetically pleasing architecture • If you take any 2 successive numbers in the sequence and divide the larger one by the smaller one, you get roughly 1.618… which is the Golden Ratio. The larger the numbers, the more accurate • Good trivia fact: the relation of miles to kilometers is nearly the same as the Golden Ratio

  9. Works cited • http://en.wikipedia.org/wiki/Spiral • http://www.mathsisfun.com/numbers/fibonacci-sequence.html • http://jsxgraph.uni-bayreuth.de/wiki/index.php/Archimedean_spiral • http://www.mathsisfun.com/numbers/golden-ratio.html • Consultation of http://www.wolframalpha.com/

More Related