(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

https://www.desmos.com/calculator/by1jmd2rig

 

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

https://www.desmos.com/calculator/cc8tpciqr8

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 

https://www.desmos.com/calculator/umabyqaarc

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

https://www.desmos.com/calculator/prwydnczt2

Right Trifolium

Polar Curve:

r = a * cos(θ) * cos(2*θ)

a > 0

Example:  a = 3.79

https://www.desmos.com/calculator/jfoueveot8

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

https://www.desmos.com/calculator/deucgq0whf

Scarabaeus

Polar Curve:

r = a * cos(2*θ) – c * cos(θ)

a > 0, c ∈ R  (real numbers)

Example:  a = 3.64, c = 1.11

https://www.desmos.com/calculator/kwzmnxveoy

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

https://www.desmos.com/calculator/uimmrc7wyx

Windmill

Polar Curve:

r = a * cot(2*θ)

or r = a/tan(2*θ)

a > 0

Example:  a = 3

https://www.desmos.com/calculator/gotjdbysgv

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).

https://www.desmos.com/calculator/10esshtmh8

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

Simple Polar Curves

Simple Polar Curves

** Updated 2/11/2015 - Thank you Mike for pointing out my mistake on the rose curve - it is much appreciated!   Eddie


Here are some of the basic curves in polar coordinates.  These are graphed using the HP Prime Emulator, where 0 ≤ θ ≤ 4*π.

Rose:  r = cos(n*θ)
If n is odd, then the rose has n pedals.
If n is even, then the rose has 2*n pedals.

r = cos(5*θ)

r = cos(5*θ)

r = cos(6*θ)

r = cos(6*θ)

Leminscate:  r = +/- sqrt(a^2 * cos (2*θ))

r = +/- sqrt(9 * cos (2*θ))

R1 = sqrt(9 * cos (2*θ)),R2 = -R1

Cardiod:  r = 2*a*(1 + cos θ)

r = 8 * (1 + cos θ)

r = 8 * (1 + cos θ)

Limacon of a Pascal:  r = b + a*cos θ

r = 6 - 7*cos θ

r = 6 - 7*cos θ

And the most basic polar curve of them all:  
Spiral of Archidemeds:  r = a * θ

r = 3 * θ

r = 3 * θ

Source:
Murray R. Spiegel, Ph.D.,  Seymour Lipschutz, Ph.D., John Liu, Ph.D.   "Schuam's Outlines:  Mathematicals Handbook of Formulas and Tables."  3rd Edition.  2009.

This blog is property of Edward Shore.  2015

Calculus Revisited #21: The Catch All

Welcome to Part 21 of our 21 part series on Calculus Series. Here are some interesting facts about calculus that were not discussed in the first 20 parts, but are worth paying attention to:

Implicit Differentiation

The key here is that when you are differentiating, you differentiate with respect to the specific variable. All other variables are treated as constants.

So for the function f(x,y):

d/dx f(x,y) means differentiate with respect to x, y is held constant

d/dy f(x,y) means differentiate with respect to y, x is held constant

Example: Let f(x,y) = ln(x + 2y) + y^2

d/dx f(x,y)
= d/dx ln(x + 2y) + d/dx y^2
= 1/(x + 2y) + 0
= 1/(x + 2y)

d/dx f(x,y)
= d/dy ln(x + 2y) + d/dy y^2
= 2/(x + 2y) + 2y

Arc Length

The arc length of a function f(x) is given by the integral:

b
∫ √( 1 + (f'(x))^2 ) dx
a

Usually, you will be finding the arc length by numerical methods and/or the calculator.

A numerical example: The arc length of f(x) = sin x from x = 0 to x = π/2 (to seven decimal places)

f(x) = sin x
f'(x) = cos x
(f'(x))^2 = cos^2 x

Then:

π/2
∫ (1 + cos^2 x) dx ≈ 1.9100989 (by calcualtor)

A Numerical Method of Integration: Simpson's Rule

For n partitions (n is even):

b
∫ f(x) dx
a

≈ (b - a)/3n * (f(a) + 4*f(x1) + 2*f(x2) + 4*f(x3) + 2*f(x4) + ... + 4*f(x_2k+1) + 2*f(x_2k+2) + .... + 4*f(x_n-1) + f(b))

where x_k = a + (b - a)/n * k

Error:

At maximum (b - a)^5/(180 * (2n)^4) * max|f''''(x)|

In general, the more complex the integral, the more terms needed

Example:
Use the Simpsons rule to calculate

2
∫ e^x dx with n = 4 (7 decimal places)
1

(b - a)/n = (2 - 1)/4 = 1/4

The integral is approximately:

≈ (2 - 1)/(3 * 4) * (f(1) + 4 * f(1.25) + 2 * f(1.5) + 4 * (1.75) + f(2))
≈ 4.6708749

(actual value is about 4.6707743)

Area Between Curves

Let f(x) and g(x) be two functions where f(x) ≥ g(x). The area between curves f(x) and g(x) is:

b
∫ f(x) - g(x) dx
a

Often a and b will be intersection points of f(x) and/or g(x).

Example:

Let f(x) = x^2 and g(x) = x. Find the area between the curves from x = 0 to x = 5.

When 0 ≤ x ≤ 5, f(x) ≥ g(x). Then the area between the curves is:

5
∫ x^2 - x dx = 175/6 ≈ 29.1666667
0

Volume of a Solid - Disk Method

The volume of a solid of revolution - method of discs revolving around the x-axis is:

b
∫ π * (f(x))^2 dx
a

Like arc length, you may be finding the volumes numerically and/or by calculator (like I will do by this example).

Example: Let f(x) = sin x from x = 0 to x = π

Then the volume is:

π
∫ π * sin^2 x dx ≈ 4.9348022 = π^2/2
0

If you are finding a volume with discs revolving around the y-axis then the volume would be:

b
∫ π * (g(y))^2 dy
a

where a and b are y-values.

Polar Equations and Polar Integral

To convert functions to their polar form and back, use the following:

x = r cos θ
y = r sin θ

r^2 = x^2 + y^2
θ = atan (y/x)

The Polar Integral is:

θ2
∫ 1/2 * (r(θ))^2 dθ
θ1

Example: Let r(θ) = 2 θ . Find the polar integral from θ = 0 to θ = 2π

Area:

2π
∫ 1/2 * (2 θ)^2 dθ = 16 π^3/ 3 ≈ 165.3668090
0


Parametric Equations

Parametric equations are in the form of ( x(t), y(t) ), where t is the independent variable and x and y are dependent variables. Each part can be separately differentiated and integrated.

Arc length of a parametric curve from t = a to t = b:

b
∫ √( (x'(t))^2 + (y'(t))^2 ) dt
a

Let:
x(t) = 2t
y(t) = t^3

Find (x, y) at t = 0 and t = 1. Find the arc length.

x(0) = 0, y(0) = 0
x(1) = 2, y(1) = 1

Arc length:

x'(t) = 2
y'(t) = 3t^2

(x'(t))^2 = 4
(y'(t))^2 = 9t^4

1
∫ √(4 + 9t^4) dt ≈ 2.3650656
0

Tips for Sketching a Curve y = f(x)

Find out where the extrema of f(x) are. Recall the critical points are found when:

f'(x) = 0

Let c be the critical points.

If f''(c) > 0, then x = c is strict minimum.

If f''(c) < 0, then x = c is a strict maximum.

If f'(c) > 0 when both f'(x) > 0 for x < c and x > c OR
f'(c) < 0 when both f'(x) < 0 for x < c and x > c,
then x = c is an inflection point.

When f'(x) > 0, f(x) is increasing.

When f'(x) < 0, f(x) is decreasing.

When f''(x) > 0, f(x) is concave up (holding water).

When f''(x) < 0, f(x) is concave down (spilling water).

If f(x) → +∞ or -∞ when x → a, then f(x) has a vertical asymptote at x = a.

If f(x) → b when x → +∞ or -∞, then f(x) has a horizontal asymptote at y = b.

Thank you as always. I hope you enjoyed the Calculus Revisited Series.

To all the students who are in calculus, good luck in finals!

Until next time,

Eddie

This blog is property of Edward Shore. © 2012