![ V = v cos (kt) + iv sin (kt)
Let's do similar calculus now:
V' = - kv sin (kt) + i kv cos (kt)
Looks even uglier, right? Well, let's rearrange V' a bit:
V' = -kv sin (kt) + i kv cos(kt)
= ik v cos(kt) - kv sin (kt) // switched terms
= (ik) [ v cos (kt) + iv sin (kt) ] // algebra I factoring
= (ik) V
So, now to take the derivative of V, we just multiply by (ik)!
Now, here's where things get strange. We had
dV/dt = CI
But now, for our half-real/half-imaginary sinusoid, we
know that V' = (ik) V. So, now, our capacitor works like this:
(ik) V = CI](https://image.slidesharecdn.com/electricalapplicationsandquaternions-170711140759/85/application-of-complex-numbers-3-320.jpg)
Uploaded byKaustubh Garud
application of complex numbers
The document discusses quaternions and their applications. It begins by providing a brief history of quaternions, defined by Sir William Rowan Hamilton in 1843. Quaternions represent rotations and orientations in 3D space and are used in computer graphics, control theory, physics, and other fields due to advantages like avoiding gimbal lock. The document then covers details of quaternion operations like addition, multiplication, and rotation representations. It provides examples of using quaternions to represent and perform rotations in three dimensions.
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application of complex numbers
- 1. Presented by: Kaustubh S.Garud Roll No. 503006
- 2. Electrical Application Let'ssay you have to analyze a circuit with a sinusoidal voltage source, a resistor, and a capacitor in parallel. Here are the rules: V = v cos (kt) // we said it was a sinusoid V = IR // Ohm's Law dV/dt = CI // the voltage across a capacitor increases as you run current through it If you solve this circuit with ordinary calculus, dealing with real voltages, you get results like the following: V = v cos (kt) V' = -k v sin (kt)
- 3. V =v cos (kt) + iv sin (kt) Let's do similar calculus now: V' = - kv sin (kt) + i kv cos (kt) Looks even uglier, right? Well, let's rearrange V' a bit: V' = -kv sin (kt) + i kv cos(kt) = ik v cos(kt) - kv sin (kt) // switched terms = (ik) [ v cos (kt) + iv sin (kt) ] // algebra I factoring = (ik) V So, now to take the derivative of V, we just multiply by (ik)! Now, here's where things get strange. We had dV/dt = CI But now, for our half-real/half-imaginary sinusoid, we know that V' = (ik) V. So, now, our capacitor works like this: (ik) V = CI
- 4. V =(C/ik) I // little i is the square root of -1, big I is current Looks a lot like Ohm's Law! In effect, in sinusoidal situations, a capacitor is mathematically just like a resistor, that happens to have an imaginary value. Strange but true. So, we can treat our circuit with the same techniques that we use for two ordinary resistors, which is just algebra, only now one of our resistors is imaginary. The algebra is a little messier, but it's still algebra! That is the beauty of complex numbers. Poof, the Calculus is out of your face!
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- 6. Quaternion - BriefHistory Invented in 1843 by Irish mathematician Sir William Rowan Hamilton Founded when attempting to extend complex numbers to the 3rd dimension Discovered on October 16 in the form of the equation: 1222 ijkkji 6 From: http://en.wikipedia.org/wiki/Quaternion
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- 8. Applications of Quaternions Used to represent rotations and orientations of objects in three-dimensional space in: Computer graphics Control theory Signal processing Attitude controls Physics Orbital mechanics Quantum Computing, quantum circuit design 8 From: http://en.wikipedia.org/wiki/Quaternion
- 9. Advantages of Quaternions Avoids Gimbal Lock Faster multiplication algorithms to combine successive rotations than using rotation matrices Easier to normalize than rotation matrices Interpolation Mathematically stable – suitable for statistics
- 10. Operators Addition Multiplication Conjugate Length kqpjqpiqpqpqp kqjqiqqq kpjpippp 33221100 3210 3210 qpqppqqpqppq 0000 qqq 0 * *** pqpq 2 3 2 2 2 1 2 0 * qqqqqqq Operators on Quaternions
- 11. Define unitquaternion as follows to represent rotation Example Rot(z,90°) q and –q represent the same rotation nqn ˆsincos),ˆ(Rot 22 1q 2 2 2 2 00q Why “unit”? DOF point of view! Unit Quaternion
- 12. q = w+ xi + yj + zk • w, x, y, z are real numbers • w is scalar part • x, y, z are vector parts • Thus it can also be represented as: q = (w, v(x,y,z)) or q = w + v Quaternion – scalar and vector parts
- 13. Scalar &Vector 4 dimensions of a quaternion: 3-dimensional space (vector) Angle of rotation (scalar) Quaternion can be transformed to other geometric algorithm: Rotation matrix ↔ quaternion Rotation axis and angle ↔ quaternion Spherical rotation angles ↔ quaternion Euler rotation angles ↔ quaternion What are relations of quaternions to other topics in kinematics? Quaternion – Dimension and Transformation
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- 15. Quaternion Operations Addition/subtraction Multiplication Division Conjugate Magnitude Normalization Transformations Concatenation
- 16. Quaternion Operations Addition: Given two quaternions: q1 = q1w + q1xi + q1yj + q1zk q2 = q2w + q2xi + q2yj + q2zk The result quaternion q3 is: q3 = q1 + q2 q3 = (q1w + q2w) + (q1x + q2x)i + (q1y + q2y)j + (q1z + q2z)k
- 17. Quaternion Operations Subtraction: Given two quaternions: q1 = q1w + q1xi + q1yj + q1zk q2 = q2w + q2xi + q2yj + q2zk The result quaternion q3 is: q3 = q1 - q2 q3 = (q1w – q2w) + (q1x – q2x)i + (q1y – q2y)j + (q1z – q2z)k
- 18. Quaternion Operations Multiplication Distributive Associative Not commutative because of the i2 =j2=k2=-1 i jk i jk (-k) (-j) (-i)
- 19. Quaternion Operations Multiplication Given two quaternions: q1 = q1w + q1xi + q1yj + q1zk q2 = q2w + q2xi + q2yj + q2zk The result quaternion q3 is: q3 = q1 * q2 q3 = q1w*q2w + q1w*q2xi + q1w*q2yj + q1w*q2zk + q1xi*q2w + q1xi*q2xi– q1x*q2x + etc…
- 20. Quaternion Operations Multiplication Resulting quaternion q3 is: q3= (q1wq2w + q1xq2x + q1yq2y + q1zq2z) + (q1wq2x + q1xq2w + q1yq2z – q1zq2y)i + (q1wq2y + q1yq2w + q1zq2x – q1xq2z)j + (q1wq2z + q1zq2w + q1xq2y – q1yq2x)k
- 21. Quaternion Operations Multiplication Or, in scalar-vector format: q3= q1q2 = (q1w, v1)(q2w, v2) = (q1wq2w - v1.v2, q1wv2 + q2v1 + v1 x v2) or q3 = q1wq2w - v1.v2 + q1wv2 + q2v1 + v1 x v2 Dot product Cross product scalar vector
- 22. Quaternion Operations Magnitude Also called modulus Is the length of the quaternion from the origin Given a quaternion: q = w + xi+ yj + zk The magnitude of quaternion q is |q|, where: 2222 *|| zyxwqqq
- 23. Quaternion Operations Normalization Normalization results in a unit quaternion where: w2 + x2 + y2 + z2 = 1 Given a quaternion: q = w + xi+ yj + zk To normalize quaternion q, divide it by its magnitude (|q|): Also referred to as quaternion sign: sgn(q) q || ˆ q q q
- 24. Quaternion Operations Conjugate: Given a quaternion: q = w + xi+ yj + zk The conjugate of quaternion q is q*, where: q* = w – xi – yj – zk
- 25. Quaternion Operations Inverse Can be used for division Given a quaternion: q = w + xi+ yj + zk The inverse of quaternion q is q-1, where: 2 1 || * q q q
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- 27. Matrix Rotationis based on 3 rotations: On axes: x, y, z Or yaw, pitch, roll (which one corresponds to which axis, depends on the orientation to the axes) Sequence matters (x-y may not equal y-x) x y z Matrix Rotation
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- 29. w =cos(θ/2) v = sin(θ/2)û Where û is a unit/normalized vector u (i, j, k) Quaternion can be represented as: q=cos(θ/2)+sin(θ/2)(xi+yj+zk) or q=cos(θ/2)+sin(θ/2) û w Parts of quaternion Matrix rotation versus quaternion rotation Quaternion Rotation
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- 32. Let’s do anotherone! x y z
- 33. Quaternion? v(x,y,z) w Can createrotation by using arbitrary axis (v(x,y,z)) and rotate the object by w amount.
- 34. Calculation isstill done in matrix form Given a quaternion: q = w + xi+ yj + zk The matrix form of quaternion q is: w 2222 2222 2222 2222 2222 2222 zyxwyzwxwyxz wxyzzyxwxywz xzwywzxyzyxw Matrix entries are taken all from quaternion Rotation of a Quaternion
- 35. Quaternion Rotation Whenit is a unit quaternion: w 222 22 22 2212222 2222122 2222221 yxyzwxwyxz wxyzzxxywz xzwywzxyzy Quaternion matrix for unit quaternion: w=1
- 36. Rotation ofvector v is v’ Where v is: v=ai+bj+ck By quaternion q=(w,u) Where w = 1 Vector (axis): u = i + j + k Rotation angle: 120° = (2π)/3 radian (θ) Length of u =√3 If we rotate a vector, the result should be a vector. 2 1 3 )( . 2 3 2 1 ˆ 2 3 2 1 ˆ60sin60cos ˆ 3 sin 3 cos ˆ 2 sin 2 cos ˆ 2 sin 2 cos 3 2 3 2 kji q kji q uq uq uq uq uq Our notation: quaternion Example of unit quaternion
- 37. So torotate v: v’ = q v q* Where q* is conjugate of q: 2 )1( * kji q So now we can substitute q v q* to matrix form of quaternion Example of quaternion rotation (cont’d)
- 38.