**(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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