# ArcSpiral

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Published on October 24, 2007

Author: Rafael

Source: authorstream.com

Archimedes Spiral:  Archimedes Spiral By Michael Liu And Tim Myers What is the spiral used for?:  What is the spiral used for? The archimedes screw, a spiral screw turned inside a cylinder, was once commonly used to lift water from canals. The screw is still used to lift water in the Nile delta in Egypt. Definition:  Definition “If a straight line drawn in a plane revolves uniformly any number of times about a fixed extremity until it returns to its original position, and if, at the same time as the line revolves, a point moves uniformly along the straight line beginning at the fixed extremity, the point will describe a spiral in the plane.” Archimedes Slide4:  The curve is defined by the polar equation r = a*Θ. When the constant a = 1,the curve is drawn by tracing a point out from the center of a fixed pole. As the curve is traced out the radius of the circle and the angle of rotation both run from 0 to 2pi with one revolution of the spiral. Slide5:  The distance between the windings of the spiral remains constant. The unique property of the Archimedian Spiral is that the radius of the spiral and the rotation angle are linearly proportional. r = 1*Ө:  r = 1*Ө r = ½* Ө:  r = ½* Ө r = 2* Ө:  r = 2* Ө r = -1* Ө:  r = -1* Ө r = 1*- Ө:  r = 1*- Ө r = -1*- Ө:  r = -1*- Ө Osculating Circle:  Osculating Circle Slide15:  Archimedes had shown in his work “On Spirals” that the area of a circle with a given radius was equal to the area of a right triangle with the two shortest sides equaling the radius and the circumference of the given circle. One more thing………:  One more thing……… You rock Dave! have an awesome summer everyone!

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