Arrow Tech Trivia - 11 - Demystify the Quaternion

Transcript

[Music] hello everybody my name is JJ Manu and I'm an application engineer at Aro Electronics today we talk about quat tanion and rotation calculation that are required in most of tracking movement applications even if it can look more intuitive at the beginning earlier angles is not the right way to compute rotations but quion many people are confused by quion because it uses a four-dimension space to move in our three-dimension world so let's demystify quanon by giving a graphical representation and showing it just a logical extension of natural real and complex numbers first numbers learn at school are natural numbers everybody learned to count 0 1 2 3 4 5 then other child we discover these numbers can be negative to create a new group called integers later in elementary school we learn that the result of the division of two integers is not necessarily an integer for example 3 / by 2 equals 1.5 any number that can be expressed as the fraction P ided by Q of integers I call rational numbers finally irrational numbers like square root of two or Pi that have an infinite numbers of decimals rational and irrational numbers together form the real numbers all these numbers can be represented along a line to move in a plane the complex numbers were contrived a Cartesian system is built with the real numbers on x axis and the imaginary part on the y axis hence any point in the plane can be represented by a complex number in the form from a + b i I represent the point on the x axis with a distance of one from the xais complex numbers are very interesting properties and are convenient to Define rotations in the plane a multiplication by I is a rotation of 90° I square is a rotation of 180° and i² = minus1 I to the power of 4 is a rotation of 36 60° this explains why a real number to the square is always positive more generally a rotation in a plane can be described by the transform 5 Z of Omega equal to Z ² mtip Omega divided by the quadrant of Z as it's convenient to work on the unit circle if we take Z equal 1 + hi i h is the half turn of the rotation we will see rotation in 3D uses the AL turn concept to the next step is logically to have numbers that can describe the rotation in 3D for many years mathematicians thought there was a solution the style of a + I + CJ three variables to move in a three dimension space in fact there is no solution like that and a space of four dimension is necessary the Irish mathematician William Rowan Hamilton got this intuition in 1843 and called this new number a quanon it has the form a + b i + C J + d k with i s = j s = k s = minus1 beside i j equal k j k equal I and so on in the end I J and and K form an olian frame a quanon q is then return in the form t + V where T is the scalar part and V the vector part and the following theorem allows simple calculation for rotation in 3D space for any quion qal t + V the map 5 q of Omega equal to Q multip by Omega multip by Q conjugate divided by quadrants of Q is a rotation about the vector v with half turn H equal to scar of V / T similar to complex numbers when you have a point in Space the quanon Omega is a pure imaginary quanon in other words t equal Z the result of the transform of five Q will give also a pure imaginary quanan and we end up with a new P that rotated around V with an half turn equal to to scale of V ided by T calculation of rotation by quion is as simple as [Music] [Applause] [Music] that