 # Math

Quaternion. Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = −k, ij = −ji History Per-Axis (Euler) Rotation Control While Avoiding Gimbal Lock? : Unity3D. Angle between two vectors. Using quaternion.AngleAxis. Quaternions: Not Satan. There are a lot of people working on cube puzzle games these days. As the developer on Six Face, I am one of them. A game like this can cause you to come across challenges that you didn't know existed. One such challenge is understanding rotation in 3D space. Quaternions: Not Satan. Visualizing Quaternions with Unity. How do you model the position and orientation of an airplane? Position is easy, just represent it with a point in 3D space. But how do you specify its orientation — which direction it’s pointing? At first glance, it seems a vector will do. After all, a vector points in some direction, right? If the plane is pointing east, represent its orientation by a unit vector pointing east. Unfortunately, we quickly run into trouble when we try to roll. Rotating Objects Using Quaternions. Last year may go down in history as The Year of the Hardware Acceleration. Much of the work rasterizing and texture-mapping polygons was off-loaded to dedicated hardware. How I learned to Stop Worrying and Love Quaternions. So you meet this cool API online somewhere. It looks pretty fun, pretty powerful. Like just what you've been looking for. You dive in, you're having a good time, things are going great, when all of a sudden, BAM, it whips out a quaternion. Uh-oh You want to be cool and use it, but you don't really know what you're doing. Don't worry. It doesn't have to be like that though. Understanding Quaternions3D Game Engine Programming. Understanding Quaternions In this article I will attempt to explain the concept of Quaternions in an easy to understand way. Avoiding Gimbal Lock. Mathematical reason behind gimbal lock in Euler angles. This article tries to explain gimbal lock from a mathematical perspective Mathematical singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability For example, the function f(x) = 1 / x on the real line has a singularity at x = 0, where it seems to “explode” to ±∞ and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it is not differentiable there. Singularity is present in (2D) polar space, it occurs when r = 0, irrespective of the value of θ the resulting point would be the same, which is the origin. Euler angles are perfectly fine for orientation represention of objects, the problem of gimbal lock only occurs when trying to rotate using Euler angles, since linear interpolation from one orientation to another by interpolating the angles isn’t well-behaved due to the gimbal lock.

Gyroscope. A gyroscope in operation. Note the freedom of rotation in all three axes. The rotor will maintain its spin axis direction regardless of the orientation of the outer frame. A gyroscope (from Greek γῦρος gûros, "circle" and σκοπέω skopéō, "to look") is a spinning wheel or disc in which the axis of rotation is free to assume any orientation. Zenith. Diagram showing the relationship between the zenith, the nadir, and different types of horizon. Note that the zenith is opposite the nadir. Zenith refers to an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e. the direction in which gravity pulls, is toward the nadir. The zenith is the "highest" point on the celestial sphere (meaning it is the farthest up from the gravitational force). Origin The word "zenith" derives from the inaccurate reading of the Arabic expression سمت الرأس (samt ar-ra's), meaning "direction of the head" or "path above the head", by Medieval Latin scribes in the Middle Ages (during the 14th century), possibly through Old Spanish. It was reduced to 'samt' ("direction") and miswritten as 'senit'/'cenit', as the "m" was misread as an "ni".

Relevance and Use See Also Euler (gimbal lock) Explained. Humane Rigging 03 - 3D Bouncy Ball 05 - Quaternion Rotation. Quaternions and 3D Rotations. The title of this tutorial is short and sweet. The tutorial itself will not be. First, an introduction. Quaternions are the things that scare all manner of mice and men. They are the things that go bump in the night. They are the reason your math teacher gave you an F. Quaternions. Quaternions. This page is an introduction to Quaternions, the pages below this have more detail about their algebra and how to use them to represent 3D rotations. Introduction Quaternions were discovered on 16 October 1843 by William Rowan Hamilton. He spent years trying to find a three dimensional number systems, but with no success, when he looked in 4 dimensions instead of 3 it worked. Quaternions form an interesting algebra where each object contains 4 scalar variables (sometimes known as Euler Parameters not to be confused with Euler angles), these objects can be added and multiplied as a single unit in a similar way to the usual algebra of numbers. Euler Angles. Understanding Euler Angles. 1. Introduction Attitude and Heading Sensors from CH Robotics can provide orientation information using both Euler Angles and Quaternions.

Compared to quaternions, Euler Angles are simple and intuitive and they lend themselves well to simple analysis and control. Matrix. Description This snippet is my matrix class. I use this class when I can't get Unity's internal rotation and 3D math functions to do exactly what I want. It's pretty easy to use and the functions are discrete. Rotating 3D objects. A cube rotating about its z-axis. Rotating things in three dimensions sounds complicated and it can be, but there are some simple rotations. For example, if we imagine rotating our cube around the z-axis (which points out of the screen), we are actually just rotating a square in two dimensions. There is a reason to learn trigonometry Rotating a point (x, 0) around the origin by θ. We can simplify things further, by just looking at a single node at position (x, 0). X' = x × cos(θ) Rotation About an Arbitrary Axis in 3 Dimensions.

Mesh from Vertices (Polygon from Spline) I am sharing Delaunay Triangulation Calculation maybe it can be helpful for others. Its useful for basic polygons. C# - Simple Voronoi/Delaunay Diagrams For Starters. Lloyd's algorithm. Example of Lloyd's algorithm. The Voronoi diagram of the current points at each iteration is shown. Delaunay - Voronoi Diagram library for Unity.

Finding Points of Intersection. Finding Points of Intersection. Geometry - Finding a point along a line a certain distance away from another point! Equation of a line - point - slope form. (Coordinate Geometry)