四元数、插值与动画：优化旋转与动画实现的深入解析

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"《四元数、插值与动画》技术报告深入探讨了四元数在计算机图形学中的重要应用。四元数是一种复数扩展形式，由四个实数组成，用于表示三维空间中的旋转。相较于传统的矩阵旋转方式，四元数具有更高的效率和精度，因为它避免了角度参数化可能导致的舍入误差，并且可以进行连续平滑的旋转。报告首先介绍了四元数的基本数学概念，包括其定义、运算规则以及它们在三维空间中的几何意义。读者将了解到四元数如何通过简洁的数学结构表示旋转，这使得它在游戏引擎、机器人控制等领域的旋转矩阵替代方案中表现出色。接下来，报告详细讨论了不同旋转插值方法，如Slerp（球面插值）和Squad插值，这两种方法在动画制作中尤为关键，因为它们能保持旋转的连续性和平滑性，从而产生自然的动画效果。作者提供了完整的证明，确保了这些插值算法的正确性。在处理这些插值方法的过程中，报告提出了一种新的插值方法——Spring，它基于一组优化的目标约束，旨在提供更加精确且符合物理预期的动画平滑度。Spring插值方法的开发是基于对已有插值技术的深入理解，展示了作者们在该领域的创新思考。《四元数、插值与动画》是一份实用且理论丰富的技术文档，不仅介绍了四元数的基础知识，还为动画制作人员提供了高效、精确的旋转管理和动画设计工具。对于任何从事游戏开发、图形渲染或计算机视觉的工程师来说，这份报告都是一个不可多得的参考资料。"

Denition 6.

Given

q; q

, subtraction is dened



The denition gives the exp ected:

Prop osition 7. (Quaternion subtraction)

Let

q; q

, where

]

and

;

]

. Then

;

]

Corresp onding to the denition of the conjugate of a complex number, we dene the conjugate

of a quaternion:

Denition 7.

Let

. Then



is cal led the conjugate of

and is dened by





[

]





[

]

The denition gives rise to the following prop erties:

Prop osition 8.

Let

p; q

. Then:

(



)



ii)

(

)



iii)

(

)



iv)



The norm of a quaternion is obtained using conjugation:

Denition 8.

Let

and let the mapping

kk

be dened by

k



. This mapping is cal led

the norm and

is the norm of

That this mapping is a norm in the usual sense is shown in the corollary to prop osition 9. The

norm mapping has a number of interesting prop erties that are summarized in:

Prop osition 9.

Let

q; q

and let

kk

be given as is denition 8. The fol lowing equations hold:



(3.1)



(3.2)

(3.3)

Pro of of prop osition 9

The equations 3.1 and 3.2 can be seen directly. Equation 3.3 follows from:

(

)



0



We will later need the inner pro duct of two quaternions. We also want to show that the norm

mapping is indeed a norm in the usual mathematical sense. From equation 3.1 in prop osition 9

it follows that the norm of a quaternion

can b e written as it is usually obtained from the inner

pro duct (if

is identied with the corresp onding vector in

). This prop erty is formalized

by:

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四元数、插值与动画：优化旋转与动画实现的深入解析

Quaternions.pdf

quaternion.pdf

Quaternion 引用呢

C++ opencv Quaterniond

for angle in euler_angles: q_angle = quaternion_from_euler(0, 0, angle, axes='sxyz') q = Quaternion(*q_angle) quaternions.append(q)

waypoints = list() waypoints.append(Pose(Point(1.480, 1.990, 0), quaternions[0]))

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