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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

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Spirals

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  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/

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