1 / 12

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

This comprehensive guide delves into classical curves such as rose curves, cardioids, limacons, lemniscates, and spirals. We explore their mathematical definitions, properties, and graphical representations, starting with rose curves using sine and cosine functions. The guide covers the transformation between polar and rectangular coordinates, as well as various equations that represent these curves. Key concepts include intercepts, distances, and the relationship between coefficients and shape characteristics, providing readers with a thorough understanding of these fascinating mathematical shapes.

wendi
Download Presentation

Exploring Classical Curves: Rose, Cardioid, Limacon, Lemniscate, and 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. 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. DESCRIBE EACH R= 3 + 3cosθ R= -2sin3 θ R= -5 + 3cosθ R2= 4cos2θ

  8. r= 2sin3θ

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

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

  11. EQUATIONS Complete the square- Substitution- Trig identities! Polar form of x2 + y2 = 16 (r2cos2θ) + (r2sin2θ)= 16 r2 (cos2θ+ sin2θ) = 16 r2 (1) = 16 r= 4 Substitute values of x & y Factor out “r2” Trig identity “cos2θ+ sin2θ=1” Simplify Rectangular form of r=-secθ r= 1 cos θ rcosθ=1 x=1 Rewrite “sec” as “1/cos” Cross multiply Substitute with “x”

  12. To Polar: R (-1,1) R (8, 8√3) X2 + (y + 6)2 = 36 To Rectangular: P(4, π/6) P (-2, π/3) r= 3sinθ

More Related