R2 – Complex Rotations

This post describes more complex rotations in R2 including rotation around points other than the origin. It also describes the formula of rotation.

R2 Rotations Around Points Other than the Origin

So far the rotations looked at have been restricted to rotations around the origin on the unit circle.  Now it’s time to look at more complex rotations where the operand can be any mix of Opposite Values with any degree of rotation.

The calculations for R2 rotations use  a Wave Number rotation formula derived from one used in classical maths for rotation around the origin:

• Classical Math formula for rotation around the origin:
  • x‘ = xcosθ – ysinθ
  • y‘ = ycosθ + xsinθ

• Wave Numbers formula for rotation around the single point axis (xa, ya):
  • x‘ = ((|x| + ^–|xa|)cosθ + (^–|y| + |ya|)sinθ + |xa|)*^{^}
  • y‘ = ((|y| +^–|ya|)cosθ + (|x| +^–|xa|)sinθ + |ya|)*i^{^}

Here x and y are the counters of the point to be rotated, xa and ya are the counters of the single point axis around which the rotation takes place and θ is the angle of rotation. Finally, x‘ and y‘ are the coordinates of the new location of the point.

Rotation Around Origin Using R2 Formula

• π^{^}/2 ↺ (3^{^} + 4i^v) – around the origin
  • => x‘ = ((3 +^–0)cos(π^{^}/2) + (^{– –}4 +0)sinπ^{^}/2 + 0)*^{^}
    • = (3*0 + 4*1)*^{^} = 4^{^}
  • => y‘ = ((^–4 +^–0)cos(π^{^}/2) + (3 +^–0)sinπ^{^}/2 + 0)i^{^}
    • = (^–4*0 + 3*1)*i^{^} = 3i^{^}
  • =   4^{^} + 3i^{^}

• π^v/2 ↺ (3^{^}+ 4i^v) – around the origin            
  • => x‘ = ((3 + ^–0)cos(π^v/2) + (^{– –}4 + 0)sinπ^v/2 + 0)*^{^}
    • = (3*0 + 4*^–1)*^{^} = 4^v
  • => y‘ = ((^–4+ ^–0)cos(π^v/2) + (3 +^–0)sinπ^v/2 + 0)*i^{^}
    • = (^–4*0 + 3*^–1)*i^{^} = 3i^v
  • =   4^v + 3i^v

• π^{^}/4 ↺ (3^{^}+ 4i^v) – around the origin
  • => x‘ = ((3 +^–0)cos(π^{^}/4) + (^{– –}4 +^–0)sinπ^{^}/4 + 0)*^{^}
    • = 3*√2/2^{^} + 4*√2/2^{^}
    • = 3√2/2^{^} + 2√2^{^}
    • = 7√2^{^}/2 = 4.95^{^}
  • => y‘ = ((^–4+ ^–0)cos(π^{^}/4) + (3 +^–0)sinπ^{^}/4 + 0)*i^{^}
    • = (^–4*√2/2)i^{^} + (3*√2/2)i^{^}
    • = 2√2i^v + 3/2√2i^{^}
    • = 0.707i^v
  • = 4.95^{^} + .707i^v

• π^v/4 ↺ (3^{^}+ 4i^v) – around the origin
  • => x‘ = ((3+^–0)cos(π^v/4) +(^{– –}4 +^–0)sinπ^v/4 + 0)*^{^}
    • = 3*√2/2^{^} + 4*√2/2^v
    • = 3√2/2^{^} + 2√2^v
    • = 1/2√2^v = 0.707^v
  • => y‘ = ((^–4+ ^–0)cos(π^v/4) + (3 +^–0)sinπ^v/4 + 0)*i^{^}
    • = ^–4*√2/2i^{^} + 3*√2/2i^v
    • = 2√2i^v + 3/2√2i^v
    • = 7/2√2i^v = 4.95i^v
  • 0.707^v +4.95i^v

Rotation Around Other Points Using R2 Formula:

• π^{^}/2 ↺^{(7v, 8iv)} (3^{^}+ 4i^v) – around point (7^v, 8i^v)
  • =>x‘ = ((3+^{– –}7)cos(π^{^}/2) + (^{– –}4 +^–8)sinπ^{^}/2 + ^–7)*^{^}
    • = (10*0 + ^–4*1 + ^–7)*^{^}= 11^v
  • =>y‘= ((^–4 +^{– –}8)cos(π^{^}/2)+(3+^{– –}7)sinπ^{^}/2 + ^–8)*i^{^}
    • = (4*0 + 10*1 + ^–8)*i^{^} = 2i^{^}
  • =   11^v + 2i^{^}

• π^v/2 ↺^{(7v, 8iv)} (3^{^}+ 4i^v) – around point (7^v, 8i^v)
  • =>x‘=  ((3+^{– –}7)cos(π^v/2) + (^{– –}4 +^–8)sinπ^v/2 +^–7)*^{^}
    • = (10*0 + ^–4 + ^–7)^{^} = 3^v
  • =>y‘= ((^–4+^{– –}8)cos(π^{^}/2) (3+^{– –}7)sinπ^v/2 + ^–8)*i^{^}
    • = (^–4*0 + 10*^–1 + ^–8)*i^{^} = 18i^v
  • =   3^v + 18i^v

• π^{^}/3 ↺^{(4^, 3i^)} (2^{^} + 5i^v ) – around point (4^{^} + 3i^{^})           
  • =>x‘= ((2+^–4)cos(π^{^}/3) +(^{– –}5 + 3)sinπ^{^}/3 + 4)*^{^}
    • = (^–2*1/2+ 8*√3/2+ 4)^{^}
    • = 1^v+4*√3^{^}+4^{^} = 9.928^{^}
  • =>y‘ = ((^–5 +^–3)cos(π^{^}/3) + (2 +^–4^{^})sinπ^{^}/3+3)*i^{^}
    • = (^–8*1/2+^–2*√3/2+3)*i^{^}
    • = 4i^v+ √3i^v+3i^{^}
    • = 1^v + √3i^v = 2.732i^v
  • =   9.928^{^}+ 2.732i^v

Conclusion

Try these examples of R2 rotation with our online calculator.

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