1. This Week

2. Short Course in: Mathematics and Analytic Geometry Week 8 Complex Numbers

3. Unreal Quadratic Roots • The roots of quadratic equations: can be found using the quadratic formula: However, if (b2 < 4ac) then the roots are not real.

4. Unreal Quadratic Roots • For example, consider roots for the quadratic equation: • The square root of (-1) is undefined in the real number set.

5. Imaginary Numbers • Terms involving the square root of (-1) form a new numbers, called Imaginary numbers: • The roots for:

6. Complex Numbers • Quite often imaginary terms appear in sums with real terms; for example: • A complex number is any number that can be expressed in this form: • This implies that real numbers are a subset of the complex numbers.

7. The Imaginary Powers • The properties of i is as follows: • These powers have a cycle length of 4:

8. The Complex Plane • Given the distinction between imaginary and real numbers, complex numbers can represent points in a plane:

9. Complex Numbers in Polar Form • Complex numbers can also be expressed in terms of radius and angle:

10. Euler’s Formula • Given the power expansions of sin x, cos x and ex: • If x is imaginary:

11. Complex Number Notations • Just as with vectors, complex numbers can be represented by a single letter (usually z): • Other notations:

12. Conjugate Pairs • Given a complex number (x + yi), its conjugate pair is (x - yi) and has the following properties:

13. Euler Rotations • Euler rotations are given by translating the centre of rotation of an object to the origin and applying rotation matrices: • Sometimes called: yaw,roll and pitch.

14. Gimbals Lock • When rotation matrices are applied in a fixed order (usually pitch, roll and yaw) their operations represent a system of rotation gimbals: • Gimbals lock occurs when any two of these frames coincide on the same plane.

15. Gimbals Lock • For example, assume we apply rotations in pitch, roll and yaw order, all rotation angles are initially zero and we choose to roll π/2 radians: • Any pitch change is independent of roll and yaw: • However, yaw depends on roll: the result is an anticlockwise pitch rotation, we have lost yaw.

16. The Quaternion* • A Quaternion can be defined as an extended complex number having the form: • The imaginary parts are as follows:

17. Quaternion Inverse • Given a quaternion q, the inverse can be found as follows:

18. Quaternion Rotation • Assume we have a unit vector u and a position vector v and we want to rotate the point described by vector v clockwise about an axis described by u:

19. Quaternion Rotation • We can define a quaternion q such that: • And θ defines the intended angle of rotation.

20. Quaternion Rotation • Given quaternion q, a clockwise rotation of θ radian, of a position vector v about the unit vector u, can be defined by the following mapping:

21. Next Week • SLERP • Curves, Splines and Patches