HP Prime CAS: Curvature

HP Prime CAS:  Curvature

Introduction

The following CAS functions calculates the curvature of:

functions, y(x)

polar functions, r(t)  (t: Θ)

parametric functions, x(t), y(t)

Let Δα be the angle of rotation angle and Δs is the slight change of distance. Then the radius of curvature is:

K = abs(Δα ÷ Δs) as Δs → 0

And the radius of curvature is the reciprocal of K.  

For circles, the radius of curvature is constant.  Wankel engines and rotary engines have their pistons traveling in a circle.

Calculating the curvature depends on the form of the function.  

Function:  y(x)

K = abs( y''(x) ) ÷ (1 + (y'(x))^2) ^(3/2)

Polar:  r(t)  (t replaces Θ)

K = abs( r(t)^2 + 2 * (r'(t))^2 - r(t) * r''(t) ) ÷ ( r(t)^2 + r'(t)^2 )^(3/2)

Parametric:  x(t), y(t) 

K = abs( x'(t) * y''(t) - y'(t) * x''(t) ) ÷ ( x'(t)^2 + y'(t)^2 )^(3/2)

Radius of Curvature:

r = 1 ÷ K

For the CAS functions, they take the form:

#cas

name(arguments):=

BEGIN

...

END;

#end

Clicking on the CAS checkbox will not put the #cas and #end delimiters.  And these programs will work in CAS mode only.

HP Prime CAS Program: crvfunc

#cas

crvfunc(y,x):=

BEGIN

// curvature

// function

// radius = 1/curvature

LOCAL a,b;

a:=diff(y,x,2);

b:=diff(y,x,1);

RETURN ABS(a)/(1+b^2)^(3/2);

END;

#end

HP Prime CAS Program: crvpol

#cas

crvpol(r,t):=

BEGIN

// curvature

// polar (t: θ)

// radius = 1/curvature

LOCAL a,b,n,d;

a:=diff(r,t,2);

b:=diff(r,t,1);

n:=simplify(r^2+2*b^2-r*a);

d:=r^2+b^2;

RETURN ABS(n)/(d)^(3/2);

END;

#end

HP Prime CAS Program: crvpar

#cas

crvpar(y,x,t):=

BEGIN

// curvature

// parametric

// radius = 1/curvature

LOCAL y1,y2,x1,x2,n,d;

y2:=diff(y,t,2);

y1:=diff(y,t,1);

x2:=diff(x,t,2);

x1:=diff(x,t,1);

n:=simplify(x1*y2-y1*x2);

d:=simplify(x1^2+y1^2); 

RETURN ABS(n)/(d)^(3/2);

END;

#end

Until next time and have a great day, 

Eddie 

Source:

Svirin, Alex Ph.D.      "Curvature and Radius of Curvature" Math24  https://math24.net/curvature-radius.html   2022.  Last Updated September 12, 2022.  

Gratitude to Arno K. and rombio for helping me with derivatives and CAS programs.  

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Proving Chebyshev Polynomial Closed Formulas for n = 0, n = 1, and n = 2

Proving Chebyshev Polynomial Closed Formulas for n = 0, n = 1, and n = 2

Chebyshev Polynomials of the First Kind

Recurrence Definition:

T_0(x) = 1

T_1(x) = x

T_n+1(x) = 2 * x * T_n(x) - T_n-1(x)

Closed Definition:

T_n(x) = 1/2 * [ (x - √(x^2 - 1))^n + (x + √(x^2 - 1))^n ]

Let: w = √(x^2 - 1)

T_n(x) = 1/2 * [ (x - w)^n + (x + w)^n ]

n = 0

T_0(x) 

= 1/2 * [ (x - w)^0 + (x + w)^0 ]

= 1/2 * [ 1 + 1 ] 

= 1

n = 1

T_1(x)

= 1/2 * [ (x - w)^1 + (x + w)^1 ]

= 1/2 * [ x - w + x + w ]

= 1/2 * [ 2 * x]

= x

n = 2

T_2(x)

= 1/2 * [ (x - w)^2 + (x + w)^2 ]

= 1/2 * [ x^2 - 2*w + w^2 + x^2 + 2*w^2 + w^2 ]

= 1/2 * [ 2 * x^2 + 2 * w^2 ]

= x^2 + x^2 - 1

= 2 * x^2 - 1

Chebyshev Polynomials of the Second Kind

Recurrence Definition:

U_0(x) = 1

U_1(x) = 2 * x

U_n+1(x) = 2 * x * U_n(x) - U_n-1(x)

Closed Definition:

U_n(x) = [ (x + √(x^2 - 1))^(n + 1) - (x - √(x^2 - 1))^(n + 1) ] ÷ [ 2 * √(x^2 - 1) ]

Let: w = √(x^2 - 1)

U_n(x) = [ (x + w)^(n + 1) - (x - w)^(n + 1) ] ÷ [ 2 * w ]

n = 0

U_0(x)

= [ (x + w)^(1) - (x - w)^(1) ] ÷ [ 2 * w ]

= [ x + w - x + w ] ÷ (2 * w)

= (2 * w) ÷ (2 * w)

= 1

n = 1

U_1(x)

= [ (x + w)^(2) - (x - w)^(2) ] ÷ [ 2 * w ]

= [ (x^2 + 2 * x * w + w^2) - (x^2 - 2 * x * w + w^2) ] ÷ (2 * w)

= [ 4 * x * w ] ÷ (2 * w)

= 2 * x

n = 2

U_2(x)

= [ (x + w)^(3) - (x - w)^(3) ] ÷ [ 2 * w ]

= [ x^3 + 3*x^2*w + 3*x*w^2 + w^3 - (x^3 - 3*x^2*w + 3*x*w^2 - w^3)] ÷ [ 2*w ]

= [ x^3 + 3*x^2*w + 3*x*w^2 + w^3 - x^3 + 3*x^2*w - 3*x*w^2 + w^3] ÷ [ 2*w ]

= [ 6*x^2*w + 2*w^3 ] ÷ [ 2*w ]

= [ 6*x*√(x^2 - 1) + 2*(x^2 - 1)^(3/2) ] ÷ [ 2*√(x^2 - 1)  ]

= [ 6*x*√(x^2 - 1) + 2*(x^2 - 1)*√(x^2- 1) ] ÷ [ 2*√(x^2 - 1)  ]

= [ 6*x*√(x^2 - 1) + 2*(x^2 - 1)*√(x^2- 1) ] ÷ [ 2*√(x^2 - 1)  ]

= [ 6*x*√(x^2 - 1) + (2*x^2 - 2)*√(x^2- 1) ] ÷ [ 2*√(x^2 - 1)  ]

= [ (8*x - 2)*√(x^2 - 1) ] ÷ [ 2*√(x^2 - 1)  ]

= 4*x^2 - 1

Good that the closed formulas hold up, at least for n = 0, 1, 2.   The closed formulas would be good if you don't want to use recurrence relations.  

Sources:

"Chebyshev polynomials"  Wikipedia.   https://en.wikipedia.org/wiki/Chebyshev_polynomials  Last Updated July 20, 2022.  Last Accessed June 21, 2022

Oldman, Keith, Jan Myland, & Jerome Spainer  An Atlas of Functions: with Equator, the Atlas Function Calculator  2nd Edition   Springer:  New York, NY.  2009  ISBN 978-0-387-48806-6

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Python - Lambda Week: Plotting Functions

Python - Lambda Week: Plotting Functions

Welcome to Python Week!  This we we're going to cover calculus and the keyword lambda.

Note:  All Python scripts presented this week were created using a TI-NSpire CX II CAS.   As of June 2022, the lambda keyword is available on all calculators (in the United States) that have Python.   If you are not sure, please check your calculator manual. 

Plotting Functions

We can use the line:

f=eval("lambda x:"+input("f(x) = "))

for multiple applications.   Remember, lambda functions are not defined so that they can be used outside of the Python script it belongs to but it lambda functions are super useful!

This code is specific to the Texas Instruments calculators (TI-Nspire CX II (CAS), TI-84 Plus CE Python, TI-83 CE Premium Python Edition, but NOT the TI-82 Advanced Python).    For other calculators, HP Prime, Casio fx-CG 50, Casio fx-9750GIII/9860GIII, Numworks, or computer Python, apply similar language. 

plotlam.py:  Plotting with Lambda

from math import *

import ti_plotlib as plt

# this is for the TI calcs

# other calculators will use their own plot syntax

print("The math module is imported.")

# input defaults to string

# use the plus sign to combine strings

f=eval("lambda x:"+input("f(x) = "))

# set up parameters

x0=eval(input("x min = "))

x1=eval(input("x max = "))

# we want n to be an integer

n=int(input("number of points = "))

# calculate step size

h=(x1-x0)/n

# calculate plot lists

x=[]

y=[]

i=x0

while i<=x1:

  x.append(i)

  y.append(f(i))

  i+=h

# choose color (not for Casio fx-9750/9850GIII)

# colors are defined using tuples

colors=((0,0,0),(255,0,0),(0,128,0),(0,0,255))

print("0: black \n1: red \n2: green \n3: blue")

c=int(input("Enter a color code: "))

# plot f(x)

# auto setup to x and y lists

plt.auto_window(x,y)

# plot axes

plt.color(128,128,128)

plt.axes("axes")

# plot the function

plt.color(colors[c])

plt.plot(x,y,".")

plt.show_plot()

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.