Guide to Game Development/Theory/Mathematics/Quaternions

About Quaternions are used for rotating a geometry and points with along multiple axes of rotation. They exist in four dimensions, and so that have four parts: w, x, y and z. As x, y and z are linked and similar to each other, people sometimes use another letter to represent all three, v is common for this. To represent them you should store them in a 4-dimensional vector like so: . Quaternions are used as an alternative to Euler angles. Benefits of Quaternionshttps://www.youtube.com/watch?v=4mXL751ko0w at 1:35: No gimbal locking Interpolation is smooth and direct Simple to do calculations with Basic creation of a quaternion So if you chose some unit length 3D-vector that you could like to rotate the geometry around () and an amount of degrees that you want to rotate the geometry () then create a rotation quaternion (). There are generally two forms for representing quaternion, short and long. In the short notation it shows it in terms of w and v (a combination of x, y and z). This can be represented in the long notation using w, x, y and z: Where https://www.youtube.com/watch?v=SCbpxiCN0U0 at 5:39: If a quaternion is not rotated, then it will have the valuehttps://www.youtube.com/watch?v=4mXL751ko0w at 4:05: Links with imaginary numbers The 4 components (w, x, y and z) can be broken down into one real number and three imaginary numbers: where , and are imaginary numbers such that . From that, you can create the following statements: Inverse Quaternions To invert a quaternion, simply multiply the x, y and z components by -1. Multiplying quaternions When multiplying two quaternions together, order matters (). The multiple is defined byhttps://www.youtube.com/watch?v=CRiR2eY5R_s at 8:28: Add and understand the statement: (rotationQuaternion*pointQuaternion)*conjugateQuaternion look into: [Seen here] Spherical Linear Interpolation (SLERP) This is how you Interpolate gradually between two quaternions, like getting a mid-point (or some other point) between two quaternions. Superscript notation for Quaternions Before you can do SLERP, you need to understand the superscript notation for a quaternion. If you have a quaternion a and you use the superscript notation to raise it to the power t, this means that it will scale the angles inside the quaternion by superscript t. As this only scales all of the angles, this means that the magnitude () of the Quaternion is still 1. Superscript notation in short notation Superscript notation in long notation The SLERP Equation If your first Quaternion was q and the second quaternion was r and you wanted to find a point in-between (p) which is t percent of the way from q to r, where 0 ≤ t ≤ 1. The final equation is defined ashttps://en.wikipedia.org/wiki/Slerp#Quaternion_Slerp: See also Creating a Quaternion class in code External Links Youtube: Math for game developers: Quaternion playlist Humane Rigging 03 - 3D Bouncy Ball 05 - Quaternion Rotation Hand Calculation of Quaternion Rotation Wikipedia pages: Quaternion Quaternions and spatial rotation Slerp - Spherical Linear Interpolation Wikibooks: Abstract_Algebra/Quaternions References 50%