November 2, 2024
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R2 – Rotation – Cartesian & Euler

This post describes the role of Cartesian Coordinates and Euler’s formula in R2 Wave Number Rotation.

Cartesian Coordinates and Euler’s Formula in R2

The Cartesian coordinates of a point are based on the intersection of perpendicular lines from the point with the x and y axes. For example: The Cartesian coordinates at π^/4 or 45^o are (√2/2^, √2/2i^). The coordinates can be expressed algebraically as the sum of the Opposite Values (√2/2^ + √2/2i^). All Opposite Values on axes are real, so there is no need for the association of Cartesian coordinates with imaginary numbers as in classical maths.

Furthermore, calculate the coordinates of a point on the unit circle using Euler’s formula:

  • ei^x = cos(x)*1^ + sin(x)*i^ 

ei^x takes the point (1^, 0) on the x-axis and then moves it to a point on a circle with radius  of 1^. Such a movement depends on the value of x, which is in radians. Note that a ^ value is a counterclockwise rotation from the x-axis and a v is a clockwise rotation.

cos(x)*1^ represents the x-axis’ coordinate of the point. sin(x)*i^ represents the y-axis’ coordinate.

For example, the coordinates for ei^x with x = 3π^/4, 135^o can be calculated using Euler’s formula as below.

  • cos (3π^/4)*1^ = √2/2*1^ = √2/2v
  • sin(3π^/4)*i^= √2/2*i^ = √2/2i^
  • Giving the coordinates (0.707v, 0.707i^)

Calculating Cartesian Coordinates using Euler’s Power Series

The Power Series definition allows ei^x to be stated as:

  • ei^x= 1^ +i^x + (i^x)2/2! + (i^x)3/3! + (i^x)4/4! + (i^x)5/5!  ….    (i^x)n/n!.

Example 1

For example, the coordinates with x = 3π^/4, 135^o can be calculated using the Power Series as in the table below.

ei^3π^/4^v  Valuesi^/iv Values
  
1^ 1^  
i^^/42.3561i^
(i^^/4)2/!22.7758v 
(i^^/4)3/!32.1801iv
(i^^/4)4/!41.2842^ 
(i^^/4)5/!50.6052i^
(i^^/4)6/!60.2376v 
(i^^/4)7/!70.0800iv
(i^^/4/!80.0238^ 
(i^^/4)9/!90.0062i^
(i^^/4)10/!100.0015v 
 (i^^/4)11/!11 
0.0003iv
:
   Total0.707v0.707i^

Example 2

For a second example, the coordinates with x = 7πv/4, 315vo can be calculated using the Power Series as in the table below.

ei^7πv/4^v  Valuesi^/iv Values
  
1^ 1^  
i^v/45.4978iv
(i^v/4)2/!215.1128v 
(i^v/4)3/!327.6957i^
(i^v/4)4/!438.0663^ 
(i^v/4)5/!541.856i^
(i^v/4)6/!638.3526v 
(i^v/4)7/!730.1221iv
(i^v/4/!820.7006^ 
(i^v/4)9/!912.6452i^
(i^v/4)10/!106.9521v 
 (i^v/4)11/!11 
3.4747iv
(i^v/4)12/!121.5919^
(i^v/4)13/!130.6732i^
(i^v/4)14/!140.2644v
(i^v/4)15/!150.0969iv
(i^v/4)16/!160.0333^
(i^v/4)17/!170.2157i^
(i^v/4)18/!180.0033v
(i^v/4)19/!190.0952iv
:
   Total0.707^0.707i^

Euler’s Formula for Coordinates

In general, the (x, y) Cartesian coordinates of any point in R2 are given by Euler’s formula:

  • x = rCos(ϕ)*1^ and y = rSin(ϕ)*i^

In this formula, ϕ (Phi) is the angle of rotation counterclockwise from the x-axis to the line joining origin with the point and r is the distance of a point from the origin.

Counterclockwise Rotation

Rotations with ^ radian values result in counterclockwise rotation: For example:

  • ei^π^/4 at 45^o    = cos(π^/4)*1^ + sin(π^/4)*i^   
    • =  √2/2*1^ + √2/2*i^ =  √2/2^ + √2/2i^
  • ei^3π^/4 at 135^o = cos(3π^/4)*1^ + sin(3π^/4)*i^ 
    • = √2/2*1^ + √2/2*i^ = √2/2v + √2/2i^           
  • ei^5π^/4 at 225^o     = cos(5π^/4)*1^ + sin(5π^/4)*i^  
    • = √2/2*1^ + √2/2*i^ = √2/2v + √2/2iv    
  • ei^7π^/4 at 315^o   = cos(7π^/4)*1^ + sin(7π^/4)*i^
    • = √2/2*1^ + √2/2*iv = √2/2^ + √2/2iv

Clockwise Rotation

Rotations with v radian values result in counterclockwise rotation: For example:

  • ei^πv/4 at 45vo   = cos(πv/4)*1^ + sin(πv/4)*i^
    • = √2/2*1^ + √2/2*iv = √2/2^ + √2/2iv  
  • ei^3πv/4 at 135vo = cos(3πv/4)*1^ + sin(3πv/4)*i^
    • = √2/2*1^ + √2/2*i^  = √2/2v + √2/2iv
  • ei^5πv/4 at 225vo    = cos(5πv/4)*1^ + sin(5πv/4)*i^  
    •  = √2/2^*1^ + √2/2*i^ = √2/2v + √2/2i^                         
  • ei^7πv/4 at 315vo     = cos(7πv/4)*1^ + sin(7πv/4)*i^
    • = √2/2*1^ + √2/2*i^  = √2/2^ + √2/2i^                         

Next: Multiplication Table

Previous: Simple Rotations

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