This Week Topic Week 1 Coordinate Systems Basic

This Week Topic Week 1 Coordinate Systems, Basic Functions Week 2 Trigonometry and Vectors (Part 1) Week 3 Vectors (Part 2) Week 4 Vectors (Part 3: Locus) Week 5 Tutorial A: Question and answer session for weeks 1 -4 Week 6 Matrices (Part 1) Week 7 Matrices (Part 2) and Transformations Week 8 Complex numbers Week 9 Curves Week 10 Tutorial B: Question and answer session focusing on weeks 6 -9

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

Unreal Quadratic Roots n The roots of quadratic equations: can be found using the quadratic formula: However, if (b 2 < 4 ac) then the roots are not real.

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

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

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

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

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

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

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

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

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

Euler Rotations n 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. n

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

Gimbals Lock n 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: n Any pitch change is independent of roll and yaw: n However, yaw depends on roll: the result is an anticlockwise pitch rotation, we have lost yaw.

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

Quaternion Inverse n Given a quaternion q, the inverse can be found as follows:

Quaternion Rotation n 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:

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

Quaternion Rotation n 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:

Next Week SLERP n Curves, Splines and Patches n