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Exploring Classical Curves: Rose, Cardiod, and Lemniscate

This resource covers various classical curves in mathematics, focusing on their equations and graphing techniques. Key topics include Rose curves, which can be defined by equations such as r = a.sin(nθ) and r = a.cos(nθ). Additionally, the Cardioid and Limacon shapes are explained, showcasing their forms based on parameters a and b. The resource also delves into the transformation between rectangular and polar coordinates, exemplified by Archimedes' Spiral and Lemniscate. A thorough understanding of these curves enriches mathematical visualization and analysis.

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Exploring Classical Curves: Rose, Cardiod, and Lemniscate

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  1. Classical Curves

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

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

  4. 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)

  5. 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)

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

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

  8. RECTANGULAR TO POLAR R (x, y) P (r, θ) R= (x2 + y2) θ = Arctan (y/x) IF X IS (+) θ = Arctan (y/x) + πIF X IS (-)

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

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

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