Casio fx-CG 50: Points on a Circle
Introduction and Derivation
Given the initial point on a circle (x, y). If the point travels on a circle a distance of arc length s, where is the new point on the circle? Assumption: The center of the circle lies on the point (0, 0).
In fact, we can have two points depending on the direction traveled: counter-clockwise or clockwise. The programs presented today will determine both points.
Points that lie on the circle have the same radius. Determine the required radius:
r = √(x^2 + y^2)
And since the circle is centered at (0, 0), we can determine the angle by:
θ = arctan(y/x) = arg(x + yi)
We will assume all angles are in radians.
The arc length can be used to determine the angle traveled. Let’s call this angle z:
s = r × z
z = s / r
The new points can be determined by:
x’ = r × cos(θ + z), y’ = r × sin(θ + z)
x’’ = r × cos(θ - z), y’’ = r × sin(θ - z)
Casio fx-CG 50 Basic: PTSONCIRC
Code:
ClrText
Blue “POINTS ON”
Blue “THE CIRCLE”
Blue “CENTER (0,0)”
Red “RADIANS”
Rad
“INITIAL X”? → X
“INITIAL Y”? → Y
“ARC LENGTH”? → S
Abs (X+Yi) → R
Arg (X+Yi) → θ
S ÷ R → Z
R × cos(θ + Z) → A
R × sin(θ + Z) → B
R × cos(θ - Z) → C
R × cos(θ - Z) → D
ClrText
Green Locate 1,3,”POINT 1”
Black Locate 1,4,A
Blue Locate 11,4,”,”
Black Locate 12,4,B
Green Locate 1,5,”POINT 2”
Black Locate 1,6,C
Blue Locate 11,6,”,”
Black Locate 12,6,D
For monochrome models, leave out the color commands (Blue, Red, Black, Green)
Casio fx-CG 50 Python: ptsoncirc.py
Code:
from math import *
print(“Points on\nthe circle.”)
print(“Center is at (0,0)”)
x=float(input(“initial x? “))
y=float(input(“initial y? “))
s=float(input(“arc length? “))
r=sqrt(x**2+y**2)
t=atan(x,y)
z=s/r
b,a=r*cos(t+z),r*sin(t+z)
d,c=r*cos(t-z),r*sin(t-z)
print(“Point 1”)
print(str(a),”,\n”,str(b))
print(“Point 2”)
print(str(c),”,\n”,str(d))
Examples
X: 5, Y: 6, S: 1.5
Point 1: (3.76280901, 6.84406811)
Point 2: (6.05333094, 4.93529983)
X: 4: Y: 5, S: 1
Point 1: (3.173620164, 5.561306956)
Point 2: (4.729016994, 4.316989492)
Python Pointers
Storing to Multiple Variables in One Line
We can store a value to multiple variables in one line. For example,
a = b = 7
Stores the value 7 to both the variables a and b.
We can store multiple values to multiple values to multiple variables in one line. For example:
a, b = 7, 8
Stores 8 to the variable b and 7 to the variable a. The expression works from the inside-out.
We could also have both uppercase and lowercase letters as different variables, such as A and a. I choose not to because I want to avoid confusion.
New Line Escape Character: \n
In a string, we can add \n (backslash, n) to create a new line.
Eddie
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