Sunday, October 30, 2016

(Maybe Not So Well Known?) Mathematical Curves

(Maybe Not So Well Known?) Mathematical Curves

Here are some mathematical curves graphed on the Desmos website (https://www.desmos.com/calculator ).   Enjoy!

For the following curves, the angle mode is radians.

The equations shown on this blog entry can be plotted on any graphing calculator that has function, polar, and parametric modes.  I will post the links to the pages on Desmos with each curve.  For parametric curves, I set the range to -4*π ≤ t ≤ 4*π.  The variables a, b, c, and n can be changed in the links for you to explore the graphs. 

Without further ado:

Arachnida

Polar Curve:
r = 2 * a * sin(n*θ)/sin((n-1)*θ)
a > 0, n N  (N:  natural numbers (1, 2, 3, …))



Example:  a = 3, n = 6
 

Conchoid of Dürer
(Dürer’s Shell Curve)

Parametric Curve:
x = (a * cos(t))/(cos(t) – sin(t)) + b * sin(t)
y = b * sin(t)
a > 0, b > 0



Example:  a = 6.1, b = 8.84

Cornoid

Parametric Curve:
x = a * cos(t) * (1 – 2 * (sin(t))^2)
y = a * sin(t) * (1 + 2 * (cos(t))^2)
a > 0



Example:  a = 5.89 

Nodal Curve

Polar Curve:
r = a * cot(n * θ)
or  r = a/(tan(n*θ))
a > 0, n N



Example:  a = 2.67, n = 8


Right Trifolium

Polar Curve:
r = a * cos(θ) * cos(2*θ)
a > 0



Example:  a = 3.79

Sand Glass Curve

Parametric Curve:
x = a * cos(2*t)/cos(t)
y = b * tan(t)
a > 0, b > 0



Example:  a = 6.6, b = 3.9

Scarabaeus

Polar Curve:
r = a * cos(2*θ) – c * cos(θ)
a > 0, c R  (real numbers)



Example:  a = 3.64, c = 1.11


Deltoid Curve
(Three-Cuspid Hypocycloid)

Parametric Curve:
x = a * (2 * cos(t) + cos(2*t))
y = a * (2 * sin(t) – sin(2*t))
a > 0



Example:  a = 3.14


Windmill

Polar Curve:
r = a * cot(2*θ)
or r = a/tan(2*θ)

a > 0



Example:  a = 3

Trichoidal Rose

Polar Curve:
r = 2 * a * (q + 1) * sin(θ/(2*q + 1))
a > 0
q = m/n where m Z, n Z, and GCD(m,n) = 1, but q ≠ 1 nor q ≠ 1/2



Example: a = 2.14, q = 2/5  (n = 2, m = 5).

My favorites are the Arachnida, Sand Glass, and Scarbaeus.  Feel to free to play with the curves and see what you get. 

Eddie

Source:

Shinkin, Eugene V.  Handbook and Atlas of Curves  CRC Press:  Boca Raton.  1995  ISBN 0-8493-8963-1

This blog is property of Edward Shore, 2016




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