This Week WeekTopic 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 The roots of quadratic equations: The roots of quadratic equations: can be found using the quadratic formula: However, if (b 2 < 4ac) then the roots are not real.

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

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

Complex Numbers Quite often imaginary terms appear in sums with real terms; for example: 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: 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. This implies that real numbers are a subset of the complex numbers.

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

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

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

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

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

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

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

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

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

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

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

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

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: 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:

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