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Exploring Classical Curves: Rose, Cardioid, Limacon, Lemniscate, and Spiral

This article delves into various classical curves, focusing on the mathematical representations and properties of Rose curves, Cardioids, Limacons, Lemniscates, and the Spiral of Archimedes. It explains how to construct these curves using polar coordinates, the relationships between their parameters, and their unique characteristics such as loops and convex shapes. The discussion also covers converting between rectangular and polar coordinates, providing essential formulas and equations for each curve. Ideal for students and math enthusiasts seeking to understand the beauty and complexity of polar curves.

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Exploring Classical Curves: Rose, Cardioid, Limacon, Lemniscate, and Spiral

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  1. Classical Curves

  2. r= a sin nθ r= a cos nθ ROSE CURVES SINE: starts Quadrant I COSINE: starts x axis

  3. CARDIOD r= a + a sin nθ r= a + a cos nθ Sine (x axis) Cosine (y axis) a + a (distance point to point) ±a (intercepts)

  4. LIMACON r= a + b sin nθ r= a + b cos nθ a + b (distance shape) b – a (distance loop) ± a intercepts a < b b < a a > 2b loop dimple convex (no shape) Cosine (x axis) Sine (y axis)

  5. LEMNISCATE r2= a sin 2θ r2= a cos 2θ Cosine (x axis) Sine (diagonal)

  6. SPIRAL OF ARCHIMEDES r = aθ More spiral (coefficient decimal/small) Less spiral (coefficient larger)

  7. RECTANGULAR TO POLAR R (x, y) P (r, θ) R= θ = Arctan (y/x) IF X IS (+) θ = Arctan (y/x) + πIF X IS (-)

  8. POLAR TO RECTANGULAR P (r, θ) R (x, y) X = r cos θ Y = r sin θ

  9. http://www.math.rutgers.edu/~greenfie/mill_courses/math152/diary3.htmlhttp://www.math.rutgers.edu/~greenfie/mill_courses/math152/diary3.html

  10. eo95@ymail.com azzedinr@aol.com turnerkayla23@gmail.com

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