题  目： Quaternions and Dual Quaternions

报告人： Ron Goldman (Rice University,USA)

时  间： 2012年6月14日下午3:30-5:30

地  点： 科研管理楼1218室

摘  要：

Quaternions are vectors in 4-dimensions endowed rules for multiplication and division as well as for addition and subtraction. Quaternion multiplication can be used to rotate vectors in 3-dimensions by sandwiching a vector between a unit quaternion and its inverse.

The purpose of this introduction to quaternions is to make four principal contributions:

1. To provide a fresh, geometric interpretation for quaternions, appropriate to contemporary Computer Graphics;

2. To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions;

3. To develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection based on the theory of complex numbers;

4. To show how to apply sandwiching to compute perspective projections.

Yet while quaternions are a powerful tool for representing rotations in 3-dimensions, quaternions cannot be used to represent translations in 3-dimensions. Therefore quaternions cannot be applied to represent all rigid motions in 3-dimensions.

Dual quaternions are vectors in 8-dimensions, represented by pairs of quaternions endowed with a special rule for multiplication. Dual quaternions can be used to represent both rotations and translations in 3-dimensions. Therefore unit dual quaternions can be used to represent all rigid motions in 3-dimensions. Thus dual quaternions extend the power of classical quaternions. We shall describe both the algebra and the geometry of dual quaternions. We will also show how to apply dual quaternions to provide an alternative approach to computing perspective projections in 3-dimensions.

主办单位：

0029cc金沙贵宾会

国家数学与交叉科学中心合肥分中心

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