It turns out there is a good way to interpolate between two pure rotations. This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The quaternion with a norm of one, jqj 1, is a unit quaternion. The most commonly used interpolation method for quaternions is called spherical linear interpolation, or slerp. Interpolating rotation quaternions mathematics stack. They provide some key advantages over the traditional way of defining generic rotational transformations using euler angles. I need to interpolate between these two angles from 0% to one side to 100% to another side. A method for orientating and animating video images using quaternions to provide smooth, predictable rotations in a digital video effect device. Cs 4620 practicum programming assignment 6 animation. Select the interpolation method from slerp, lerp, or nlerp. W ein tro duce quaternion mathematics and discuss wh y quaternions are a b etter c hoice for implemen ting rotation than the w ellkno wn matrix implemen tations. By comparing them to two other kinds of rigid transformations, we examine their properties. When the initial end point is the identity quaternion, slerp gives a segment of a oneparameter subgroup of both the lie group of 3d rotations, so3, and its universal covering group of unit quaternions, s 3.
We see that the product of two quaternions is still a quaternion with scalar part p0q0. Note that, just like complex numbers, two unit quaternions multiplied together will result in another unit quaternion. Point m and a vector v the application of the vector over the point results in a. Therefore, the general idea for a more accurate interpolation is. Interpolating in quaternion space ensures a unique path under all circumstances. There are two special quaternions, the unit element 1 1. It is shown that for any two given points and two tangents specified at the. In the same way that a vector represent the relationship between two points. Suppose ive got two quaternions that each represent an angle. Abstract the main topics of this tec hnical rep ort are quaternions, their mathematical properties, and ho w they can b e used to rotate ob jects. Tutorial on quaternions part i computer action team. If we want to do a linear interpolation between two points a and b in normal space where t ranges from 0 to 1 this is a convex. Enter the values into the top two quaternion and t then press slerp to.
Since the unit quaternions form the unit sphere 83 in p. This document describes how to intepolate between two rigid transformations, each involving rotation and translation. A naive, straightforward, linear interpolation between the two keys results in a motion. An appropriate approach to interpolating quaternions will ensure a consistent angular velocity between a pair of quaternions. This work challenges that assumption, proposing instead an integrated set of equations which are all based on the same 4. The slerp formula for interpolating between two quaternions q 1 and q 2 is. Interpolating euler angles and interpolating quaternions are intuitively very different operations.
When interpolating between animation key frames quaternions provide an efficient means to interpolate the general rotations of joints in a skeleton. Orientation interpolation in quatemion space using. For these reasons quaternion interpolation of the rotational parameters is performed. Set the value of the inbetween rotation to be the normalized quaternion. From quaternion to matrix and back fabien sanglard. Rigid multibody systems are characterized by two dis. Applications of dual quaternions in three dimensional transformation and interpolation november 11, 20 matthew smith. Introducing the quaternions rotations using quaternions but there are many more unit quaternions than these. Aerospace blockset uses quaternions that are defined using the scalarfirst convention. Pufferfish does have it but i did it the hard way before i knew about the rhinocommon quaternions i coded what a quaternion does behind the scenes ala a reference from daniel piker, not actually using rhinocommon quaternions.
Although rotations can be represented as a 3 3 matrix, directly interpolating corresponding entries is not viable due to the partial dependency of entries in a rotation matrix. Yanghsing liwc10 calculated the relative orientation using dualquaternions. Because of its spherical nature, this method guarantees that any intermediate quaternions will also be unit quaternions. Quaternions as a solution to determining the angular. How do we interpolate between two quaternions representing rotations. An overview of doing physics with quaternions it has been said that one reason physics succeeds is because all the terms in an equation are tensors of the same rank. Ge gvmc98 demonstrated dualquaternions to be an efficient and practical method for interpolating threedimensional motions. Since the system is strictly diagonally dominant, there exists a unique solution. We will define orientation to mean an objects instantaneous rotational. An algorithm is presented to construct a c 2continuous bspline quaternion curve which interpolates a given sequence of unit quaternions on the rotation group so3. Euler angles are used in a lot of applications, but they tend to require some rather arbitrary.
Pdf interpolation of rotation and motion researchgate. Pdf the problem of smoothly interpolating between a given sequence of orientations is discussed. The result quaternion has a vector that is the cross product of the two vector parts, and a scalar that is their dot product, negated. The dot product of the quaternions is similar to that of vectors, it measures the angle formed between the two quaternions.
I havent encountered a need for this yet, so i havent researched it, but perhaps someday ill research it and write a tutorial about it to add to this series. Since i work a lot with complex numbers, id thought about getting the arg of these quaternions, averaging them, and creating a new quaternion. Then i just wrote an expression for linearly interpolating between them by t. This matlab function calculates the quaternion interpolation between two normalized quaternions p and q by interval fraction f.
However, matrices are more efficient when many points or vertices need to be. In computer graphics applications, quaternions are used to represent threedimensional rotations. Its done via a method known as slerp, or spherical linear interpolation. Interpolating between two orientations using quaternions is also the smoothest way to interpolate angles. Applications of dual quaternions in three dimensional. Animating rotation with quaternion curves ken shoemaker. For incremental orientation changes the appropriate quaternion is combined with the current quaternion of the video image in either source or. Quaternion interpolation between two quaternions simulink. Slerp gives a straightest and shortest path between its quaternion end points, and maps to a rotation through an angle of 2. Malte schi11 used a mean of multiple utational mmc model with dualquaternions to model bodies. An introduction to quaternions with application to rotations 3 this property is important because without it the product of two or more terms, such as vwzq, is ambiguous. The resulting interpolations are also visually very jerky, which simply is not acceptable in games any more.
Slerp yields a nondifferentiable curve whose angular velocity tends to infinity. In 1758, euler 1 introduced a set of independent coordinates, called euler angles, which describe a three. If a quaternion is looked upon as a fourdimensional vector, the quaternion product can be described by a matrixvector product as. The quaternion interpolation block calculates the quaternion interpolation between two normalized quaternions by an interval fraction. The dot product of two quaternions works in the same way as the dot product of two vectors. One must then include a lot of parentheses to dictate order. The procedure typically used for quaternion interpolation is called slerp, a name which derived from spherical linear interpolation. Interpolating solid orientations with a continuous b. Interpolating solid orientations with a c 2continuous b.
Consider the simplest case of interpolating between just two key quaternions. The spherical biarc is a curve on a sphere consisting of two smoothly joining circular arcs. Understanding quaternions 3d game engine programming. The identity quaternion has real part 1 and vector part 0. I i, j, and k are just three special unit imaginary quaternions.
The set of quaternions is closed under multiplication and addition. Moreover, the interpolating function is a curve that is required to be a shortestdistance path between the transformations, in which case the path is geodesic. Quaternions are often used in skeletal animation systems for the interpolation between general rotations. Quaternion multiplication an overview sciencedirect topics. Euler angles for a new orientation are input by an operator and are converted into an appropriate quaternion. To avoid confusion and enable the reader to easily distinguish a quaternion from a dualquaternion we use two discernible symbols to. Quaternions are also extremely useful for interpolating between two orientations in threedimensional space. In this case, its the linear angle between the two vectors \\mathbfp\ and \\mathbfp\prime\ but the the rotation is measured about the circle that is swept out by the rotation. The following calculator allows you to interpolate between two quaternions using the slerp algorithm. The angle between two quaternions in 4d space is half the angle one would need to rotate from one orientation to the other in 3d space p q p 0 q 0 p 1 q 1 p 2 q 2 p 3 q 3 p q cosm. Quaternion interpolation between two quaternions matlab. Interpolating solid orientations with a c2continuous bspline quaternion curve 609 where x i i0,1,n are the given points to be interpolated and x i i0,1,n are the control points we want to compute.
With quaternions we interpolate in the 4dimensional space of quaternions and normalize. I figured this out just by drawing a 2d graph like figure 1, where one of my vectors is the x axis 1, 0 and the other one is cos. If it is necessary to interpolate a sequence of two or more quaternions, the usage of. Interpolating quaternions is what you expect it to be. Intuitive interpolation between different orientations.