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:

• e^{i^x} = cos(x)*1^{^} + sin(x)*i^{^} 

e^{i^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 e^{i^x} with x = 3π^{^}/4, 135^{^o} can be calculated using Euler’s formula as below.

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

Calculating Cartesian Coordinates using Euler’s Power Series

The Power Series definition allows e^{i^x} to be stated as:

• e^{i^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.

Example 2

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

ei^7πv/4	^v  Values	i^/iv Values
1^	1^
i^7πv/4		5.4978iv
(i^7πv/4)2/!2	15.1128v
(i^7πv/4)3/!3		27.6957i^
(i^7πv/4)4/!4	38.0663^
(i^7πv/4)5/!5		41.856i^
(i^7πv/4)6/!6	38.3526v
(i^7πv/4)7/!7		30.1221iv
(i^7πv/4/!8	20.7006^
(i^7πv/4)9/!9		12.6452i^
(i^7πv/4)10/!10	6.9521v
(i^7πv/4)11/!11		3.4747iv
(i^7πv/4)12/!12	1.5919^
(i^7πv/4)13/!13		0.6732i^
(i^7πv/4)14/!14	0.2644v
(i^7πv/4)15/!15		0.0969iv
(i^7πv/4)16/!16	0.0333^
(i^7πv/4)17/!17		0.2157i^
(i^7πv/4)18/!18	0.0033v
(i^7πv/4)19/!19		0.0952iv
	:
Total	0.707^	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:

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

Clockwise Rotation

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

• e^{i^πv/4} at 45^vo   = cos(π^v/4)*1^{^} + sin(π^v/4)*i^{^}
  • = √2/2*1^{^} + ^–√2/2*i^v = √2/2^{^} + √2/2i^v  
• e^{i^3πv/4} at 135^vo = cos(3π^v/4)*1^{^} + sin(3π^v/4)*i^{^}
  • = ^–√2/2*1^{^} + ^–√2/2*i^{^}  = √2/2^v + √2/2i^v
• e^{i^5πv/4} at 225^vo    = cos(5π^v/4)*1^{^} + sin(5π^v/4)*i^{^}  
  •  = ^–√2/2^{^}*1^{^} + √2/2*i^{^} = √2/2^v + √2/2i^{^}                         
• e^{i^7πv/4} at 315^vo     = cos(7π^v/4)*1^{^} + sin(7π^v/4)*i^{^}
  • = √2/2*1^{^} + √2/2*i^{^}  = √2/2^{^} + √2/2i^{^}                         

Next: Multiplication Table

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