How can this indicate rotation?

  • Thread starter Thread starter yicong2011
  • Start date Start date
  • Tags Tags
    Rotation
AI Thread Summary
The discussion explores the mathematical representation of displacement and rotation using equations involving real and imaginary components. It highlights that while r² = (x-a)² + (y-b)² + (z-c)² signifies displacement, the inclusion of imaginary units (i) in r² = (x-ia)² + (y-ib)² + (z-ic)² indicates rotation. The concept of representing 2D coordinates in the imaginary plane shows that multiplying by i results in a 90-degree rotation. Furthermore, it explains that in 3D space, quaternions can represent coordinates, where multiplication with another quaternion results in a rotation. This mathematical framework connects displacement and rotation through complex and quaternion numbers.
yicong2011
Messages
75
Reaction score
0
If we say

r2= (x-a)2 + (y-b)2 + (z-c)2

can generally represent displacement,

why can

r2= (x- ia)2 + (y- ib)2 + (z- ic)2

generally represent rotation?

(i2 = -1)


And why

r2= x2 + y2 + (z - ia)2

represents rotation with respect to z axis
 
Physics news on Phys.org
I'm not quite sure what you're saying here.

What I see is the equation of a sphere of which the center has been translated.
First over real coordinates, and then over imaginary coordinates.

What I can tell you, is that if you can for instance represent 2-dimensional coordinates by a number in the imaginary plane, that multiplication by i equals rotation over an angle of 90 degrees.

More generally if you have 3-dimensional coordinates, you can represent them by a number in quaternion space. Multiplication with an other quaternion number comes out as a rotation.
 

Thread 'Is calling fictitious forces "not real" just about terminology?'

Hi there, im studying nanoscience at the university in Basel. Today I looked at the topic of intertial and non-inertial reference frames and the existence of fictitious forces. I understand that you call forces real in physics if they appear in interplay. Meaning that a force is real when there is the "actio" partner to the "reactio" partner. If this condition is not satisfied the force is not real. I also understand that if you specifically look at non-inertial reference frames you can...
View full post »

Thread 'Derivation of Hamilton's Principle: Questions'

I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
View full post »

Similar threads

I Question on analytic mechanics
Replies
3
Views
1K
Observation about the rotation of a disc
Replies
4
Views
1K
I Lagrangian mechanics - generalised coordinates question
Replies
4
Views
1K
Bar hanging from two springs with unequal masses
Replies
7
Views
3K
I Moment of inertia of human body with limbs at an angle
Replies
12
Views
2K
I Change of coordinates in a potential energy field
Replies
2
Views
1K
A Displacement-Rotation Algorithm
Replies
2
Views
916
Can you find orbital velocity from circle equation y^2+x^2=r^2?
Replies
9
Views
2K
I How can angular displacement be larger than 2pi?
Replies
20
Views
2K
I Rotating plate thrown parallel to ground
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top