(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