Casio Python with fx-CG 50: Drawing Shapes with Casioplot (draws.py)

Casio Python with fx-CG 50: Drawing Shapes with Casioplot (draws.py)

The Casio file draws.py is a file that use both the casioplot and math modules and contains four drawing functions.

The plot functions uses the pixel system.

Upper left hand corner: (0, 0)	Upper right hand corner: (383, 0)
Lower left hand corner: (0, 191)	Lower right hand corner: (383, 191)

The x axis increases going right and the y axis increases going down. This orientation is common for a pixel-orientated coordinate system.

Note that the pixels must be integers. A non-integer value for a pixel will cause an error.

Importing the draws module

After copying the file draws.py to your calculator, type: from draws import *

The casioplot and math modules will also be imported because the draws module is imported. This is the case of the fx-CG 50, and I’m pretty sure it will work with the other Casio calculators with Python (fx-9750GIII, fx-9860GIII, fx-CG 100, Graph Math+).

To clear the drawing screen, use the casioplot’s command clear_screen().

To show the picture, use the casioplot’s command show_screen().

The functions from the draws must be manually typed, as they will not appear in the catalog or VARS menu.

sline(x,sx,y,sy,l,c)

Draws a line from (x,y) of length l and color c. The color is a three-element tuple in the RGB format ((red, green, blue)).

The arguments sx and sy are direction/slope arguments which dictate the direction of the line.

To draw a line going left (←)	Set sx = -1 and sy = 0
To draw a line going right (→)	Set sx = 1 and sy = 0
To draw a line going up (↑)	Set sx = 0 and sy = -1
To draw a line going down (↓)	Set sx = 0 and sy = 1
To draw a line going right and up (↗)	Set sx = 1 and sy = -1
To draw a line going left and up (↖)	Set sx = -1 and sy = -1
To draw a line going right and down (↘)	Set sx = 1 and sy = 1
To draw a line going left and down (↙)	Set sx = -1 and sy = 1

Examples:

sline(1,1,1,0,140,(255,0,0)) draws a red line starting from (1,1) going right with length of 140 pixels

sline(180,0,100,-1,50,(0,255,0)) draws a green line starting from (180,100) going up with length of 50 pixels

The sample file draw1.py is a demonstration of the sline command.

box(x,xl,y,yl,c)

The box function draws a box with the upper left hand corner (x,y) with color c, horizontal length (width) xl, and vertical length (height) yl. The box draws to the right and down. The box is filled with the specified color. If xl = yl, the function will draw a square.

Example:

box(20,100,40,150,(128,128,128)) draws a gray rectangle with upper-left corner at (20,40) with horizontal length of100 and vertical length of 150.

box(300,60,0,60,(0,0,0)) draws a black square with upper-left corner at (300, 0) with the side length of 60.

The sample file draw2.py uses the box function to generate a random game map of land (green) and water (blue).

tri(x,xd,y,yd,l,c)

The tri function draws a 45-45-90 right triangle. The point (x,y) is the corner point that contains the right angle of 90°. The arguments xd and yd dictate the direction of the triangle (see table below).

The triangle is filled with color c. The sides of the triangle will be drawn with length l.

Example:

tri(80,1,80,-1,75,(0,128,128)) draws a teal 45-45-90 right triangle with the right angle located at (80, 80). The length is 75 pixels.

The sample file draw3.py uses the tri function to draw four triangles, one with each proper orientation.

circ(x,y,r,c)

The circ function draws a circle centered at (x, y) with radius r and color c. The circle is drawn as outline and not filled. To make the circle appear as smooth as possible, 720 points are plotted.

Example:

circ(190,100,85,(255,128,0)) draws an orange circle with radius 85, centered at (190, 100).

The file draw4.py demonstrates the circ function.

Casio fx-CG 50 Python: draws.py code

# drawing utilities

# 12-04-2024

from casioplot import *

from math import *

def sline(x,sx,y,sy,l,c):

  for i in range(l+1):

    set_pixel(x+sx*i,y+sy*i,c)

# upper left corner

def box(x,xl,y,yl,c):

  for i in range(xl+1):

    for j in range(yl+1):

      set_pixel(x+i,y+j,c)

# right triangle

# (x,y) point with right angle

# l=length

# xd=1 right, xd=-1 left

# yd=1 down, yd=-1 up

def tri(x,xd,y,yd,l,c):

  for i in range(l+1):

    k=l-i

    for j in range(k+1):

      set_pixel(x+xd*i,y+yd*j,c)

# hallow circle

def circ(x,y,r,c):

for i in range(721):

  t=2*pi*i/720

  xr=int(x+r*cos(t))

  yr=int(y+r*sin(t))

  set_pixel(xr,yr,c)

You can download the draws.py and the example Python files here:

https://drive.google.com/file/d/1rGQHa60V7ZCE9Vs0kYy0cfIe0pOXqVaZ/view?usp=sharing

I hope you enjoy this program as much as I have making it,

Eddie

All original content copyright, © 2011-2025. 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.

Casio fx-9750GIII and fx-CG 50: Playing Games with the Probability Simulation Mode

Casio fx-9750GIII and fx-CG 50: Playing Games with the Probability Simulation Mode

The Probability Simulation add-in has six types of probability simulations:

* Coin Toss

* Dice Roll

* Spinner

* Marble Grab

* Card Draw

* Random Numbers

The add-in application is available for the following calculators:

* Casio fx-9750GIII and fx-9860GIII (and Graph 75/85/95 series, Graph 35+ E II)

* Casio fx-CG 10/20 and fx-CG 50 (and Graph 90+E)

* Casio fx-9860G

I believe on the fx-9750GIII and fx-9860GIII, the Probability Simulation Add-In is available out of the box. For others, the add-in can be downloaded through Casio’s Worldwide Education Website: https://edu.casio.com/download/index.php.

Let’s look at three ways we can use the Probability Simulation add-in in games of chance. This is a great app when you don’t have a pair of dice, playing cards, or a bag of marbles around.

In the Set Up menu, there is an option for seed from 1 to 99999.

Screen shots are from the fx-CG 50.

Interaction with Other Modes

* Data can be stored into global lists 1-26. Lists in these Casio calculators contain only numerical information. Numerical codes are used for card suits and face cards.

* There are no commands from the Add-In that can be used in programming. The simulation is mean to be a stand-alone app.

Drawing a Poker Hand

From the main screen, press F5 for Card Draw. To simulate poker, go into set up by pressing [SHIFT] {SET UP}. We can set either a 52 playing card deck, which is the standard deck without Jokers, or a reduced deck of 32 cards (sevens through Aces only). We don’t want Replacement, so turn that off. Press [ EXIT ] to go back to the simulation.

To draw a single card, press [ F1 ]. To draw multiple cards, press [ F2 ] for { +n }. At the prompt, press [AC/ON] and enter the number of cards.

We will have to memorize the cards or note the down on paper or another writing device.

To save the cards drawn, select [ F3 ] (STORE). There are three lists:

Draw: Draw number

Value: Card value. 1 = Ace, 11 = Jack, 12 = Queen, 13 = King

Suit: 1 = Heart, 2 = Club, 3 = Spade, 4 = Diamond

Lists can be allocated to the global list variables List 1 to List 26. Press [ F6 ] { EXE } to store the cards. Storing results works similarly in other applications.

Rolling Dice in Adventure Games

In adventure and fantasy games such as Dungeon and Dragons, sometimes dice beyond the standard six-sided die is needed. The Dice Roll (F2 from the Main Menu) has dice that are four-sided, six-sided, eight-sided, twelve-sided, and twenty-sided. Up to three dice can be thrown at once.

A Simple Lottery

Random integers from 0 to 99 can be drawn with the Random Numbers. Again, the set up menu is the key. For the lottery, turn the Repeat option off. Above are four draws from a simple lottery from 63 numbers.

This has been a look at Casio’s Probability Simulation Add-In. Until next time,

Eddie

All original content copyright, © 2011-2024. 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.

Casio Graphing Calculators: Equation Variables

Casio Graphing Calculators: Equation Variables

Introduction

Most modern Casio graphing calculators have an Equation solver mode that solves linear systems (Simultaneous), polynomials (Polynomial), and general equations.

To find run Equation mode, press [ MENU ] { A }*.

(*Equation mode is currently option A (fx-9750GIII, fx-CG 50, fx-CG 10/20, fx-9860 Slim))

Simultaneous Equations: SimRes and SimCoef

The simultaneous equation mode solves linear systems up to 6 x 6. The coefficients of the equations are stored in the variable Sim Coef (labeled SimCoef in VARS-EQUATION menu) while the results are stored in the variable Sim Result (labeled SimRes).

Polynomials: PlyRes and PlyCoef

The polynomial equation mode solves polynomials with real coefficients up to degree 6 (3 in early graphing models). The coefficients of the polynomial are stored in the variable Poly Coef (labeled PolyCoef in the VARS-EQUATION menu) while the results are stored in the variable Poly Result (labeled PolyRes).

Accessing Equation Variables Outside of Equation Mode

The four equation variables can be recalled by pressing [ VARS ], [ F6 ] { > }, [ EQUA ] { F3 }.

F1: SimRes (S·Rlt on older models)

F2: SimCoef (S·Cof on older models)

F3: PlyRes (P·Rlt on older models)

F4: PlyCof (P·Cof on older models)

The four variables are read-only matrices. We can not store values or information to read-only variables.

If we want to extract specific values from the equations variables, we must store them into a Matrix variable (A – Z, θ). For example, to retrieve the number from the 1^st row and column of Sim Coef:

Sim Coef → Mat A

Mat A[1,1]

We can execute operations on equation variables. An interesting check is to run the simultaneous solver of the variable by typing:

Rref Sim Coef

and reading the last column.

Until next time,

Eddie

 All original content copyright, © 2011-2024. 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.

Casio fx-9750GIII: Integrals with Infinite Limits

Casio fx-9750GIII:  Integrals with Infinite Limits

A Substitution to Get to Infinity... 

It is quite a challenge to calculate numerical integrals with infinite limits such as 

∫( f(x) dx, a, ∞)

∫( f(x) dx, -∞, a)

∫( f(x) dx, -∞, ∞)

A trick is to substitute x = tan Θ.  Then:

Θ = arctan x

dx = sec^2 Θ dΘ = cos^-2 Θ dΘ

Note that

lim t→∞ arctan t = π/2

lim t→-∞ arctan t = -π/2  

In this blog, assume that radian angle mode is used in all calculations.

Let's go over some examples and see how it works.  I used a Casio fx-9750GIII, however, this technique should work with all calculators with numerical integral calculations.  Furthermore, changing the integral will allow for Simpson's Rule or Trapezoid Rule approximation.

For the upper limit, I approximate π/2.

A = 1.5708  (π/2 to 4 places)

B = 1.570796  (π/2 to 6 places)

C = 1.57079633 (π/2 to 8 places)

Example 1:  

∫( e^(-x^2) dx, 0, ∞)  

transforms to

∫( e^(-tan^2 Θ)/cos^2 Θ dΘ, 0, ≈π/2)

Calculations (fx-9750GIII):

Upper Limit: A (see above),  Result:  0.8862269255

Upper Limit: B,  Result:  0.8862269255

Upper Limit: C,  Result:  0.8862269255

Example 2:  

∫(x^0.5 * e^(-x) dx, 0, ∞)

transforms to

∫((tan Θ)^0.5 * e^(-tan Θ))/cos^2 Θ dΘ, 0, π/2)

Calculations:

Upper limits A, B, C:  0.862269255

Coincidently, ∫( e^(-x^2) dx, 0, ∞)  = ∫(x^0.5 * e^(-x) dx, 0, ∞) = Γ(1.5) = √(π)/2

Example 3:

∫( e^x*(x^2 + 1) dx, -∞, 0)

transforms to

∫( e^(tan Θ) * (tan^2 Θ + 1)/cos^2 Θ dΘ, -π/2, 0)

=  ∫( e^(tan Θ) * 1/cos^2 Θ * 1/cos^2 Θ dΘ, -π/2, 0)

∫( e^(tan Θ)/cos^4 Θ dΘ, -π/2, 0)

For upper limits A, B, C, the answer returned is 3, which is the exact answer.  

For integrals like this all is needed is a four digit approximation of π/2 = 1.5708.

Let's keep going:

Example 4:

∫( (x^2 + 1)/(x^4 - 1) dx, 0, ∞) 

transforms to

∫( 1/cos^Θ * (tan^2 Θ + 1)/(tan^4 Θ - 1) dΘ, 0, π/2)

= ∫( 1/(sin^4 Θ - cos^4 Θ) dΘ, 0, π/2)

On the fx-9750GIII get MA Error.  

Example 5:

∫( 1/√(x^2 + 3*x + 1) dx, 0, ∞)

transforms to 

∫( 1/(cos^2 Θ * √(tan^2 Θ + 3 * tan Θ + 1)) dΘ, 0, π/2)

like example 4, I get the MA Error.

Observation:  the transformation works best if the integral involves some form of either of the following:

f(x) * e^(g(x))

or 

f(x) * e^(-g(x))

Gamma

The Gamma Function is an excellent candidate for this transformation.  

Γ(t) = ∫( x^(t-1) * e^-x dx, 0, ∞)

transforms to

∫( tan^(t-1) Θ * e^(-tan Θ)/cos^2 Θ dΘ, 0, π/2)

Source

Mier-Jedrzejuwicz, W.A.C. Ph.D.    Tips And Programs for the HP 32S   Synthetix Publication.  Berkeley, CA.  September 1988.  ISBN 0-937637-05-X

Download the book here:  https://literature.hpcalc.org/items/1756

Happy calculating,

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. 

Heron's Formula vs Area by Vertices

Heron's Formula vs Area by Vertices

There are several ways to determine the area of a triangle.

Heron's Formula:

With side lengths A, B, and C, the area is:

Area = √( S * (S - A) * (S - B) * (S - C) ) 

where S = (A + B + C) / 2

Area of Vertices:

With vertices (x1, y1), (x2, y2), and (x3, y3), the area is:

Area = 1/2 * abs( (x1 - x2) * (y1 + y2) + (x2 - x3) * (y2 + y3) + (x3 - x1) * (y3 + y1) )

Testing The Data

I ran a test of 50 randomly selected sets of three points that form the triangle.  The three points are:

*  The origin: (0,0)

*  The second point has 0 ≤ x ≤ 20 and -20 ≤ y ≤ 20, x and y are integers

*  The third point has 0 ≤ x ≤ 20 and 0 ≤ y ≤ 20, x and y are integers

I used the Casio fx-9750gIII Spreadsheet application.    

Casio Spreadsheet Set Up

Column Titles - Row 1:

Column A:  "X1"   x coordinate of point 1

Column B:  "Y1"   y coordinate of point 1

Column C:  "X2"   x coordinate of point 2. 

Column D:  "Y2"  y coordinate of point 2

Column E:  "X3"  x coordinate of point 3

Column F:  "Y3"  y coordinate of point 3

Column G:  "XY1"  distance between points 1 and 2

Column H:  "XY2"  distance between points 2 and 3

Column I:  "XY3"  distance between 3 and 1

Column J:  "S"   semi-perimeter of the triangle

Column K:  "HERON"  Area by Heron's Formula

Column L: "VERT"  Area by Vertices 

Column M:  "DIFF"  Difference between two calculation methods

Formulas for Rows 2 through 51

Column A:  A2 = Fill(0,50)

Column B:  B2 =Fill(0,50)

Column C:  C2 =Seq(Int(22 Ran#) - 1,X,1,50,1)

Column D:  D2 =Seq(Int(41 Ran#) - 21,X,1,50,1)

Column E:  E2 =Seq(Int(22 Ran#) - 1,X,1,50,1)

Column F:  F2 =Seq(Int(22 Ran#)-1, X,1,50,1)

Column G:  G2 Fill:  =√(C2^2 + D2^2) for range G2:G51*

Column H:  H2 Fill:  =√((C2-E2)^2 + (D2-F2)^2) for range H2:H51

Column I:  I2 Fill:  =√(E2^2 + F2^2) for range I2:I51*

Column J: J2  Fill:  =(G2 + H2 + I2) ÷ 2 for range J2:J51

Column K:  K2 Fill =√(J2 (J2 - G2) (J2 - H2) (J2 - I2) ) for range K2:K51

Column L:  L2 Fill =0.5 × Abs( (A2 - C2)(B2 + D2) + (C2 - E2)(D2 + F2) 

+ (E2 - A2)(F2 + B2)) for range L2: L51

Column M:  M2  Fill =K2-L2

 * To take advantage that the first point is (0, 0).  The full distance formula would be needed otherwise.  

This spreadsheet implies that there is no "rounding" the middle results.  

If you want to download the spreadsheet results, click here:

https://drive.google.com/file/d/1gtflIVWmc-rfhK2HT5AeWXwwV6rEKORR/view?usp=sharing

The zip file contains two images, a csv file, and a Casio spreadsheet file that can be ran on Casio calculators with a spreadsheet application. 

Of the sample taken, the areas determined by Heron's Formula and Area by Vertices are substantially equal; any difference is the order of 10^-12.

Happy Thanksgiving,

Eddie

All original content copyright, © 2011-2020.  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. 

Casio fx-9750GIII and fx-CG50: System of Two Differential Equations, Runge Kutta 4th Order

Casio fx-9750GIII and fx-CG50: System of Two Differential Equations, Runge Kutta 4th Order

Introduction

The program TWORK4 uses the Runge Kutta 4th Order method to solve the following system of differential equations:

dx/dt = f(t, x, y)
dy/dt = g(t, x, y)

with initial conditions x0 = x(t0) and y0 = y(t0)

The next step is calculated with step h from:

x1 ≈ x0 + (k1 + 2 * k2 + 2 * k3 + k4) / 6
y1 ≈ y0 + (l1 + 2 * l2 + 2 * l3 + l4) / 6

k1 = h * f(t0, x0, y0)
l1 = h * g(t0, x0, y0)

k2 = h * f(t0 + h / 2, x0 + k1 / 2, y0 + l1 / 2)
l2 = h * g(t0 + h / 2, x0 + k1 / 2, y0 + l1 / 2)

k3 = h * f(t0 + h / 2, x0 + k2 / 2, y0 + l2 / 2)
l3 = h * g(t0 + h / 2, x0 + k2 / 2, y0 + l2 / 2)

k4 = h * f(t0 + h, x0 + k3, y0 + l3)
l4 = h * g(t0 + h, x0 + k3, y0 + l3)

For the next step set t0 = t0 + h, x0 = x1, and y0 = y1. 

Inputs:

DX/DT:  Enter dx/dt as a function of T, X, and Y.   T is the independent variable.

DY/DT:  Enter dy/dt as a function of T, X, and Y.   T is the independent variable.

T0, X0, Y0:  Enter the initial conditions

STEP:  Enter the step size

ITERATIONS: Enter the number of iterations desired.  This allows you to calculate a far point in one leap.  Example, if your initial condition is to = 0 and you want to find the point when t = 1 using the step size of 0.1, enter 0.1 for STEP and 10 for ITERATIONS.

Casio fx-9750GIII and fx-CG 50 Program: TWORK4

Notes: 

*  Although the code for the two calculators are the same, the programs will need to be programmed on each calculator separately.

*  The slash character (/) is accessed from the CHAR submenu.  The submenu shows up at the top-level menu which would read:
TOP/BOTTOM/SEARCH/MENU/A ←→ a/CHAR

* fn1 and fn2 are stored as function memories. 

"2020-06-18 EWS"
Rad
"DX/DT="? → fn1
"DY/DT="? → fn2
"T0"? → U
"X0"? → A
"Y0"? → B
"STEP?" → H
4 → Dim List 25
4 → Dim List 26
Lbl 0
"ITERATIONS"? → N
For 1 → I To N
A → X: B → Y: U → T
H * fn1 → List 25[1]
H * fn2 → List 26[1]
A + List 25[1] ÷ 2 → X
B + List 26[1] ÷ 2 → Y
U + H ÷ 2 → T
H * fn1 → List 25[2]
H * fn2 → List 26[2]
A + List 25[2] ÷ 2 → X
B + List 26[2] ÷ 2 → Y
H * fn1 → List 25[3]
H * fn2 → List 26[3]
A + List 25[3] → X
B + List 26[3] → Y
U + H → T
H * fn1 → List 25[4]
H * fn2 → List 26[4]
A + (List 25[2] + List 25[3] + Sum List 25) ÷ 6 → A
B + (List 26[2] + List 26[3] + Sum List 26) ÷ 6 → B
U + H → U
Next
ClrText
"(T, X, Y)"
{U, A, B} ◢
Menu "NEXT?", "YES", 0, "NO", 1
Lbl 1
"DONE"

Example

dx/dt = x sin y

dy/dt = y^2/500 + y/3 - y

Initial conditions: x(0) = 0.1, y(0) = 0.1, h = 0.1

T0 = 0
STEP = 0.1

ITERATIONS: 10
Results:  (T, X, Y): (1, 0.1075645239, 0.05134921355)

ITERATIONS: 10
Results:  (T, X, Y): (2, 0.1116714419, 0.02636554462)

Source:

John W. Harris and Horst Stocker.  Handbook of Mathematics and Computational Science.  Springer: New York.  2006.  ISBN 978-0-387-94746-4

Stack Exchange.  "Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE'S"  Asked March 2014.   Last updated February 2019.  https://math.stackexchange.com/questions/721076/help-with-using-the-runge-kutta-4th-order-method-on-a-system-of-2-first-order-od  Retrieved June 17, 2020

Eddie

All original content copyright, © 2011-2020.  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.

Casio fx-9750GIII: Sequence of Rotated Points

Casio fx-9750GIII: Sequence of Rotated Points

Introduction

The program ROTSEQ generates a 2 row matrix from the sequence:

[ [ x_n+1 ] [ y_n+1 ] ] = A * [ [ cos θ, -sin θ ] [ sin θ, cos θ ] ] * [ [ x_n ] [ y_n ] ]

The required inputs are:
You will set the angle mode to Degree, Radian, or Gradian
A = multiplier
θ = angle
x1 = initial x point
y1 = initial y point
N = number of steps

Casio fx-9750GIII Program ROTSEQ

This can be used on most if not every modern Casio Graphing calculator.

"EWS 2020-06-07"
Menu "ANGLE","DEGREE",1,"RADIAN",2,"GRADIAN",3
Lbl 1:Deg:Goto 4
Lbl 2:Rad:Goto 4
Lbl 3:Gra:Goto 4
Lbl 4
"F=A*MAT*[[X][Y]]"
"A"?->A
"θ"?->θ
"X1"?->X
"Y1"?->Y
"STEPS"?->N
[[X][Y]]->Mat A
Mat A->Mat B
For 1->I To N
[[cos θ,-sin θ][sin θ,cos θ]]*Mat A->Mat A
Augment(Mat B,Mat A)->Mat B
Next
"FINAL RESULTS:"◢
Mat B

Example

A = 0.5
θ = 10 grads  (Gradian mode)
x1 = 1
y1 = -1
N = 5  (5 steps)

I don't think I ever used gradian angle units on this blog before, so why not?

Results are shown and rounded to 2 decimal places

Mat B:

[ 1.00   1.14   1.26   1.34   1.40   1.41 ]
[ -1.00  -0.83  -0.64 -0.44  -0.22  0.00 ]

The next blog post will be on July 5 since tomorrow will be the 4th of July (Happy Birthday, United States). 

Eddie

All original content copyright, © 2011-2020.  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.