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980 lines
37 KiB
C++
980 lines
37 KiB
C++
// This file is part of OpenCV project.
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// It is subject to the license terms in the LICENSE file found in the top-level directory
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// of this distribution and at http://opencv.org/license.html.
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//
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//
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// License Agreement
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// For Open Source Computer Vision Library
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//
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// Copyright (C) 2020, Huawei Technologies Co., Ltd. All rights reserved.
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// Third party copyrights are property of their respective owners.
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//
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// Licensed under the Apache License, Version 2.0 (the "License");
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// you may not use this file except in compliance with the License.
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// You may obtain a copy of the License at
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//
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// http://www.apache.org/licenses/LICENSE-2.0
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//
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// Unless required by applicable law or agreed to in writing, software
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// distributed under the License is distributed on an "AS IS" BASIS,
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// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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// See the License for the specific language governing permissions and
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// limitations under the License.
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//
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// Author: Liangqian Kong <kongliangqian@huawei.com>
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// Longbu Wang <wanglongbu@huawei.com>
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#ifndef OPENCV_CORE_DUALQUATERNION_HPP
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#define OPENCV_CORE_DUALQUATERNION_HPP
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#include <opencv2/core/quaternion.hpp>
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#include <opencv2/core/affine.hpp>
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namespace cv{
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//! @addtogroup core
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//! @{
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template <typename _Tp> class DualQuat;
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template <typename _Tp> std::ostream& operator<<(std::ostream&, const DualQuat<_Tp>&);
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/**
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* Dual quaternions were introduced to describe rotation together with translation while ordinary
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* quaternions can only describe rotation. It can be used for shortest path pose interpolation,
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* local pose optimization or volumetric deformation. More details can be found
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* - https://en.wikipedia.org/wiki/Dual_quaternion
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* - ["A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D character hierarchies", Ben Kenwright, 2012](https://borodust.org/public/shared/beginner_dual_quats.pdf)
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* - ["Dual Quaternions", Yan-Bin Jia, 2013](http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pdf)
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* - ["Geometric Skinning with Approximate Dual Quaternion Blending", Kavan, 2008](https://www.cs.utah.edu/~ladislav/kavan08geometric/kavan08geometric)
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* - http://rodolphe-vaillant.fr/?e=29
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*
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* A unit dual quaternion can be classically represented as:
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* \f[
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* \begin{equation}
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* \begin{split}
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* \sigma &= \left(r+\frac{\epsilon}{2}tr\right)\\
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* &= [w, x, y, z, w\_, x\_, y\_, z\_]
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* \end{split}
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* \end{equation}
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* \f]
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* where \f$r, t\f$ represents the rotation (ordinary unit quaternion) and translation (pure ordinary quaternion) respectively.
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*
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* A general dual quaternions which consist of two quaternions is usually represented in form of:
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* \f[
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* \sigma = p + \epsilon q
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* \f]
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* where the introduced dual unit \f$\epsilon\f$ satisfies \f$\epsilon^2 = \epsilon^3 =...=0\f$, and \f$p, q\f$ are quaternions.
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*
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* Alternatively, dual quaternions can also be interpreted as four components which are all [dual numbers](https://www.cs.utah.edu/~ladislav/kavan08geometric/kavan08geometric):
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* \f[
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* \sigma = \hat{q}_w + \hat{q}_xi + \hat{q}_yj + \hat{q}_zk
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* \f]
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* If we set \f$\hat{q}_x, \hat{q}_y\f$ and \f$\hat{q}_z\f$ equal to 0, a dual quaternion is transformed to a dual number. see normalize().
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*
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* If you want to create a dual quaternion, you can use:
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*
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* ```
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* using namespace cv;
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* double angle = CV_PI;
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*
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* // create from eight number
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* DualQuatd dq1(1, 2, 3, 4, 5, 6, 7, 8); //p = [1,2,3,4]. q=[5,6,7,8]
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*
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* // create from Vec
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* Vec<double, 8> v{1,2,3,4,5,6,7,8};
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* DualQuatd dq_v{v};
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*
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* // create from two quaternion
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* Quatd p(1, 2, 3, 4);
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* Quatd q(5, 6, 7, 8);
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* DualQuatd dq2 = DualQuatd::createFromQuat(p, q);
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*
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* // create from an angle, an axis and a translation
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* Vec3d axis{0, 0, 1};
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* Vec3d trans{3, 4, 5};
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* DualQuatd dq3 = DualQuatd::createFromAngleAxisTrans(angle, axis, trans);
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*
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* // If you already have an instance of class Affine3, then you can use
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* Affine3d R = dq3.toAffine3();
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* DualQuatd dq4 = DualQuatd::createFromAffine3(R);
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*
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* // or create directly by affine transformation matrix Rt
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* // see createFromMat() in detail for the form of Rt
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* Matx44d Rt = dq3.toMat();
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* DualQuatd dq5 = DualQuatd::createFromMat(Rt);
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*
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* // Any rotation + translation movement can
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* // be expressed as a rotation + translation around the same line in space (expressed by Plucker
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* // coords), and here's a way to represent it this way.
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* Vec3d axis{1, 1, 1}; // axis will be normalized in createFromPitch
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* Vec3d trans{3, 4 ,5};
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* axis = axis / std::sqrt(axis.dot(axis));// The formula for computing moment that I use below requires a normalized axis
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* Vec3d moment = 1.0 / 2 * (trans.cross(axis) + axis.cross(trans.cross(axis)) *
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* std::cos(rotation_angle / 2) / std::sin(rotation_angle / 2));
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* double d = trans.dot(qaxis);
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* DualQuatd dq6 = DualQuatd::createFromPitch(angle, d, axis, moment);
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* ```
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*
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* A point \f$v=(x, y, z)\f$ in form of dual quaternion is \f$[1+\epsilon v]=[1,0,0,0,0,x,y,z]\f$.
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* The transformation of a point \f$v_1\f$ to another point \f$v_2\f$ under the dual quaternion \f$\sigma\f$ is
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* \f[
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* 1 + \epsilon v_2 = \sigma * (1 + \epsilon v_1) * \sigma^{\star}
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* \f]
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* where \f$\sigma^{\star}=p^*-\epsilon q^*.\f$
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*
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* A line in the \f$Pl\ddot{u}cker\f$ coordinates \f$(\hat{l}, m)\f$ defined by the dual quaternion \f$l=\hat{l}+\epsilon m\f$.
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* To transform a line, \f[l_2 = \sigma * l_1 * \sigma^*,\f] where \f$\sigma=r+\frac{\epsilon}{2}rt\f$ and
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* \f$\sigma^*=p^*+\epsilon q^*\f$.
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*
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* To extract the Vec<double, 8> or Vec<float, 8>, see toVec();
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*
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* To extract the affine transformation matrix, see toMat();
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*
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* To extract the instance of Affine3, see toAffine3();
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*
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* If two quaternions \f$q_0, q_1\f$ are needed to be interpolated, you can use sclerp()
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* ```
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* DualQuatd::sclerp(q0, q1, t)
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* ```
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* or dqblend().
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* ```
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* DualQuatd::dqblend(q0, q1, t)
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* ```
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* With more than two dual quaternions to be blended, you can use generalize linear dual quaternion blending
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* with the corresponding weights, i.e. gdqblend().
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*
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*/
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template <typename _Tp>
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class CV_EXPORTS DualQuat{
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static_assert(std::is_floating_point<_Tp>::value, "Dual quaternion only make sense with type of float or double");
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using value_type = _Tp;
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public:
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static constexpr _Tp CV_DUAL_QUAT_EPS = (_Tp)1.e-6;
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DualQuat();
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/**
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* @brief create from eight same type numbers.
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*/
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DualQuat(const _Tp w, const _Tp x, const _Tp y, const _Tp z, const _Tp w_, const _Tp x_, const _Tp y_, const _Tp z_);
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/**
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* @brief create from a double or float vector.
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*/
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DualQuat(const Vec<_Tp, 8> &q);
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_Tp w, x, y, z, w_, x_, y_, z_;
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/**
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* @brief create Dual Quaternion from two same type quaternions p and q.
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* A Dual Quaternion \f$\sigma\f$ has the form:
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* \f[\sigma = p + \epsilon q\f]
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* where p and q are defined as follows:
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* \f[\begin{equation}
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* \begin{split}
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* p &= w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}\\
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* q &= w\_ + x\_\boldsymbol{i} + y\_\boldsymbol{j} + z\_\boldsymbol{k}.
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* \end{split}
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* \end{equation}
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* \f]
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* The p and q are the real part and dual part respectively.
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* @param realPart a quaternion, real part of dual quaternion.
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* @param dualPart a quaternion, dual part of dual quaternion.
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* @sa Quat
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*/
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static DualQuat<_Tp> createFromQuat(const Quat<_Tp> &realPart, const Quat<_Tp> &dualPart);
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/**
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* @brief create a dual quaternion from a rotation angle \f$\theta\f$, a rotation axis
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* \f$\boldsymbol{u}\f$ and a translation \f$\boldsymbol{t}\f$.
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* It generates a dual quaternion \f$\sigma\f$ in the form of
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* \f[\begin{equation}
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* \begin{split}
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* \sigma &= r + \frac{\epsilon}{2}\boldsymbol{t}r \\
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* &= [\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})]
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* + \frac{\epsilon}{2}[0, \boldsymbol{t}][[\cos(\frac{\theta}{2}),
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* \boldsymbol{u}\sin(\frac{\theta}{2})]]\\
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* &= \cos(\frac{\theta}{2}) + \boldsymbol{u}\sin(\frac{\theta}{2})
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* + \frac{\epsilon}{2}(-(\boldsymbol{t} \cdot \boldsymbol{u})\sin(\frac{\theta}{2})
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* + \boldsymbol{t}\cos(\frac{\theta}{2}) + \boldsymbol{u} \times \boldsymbol{t} \sin(\frac{\theta}{2})).
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* \end{split}
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* \end{equation}\f]
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* @param angle rotation angle.
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* @param axis rotation axis.
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* @param translation a vector of length 3.
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* @note Axis will be normalized in this function. And translation is applied
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* after the rotation. Use @ref createFromQuat(r, r * t / 2) to create a dual quaternion
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* which translation is applied before rotation.
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* @sa Quat
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*/
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static DualQuat<_Tp> createFromAngleAxisTrans(const _Tp angle, const Vec<_Tp, 3> &axis, const Vec<_Tp, 3> &translation);
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/**
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* @brief Transform this dual quaternion to an affine transformation matrix \f$M\f$.
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* Dual quaternion consists of a rotation \f$r=[a,b,c,d]\f$ and a translation \f$t=[\Delta x,\Delta y,\Delta z]\f$. The
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* affine transformation matrix \f$M\f$ has the form
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* \f[
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* \begin{bmatrix}
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* 1-2(e_2^2 +e_3^2) &2(e_1e_2-e_0e_3) &2(e_0e_2+e_1e_3) &\Delta x\\
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* 2(e_0e_3+e_1e_2) &1-2(e_1^2+e_3^2) &2(e_2e_3-e_0e_1) &\Delta y\\
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* 2(e_1e_3-e_0e_2) &2(e_0e_1+e_2e_3) &1-2(e_1^2-e_2^2) &\Delta z\\
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* 0&0&0&1
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* \end{bmatrix}
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* \f]
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* if A is a matrix consisting of n points to be transformed, this could be achieved by
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* \f[
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* new\_A = M * A
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* \f]
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* where A has the form
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* \f[
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* \begin{bmatrix}
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* x_0& x_1& x_2&...&x_n\\
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* y_0& y_1& y_2&...&y_n\\
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* z_0& z_1& z_2&...&z_n\\
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* 1&1&1&...&1
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* \end{bmatrix}
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* \f]
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* where the same subscript represent the same point. The size of A should be \f$[4,n]\f$.
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* and the same size for matrix new_A.
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* @param _R 4x4 matrix that represents rotations and translation.
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* @note Translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create
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* a dual quaternion which translation is applied before rotation.
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*/
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static DualQuat<_Tp> createFromMat(InputArray _R);
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/**
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* @brief create dual quaternion from an affine matrix. The definition of affine matrix can refer to createFromMat()
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*/
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static DualQuat<_Tp> createFromAffine3(const Affine3<_Tp> &R);
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/**
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* @brief A dual quaternion is a vector in form of
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* \f[
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* \begin{equation}
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* \begin{split}
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* \sigma &=\boldsymbol{p} + \epsilon \boldsymbol{q}\\
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* &= \cos\hat{\frac{\theta}{2}}+\overline{\hat{l}}\sin\frac{\hat{\theta}}{2}
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* \end{split}
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* \end{equation}
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* \f]
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* where \f$\hat{\theta}\f$ is dual angle and \f$\overline{\hat{l}}\f$ is dual axis:
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* \f[
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* \hat{\theta}=\theta + \epsilon d,\\
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* \overline{\hat{l}}= \hat{l} +\epsilon m.
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* \f]
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* In this representation, \f$\theta\f$ is rotation angle and \f$(\hat{l},m)\f$ is the screw axis, d is the translation distance along the axis.
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*
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* @param angle rotation angle.
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* @param d translation along the rotation axis.
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* @param axis rotation axis represented by quaternion with w = 0.
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* @param moment the moment of line, and it should be orthogonal to axis.
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* @note Translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create
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* a dual quaternion which translation is applied before rotation.
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*/
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static DualQuat<_Tp> createFromPitch(const _Tp angle, const _Tp d, const Vec<_Tp, 3> &axis, const Vec<_Tp, 3> &moment);
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/**
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* @brief return a quaternion which represent the real part of dual quaternion.
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* The definition of real part is in createFromQuat().
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* @sa createFromQuat, getDualPart
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*/
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Quat<_Tp> getRealPart() const;
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/**
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* @brief return a quaternion which represent the dual part of dual quaternion.
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* The definition of dual part is in createFromQuat().
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* @sa createFromQuat, getRealPart
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*/
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Quat<_Tp> getDualPart() const;
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/**
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* @brief return the conjugate of a dual quaternion.
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* \f[
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* \begin{equation}
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* \begin{split}
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* \sigma^* &= (p + \epsilon q)^*
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* &= (p^* + \epsilon q^*)
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* \end{split}
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* \end{equation}
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* \f]
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* @param dq a dual quaternion.
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*/
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template <typename T>
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friend DualQuat<T> conjugate(const DualQuat<T> &dq);
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/**
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* @brief return the conjugate of a dual quaternion.
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* \f[
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* \begin{equation}
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* \begin{split}
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* \sigma^* &= (p + \epsilon q)^*
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* &= (p^* + \epsilon q^*)
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* \end{split}
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* \end{equation}
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* \f]
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*/
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DualQuat<_Tp> conjugate() const;
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/**
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* @brief return the rotation in quaternion form.
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*/
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Quat<_Tp> getRotation(QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
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/**
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* @brief return the translation vector.
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* The rotation \f$r\f$ in this dual quaternion \f$\sigma\f$ is applied before translation \f$t\f$.
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* The dual quaternion \f$\sigma\f$ is defined as
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* \f[\begin{equation}
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* \begin{split}
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* \sigma &= p + \epsilon q \\
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* &= r + \frac{\epsilon}{2}{t}r.
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* \end{split}
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* \end{equation}\f]
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* Thus, the translation can be obtained as follows
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* \f[t = 2qp^*.\f]
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* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion
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* and this function will save some computations.
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* @note This dual quaternion's translation is applied after the rotation.
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*/
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Vec<_Tp, 3> getTranslation(QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
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/**
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* @brief return the norm \f$||\sigma||\f$ of dual quaternion \f$\sigma = p + \epsilon q\f$.
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* \f[
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* \begin{equation}
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* \begin{split}
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* ||\sigma|| &= \sqrt{\sigma * \sigma^*} \\
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* &= ||p|| + \epsilon \frac{p \cdot q}{||p||}.
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* \end{split}
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* \end{equation}
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* \f]
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* Generally speaking, the norm of a not unit dual
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* quaternion is a dual number. For convenience, we return it in the form of a dual quaternion
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* , i.e.
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* \f[ ||\sigma|| = [||p||, 0, 0, 0, \frac{p \cdot q}{||p||}, 0, 0, 0].\f]
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*
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* @note The data type of dual number is dual quaternion.
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*/
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DualQuat<_Tp> norm() const;
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/**
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* @brief return a normalized dual quaternion.
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* A dual quaternion can be expressed as
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* \f[
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* \begin{equation}
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* \begin{split}
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* \sigma &= p + \epsilon q\\
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* &=||\sigma||\left(r+\frac{1}{2}tr\right)
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* \end{split}
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* \end{equation}
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* \f]
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* where \f$r, t\f$ represents the rotation (ordinary quaternion) and translation (pure ordinary quaternion) respectively,
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* and \f$||\sigma||\f$ is the norm of dual quaternion(a dual number).
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* A dual quaternion is unit if and only if
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* \f[
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* ||p||=1, p \cdot q=0
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* \f]
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* where \f$\cdot\f$ means dot product.
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* The process of normalization is
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* \f[
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* \sigma_{u}=\frac{\sigma}{||\sigma||}
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* \f]
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* Next, we simply proof \f$\sigma_u\f$ is a unit dual quaternion:
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* \f[
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* \renewcommand{\Im}{\operatorname{Im}}
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* \begin{equation}
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* \begin{split}
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* \sigma_{u}=\frac{\sigma}{||\sigma||}&=\frac{p + \epsilon q}{||p||+\epsilon\frac{p\cdot q}{||p||}}\\
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* &=\frac{p}{||p||}+\epsilon\left(\frac{q}{||p||}-p\frac{p\cdot q}{||p||^3}\right)\\
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* &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\left(qp^{*}-p\cdot q\right)\frac{p}{||p||}\\
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* &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\Im(qp^*)\frac{p}{||p||}.\\
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* \end{split}
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* \end{equation}
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* \f]
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* As expected, the real part is a rotation and dual part is a pure quaternion.
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*/
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DualQuat<_Tp> normalize() const;
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/**
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* @brief if \f$\sigma = p + \epsilon q\f$ is a dual quaternion, p is not zero,
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* the inverse dual quaternion is
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* \f[\sigma^{-1} = \frac{\sigma^*}{||\sigma||^2}, \f]
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* or equivalentlly,
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* \f[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\f]
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* @param dq a dual quaternion.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> inv(const DualQuat<T> &dq, QuatAssumeType assumeUnit);
|
|
|
|
/**
|
|
* @brief if \f$\sigma = p + \epsilon q\f$ is a dual quaternion, p is not zero,
|
|
* the inverse dual quaternion is
|
|
* \f[\sigma^{-1} = \frac{\sigma^*}{||\sigma||^2}, \f]
|
|
* or equivalentlly,
|
|
* \f[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\f]
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
DualQuat<_Tp> inv(QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
|
|
|
|
/**
|
|
* @brief return the dot product of two dual quaternion.
|
|
* @param p other dual quaternion.
|
|
*/
|
|
_Tp dot(DualQuat<_Tp> p) const;
|
|
|
|
/**
|
|
** @brief return the value of \f$p^t\f$ where p is a dual quaternion.
|
|
* This could be calculated as:
|
|
* \f[
|
|
* p^t = \exp(t\ln p)
|
|
* \f]
|
|
* @param dq a dual quaternion.
|
|
* @param t index of power function.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> power(const DualQuat<T> &dq, const T t, QuatAssumeType assumeUnit);
|
|
|
|
/**
|
|
** @brief return the value of \f$p^t\f$ where p is a dual quaternion.
|
|
* This could be calculated as:
|
|
* \f[
|
|
* p^t = \exp(t\ln p)
|
|
* \f]
|
|
*
|
|
* @param t index of power function.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
DualQuat<_Tp> power(const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
|
|
|
|
/**
|
|
* @brief return the value of \f$p^q\f$ where p and q are dual quaternions.
|
|
* This could be calculated as:
|
|
* \f[
|
|
* p^q = \exp(q\ln p)
|
|
* \f]
|
|
* @param p a dual quaternion.
|
|
* @param q a dual quaternion.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, dual quaternion p assume to be a dual unit quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> power(const DualQuat<T>& p, const DualQuat<T>& q, QuatAssumeType assumeUnit);
|
|
|
|
/**
|
|
* @brief return the value of \f$p^q\f$ where p and q are dual quaternions.
|
|
* This could be calculated as:
|
|
* \f[
|
|
* p^q = \exp(q\ln p)
|
|
* \f]
|
|
*
|
|
* @param q a dual quaternion
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a dual unit quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
DualQuat<_Tp> power(const DualQuat<_Tp>& q, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
|
|
|
|
/**
|
|
* @brief return the value of exponential function value
|
|
* @param dq a dual quaternion.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> exp(const DualQuat<T> &dq);
|
|
|
|
/**
|
|
* @brief return the value of exponential function value
|
|
*/
|
|
DualQuat<_Tp> exp() const;
|
|
|
|
/**
|
|
* @brief return the value of logarithm function value
|
|
*
|
|
* @param dq a dual quaternion.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> log(const DualQuat<T> &dq, QuatAssumeType assumeUnit);
|
|
|
|
/**
|
|
* @brief return the value of logarithm function value
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*/
|
|
DualQuat<_Tp> log(QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
|
|
|
|
/**
|
|
* @brief Transform this dual quaternion to a vector.
|
|
*/
|
|
Vec<_Tp, 8> toVec() const;
|
|
|
|
/**
|
|
* @brief Transform this dual quaternion to a affine transformation matrix
|
|
* the form of matrix, see createFromMat().
|
|
*/
|
|
Matx<_Tp, 4, 4> toMat(QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
|
|
|
|
/**
|
|
* @brief Transform this dual quaternion to a instance of Affine3.
|
|
*/
|
|
Affine3<_Tp> toAffine3(QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const;
|
|
|
|
/**
|
|
* @brief The screw linear interpolation(ScLERP) is an extension of spherical linear interpolation of dual quaternion.
|
|
* If \f$\sigma_1\f$ and \f$\sigma_2\f$ are two dual quaternions representing the initial and final pose.
|
|
* The interpolation of ScLERP function can be defined as:
|
|
* \f[
|
|
* ScLERP(t;\sigma_1,\sigma_2) = \sigma_1 * (\sigma_1^{-1} * \sigma_2)^t, t\in[0,1]
|
|
* \f]
|
|
*
|
|
* @param q1 a dual quaternion represents a initial pose.
|
|
* @param q2 a dual quaternion represents a final pose.
|
|
* @param t interpolation parameter
|
|
* @param directChange if true, it always return the shortest path.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*
|
|
* For example
|
|
* ```
|
|
* double angle1 = CV_PI / 2;
|
|
* Vec3d axis{0, 0, 1};
|
|
* Vec3d t(0, 0, 3);
|
|
* DualQuatd initial = DualQuatd::createFromAngleAxisTrans(angle1, axis, t);
|
|
* double angle2 = CV_PI;
|
|
* DualQuatd final = DualQuatd::createFromAngleAxisTrans(angle2, axis, t);
|
|
* DualQuatd inter = DualQuatd::sclerp(initial, final, 0.5);
|
|
* ```
|
|
*/
|
|
static DualQuat<_Tp> sclerp(const DualQuat<_Tp> &q1, const DualQuat<_Tp> &q2, const _Tp t,
|
|
bool directChange=true, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT);
|
|
/**
|
|
* @brief The method of Dual Quaternion linear Blending(DQB) is to compute a transformation between dual quaternion
|
|
* \f$q_1\f$ and \f$q_2\f$ and can be defined as:
|
|
* \f[
|
|
* DQB(t;{\boldsymbol{q}}_1,{\boldsymbol{q}}_2)=
|
|
* \frac{(1-t){\boldsymbol{q}}_1+t{\boldsymbol{q}}_2}{||(1-t){\boldsymbol{q}}_1+t{\boldsymbol{q}}_2||}.
|
|
* \f]
|
|
* where \f$q_1\f$ and \f$q_2\f$ are unit dual quaternions representing the input transformations.
|
|
* If you want to use DQB that works for more than two rigid transformations, see @ref gdqblend
|
|
*
|
|
* @param q1 a unit dual quaternion representing the input transformations.
|
|
* @param q2 a unit dual quaternion representing the input transformations.
|
|
* @param t parameter \f$t\in[0,1]\f$.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion
|
|
* and this function will save some computations.
|
|
*
|
|
* @sa gdqblend
|
|
*/
|
|
static DualQuat<_Tp> dqblend(const DualQuat<_Tp> &q1, const DualQuat<_Tp> &q2, const _Tp t,
|
|
QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT);
|
|
|
|
/**
|
|
* @brief The generalized Dual Quaternion linear Blending works for more than two rigid transformations.
|
|
* If these transformations are expressed as unit dual quaternions \f$q_1,...,q_n\f$ with convex weights
|
|
* \f$w = (w_1,...,w_n)\f$, the generalized DQB is simply
|
|
* \f[
|
|
* gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n}
|
|
* {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}.
|
|
* \f]
|
|
* @param dualquat vector of dual quaternions
|
|
* @param weights vector of weights, the size of weights should be the same as dualquat, and the weights should
|
|
* satisfy \f$\sum_0^n w_{i} = 1\f$ and \f$w_i>0\f$.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions
|
|
* and this function will save some computations.
|
|
* @note the type of weights' element should be the same as the date type of dual quaternion inside the dualquat.
|
|
*/
|
|
template <int cn>
|
|
static DualQuat<_Tp> gdqblend(const Vec<DualQuat<_Tp>, cn> &dualquat, InputArray weights,
|
|
QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT);
|
|
|
|
/**
|
|
* @brief The generalized Dual Quaternion linear Blending works for more than two rigid transformations.
|
|
* If these transformations are expressed as unit dual quaternions \f$q_1,...,q_n\f$ with convex weights
|
|
* \f$w = (w_1,...,w_n)\f$, the generalized DQB is simply
|
|
* \f[
|
|
* gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n}
|
|
* {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}.
|
|
* \f]
|
|
* @param dualquat The dual quaternions which have 8 channels and 1 row or 1 col.
|
|
* @param weights vector of weights, the size of weights should be the same as dualquat, and the weights should
|
|
* satisfy \f$\sum_0^n w_{i} = 1\f$ and \f$w_i>0\f$.
|
|
* @param assumeUnit if @ref QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions
|
|
* and this function will save some computations.
|
|
* @note the type of weights' element should be the same as the date type of dual quaternion inside the dualquat.
|
|
*/
|
|
static DualQuat<_Tp> gdqblend(InputArray dualquat, InputArray weights,
|
|
QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT);
|
|
|
|
/**
|
|
* @brief Return opposite dual quaternion \f$-p\f$
|
|
* which satisfies \f$p + (-p) = 0.\f$
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd q{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* std::cout << -q << std::endl; // [-1, -2, -3, -4, -5, -6, -7, -8]
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp> operator-() const;
|
|
|
|
/**
|
|
* @brief return true if two dual quaternions p and q are nearly equal, i.e. when the absolute
|
|
* value of each \f$p_i\f$ and \f$q_i\f$ is less than CV_DUAL_QUAT_EPS.
|
|
*/
|
|
bool operator==(const DualQuat<_Tp>&) const;
|
|
|
|
/**
|
|
* @brief Subtraction operator of two dual quaternions p and q.
|
|
* It returns a new dual quaternion that each value is the sum of \f$p_i\f$ and \f$-q_i\f$.
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* std::cout << p - q << std::endl; //[-4, -4, -4, -4, 4, -4, -4, -4]
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp> operator-(const DualQuat<_Tp>&) const;
|
|
|
|
/**
|
|
* @brief Subtraction assignment operator of two dual quaternions p and q.
|
|
* It subtracts right operand from the left operand and assign the result to left operand.
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* p -= q; // equivalent to p = p - q
|
|
* std::cout << p << std::endl; //[-4, -4, -4, -4, 4, -4, -4, -4]
|
|
*
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp>& operator-=(const DualQuat<_Tp>&);
|
|
|
|
/**
|
|
* @brief Addition operator of two dual quaternions p and q.
|
|
* It returns a new dual quaternion that each value is the sum of \f$p_i\f$ and \f$q_i\f$.
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* std::cout << p + q << std::endl; //[6, 8, 10, 12, 14, 16, 18, 20]
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp> operator+(const DualQuat<_Tp>&) const;
|
|
|
|
/**
|
|
* @brief Addition assignment operator of two dual quaternions p and q.
|
|
* It adds right operand to the left operand and assign the result to left operand.
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* p += q; // equivalent to p = p + q
|
|
* std::cout << p << std::endl; //[6, 8, 10, 12, 14, 16, 18, 20]
|
|
*
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp>& operator+=(const DualQuat<_Tp>&);
|
|
|
|
/**
|
|
* @brief Multiplication assignment operator of two quaternions.
|
|
* It multiplies right operand with the left operand and assign the result to left operand.
|
|
*
|
|
* Rule of dual quaternion multiplication:
|
|
* The dual quaternion can be written as an ordered pair of quaternions [A, B]. Thus
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p * q &= [A, B][C, D]\\
|
|
* &=[AC, AD + BC]
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* p *= q;
|
|
* std::cout << p << std::endl; //[-60, 12, 30, 24, -216, 80, 124, 120]
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp>& operator*=(const DualQuat<_Tp>&);
|
|
|
|
/**
|
|
* @brief Multiplication assignment operator of a quaternions and a scalar.
|
|
* It multiplies right operand with the left operand and assign the result to left operand.
|
|
*
|
|
* Rule of dual quaternion multiplication with a scalar:
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\
|
|
* &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s].
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* double s = 2.0;
|
|
* p *= s;
|
|
* std::cout << p << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]
|
|
* ```
|
|
* @note the type of scalar should be equal to the dual quaternion.
|
|
*/
|
|
DualQuat<_Tp> operator*=(const _Tp s);
|
|
|
|
|
|
/**
|
|
* @brief Multiplication operator of two dual quaternions q and p.
|
|
* Multiplies values on either side of the operator.
|
|
*
|
|
* Rule of dual quaternion multiplication:
|
|
* The dual quaternion can be written as an ordered pair of quaternions [A, B]. Thus
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p * q &= [A, B][C, D]\\
|
|
* &=[AC, AD + BC]
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* std::cout << p * q << std::endl; //[-60, 12, 30, 24, -216, 80, 124, 120]
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp> operator*(const DualQuat<_Tp>&) const;
|
|
|
|
/**
|
|
* @brief Division operator of a dual quaternions and a scalar.
|
|
* It divides left operand with the right operand and assign the result to left operand.
|
|
*
|
|
* Rule of dual quaternion division with a scalar:
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p / s &= [w, x, y, z, w\_, x\_, y\_, z\_] / s\\
|
|
* &=[w/s, x/s, y/s, z/s, w\_/s, x\_/s, y\_/s, z\_/s].
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* double s = 2.0;
|
|
* p /= s; // equivalent to p = p / s
|
|
* std::cout << p << std::endl; //[0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4]
|
|
* ```
|
|
* @note the type of scalar should be equal to this dual quaternion.
|
|
*/
|
|
DualQuat<_Tp> operator/(const _Tp s) const;
|
|
|
|
/**
|
|
* @brief Division operator of two dual quaternions p and q.
|
|
* Divides left hand operand by right hand operand.
|
|
*
|
|
* Rule of dual quaternion division with a dual quaternion:
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p / q &= p * q.inv()\\
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* std::cout << p / q << std::endl; // equivalent to p * q.inv()
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp> operator/(const DualQuat<_Tp>&) const;
|
|
|
|
/**
|
|
* @brief Division assignment operator of two dual quaternions p and q;
|
|
* It divides left operand with the right operand and assign the result to left operand.
|
|
*
|
|
* Rule of dual quaternion division with a quaternion:
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p / q&= p * q.inv()\\
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
|
|
* p /= q; // equivalent to p = p * q.inv()
|
|
* std::cout << p << std::endl;
|
|
* ```
|
|
*/
|
|
DualQuat<_Tp>& operator/=(const DualQuat<_Tp>&);
|
|
|
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/**
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* @brief Division assignment operator of a dual quaternions and a scalar.
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* It divides left operand with the right operand and assign the result to left operand.
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*
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* Rule of dual quaternion division with a scalar:
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* \f[
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* \begin{equation}
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* \begin{split}
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* p / s &= [w, x, y, z, w\_, x\_, y\_ ,z\_] / s\\
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* &=[w / s, x / s, y / s, z / s, w\_ / \space s, x\_ / \space s, y\_ / \space s, z\_ / \space s].
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* \end{split}
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* \end{equation}
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* \f]
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*
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* For example
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* ```
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* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
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* double s = 2.0;;
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* p /= s; // equivalent to p = p / s
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* std::cout << p << std::endl; //[0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0]
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* ```
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* @note the type of scalar should be equal to the dual quaternion.
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*/
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Quat<_Tp>& operator/=(const _Tp s);
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/**
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* @brief Addition operator of a scalar and a dual quaternions.
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* Adds right hand operand from left hand operand.
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*
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* For example
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* ```
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* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
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* double scalar = 2.0;
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* std::cout << scalar + p << std::endl; //[3.0, 2, 3, 4, 5, 6, 7, 8]
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* ```
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* @note the type of scalar should be equal to the dual quaternion.
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*/
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template <typename T>
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friend DualQuat<T> cv::operator+(const T s, const DualQuat<T>&);
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/**
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* @brief Addition operator of a dual quaternions and a scalar.
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* Adds right hand operand from left hand operand.
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*
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* For example
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* ```
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* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
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* double scalar = 2.0;
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* std::cout << p + scalar << std::endl; //[3.0, 2, 3, 4, 5, 6, 7, 8]
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* ```
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* @note the type of scalar should be equal to the dual quaternion.
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*/
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template <typename T>
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friend DualQuat<T> cv::operator+(const DualQuat<T>&, const T s);
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/**
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* @brief Multiplication operator of a scalar and a dual quaternions.
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* It multiplies right operand with the left operand and assign the result to left operand.
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*
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* Rule of dual quaternion multiplication with a scalar:
|
|
* \f[
|
|
* \begin{equation}
|
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* \begin{split}
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* p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\
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* &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s].
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* \end{split}
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* \end{equation}
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* \f]
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*
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* For example
|
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* ```
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* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
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* double s = 2.0;
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|
* std::cout << s * p << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]
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* ```
|
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* @note the type of scalar should be equal to the dual quaternion.
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|
*/
|
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template <typename T>
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friend DualQuat<T> cv::operator*(const T s, const DualQuat<T>&);
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|
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/**
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* @brief Subtraction operator of a dual quaternion and a scalar.
|
|
* Subtracts right hand operand from left hand operand.
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|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* double scalar = 2.0;
|
|
* std::cout << p - scalar << std::endl; //[-1, 2, 3, 4, 5, 6, 7, 8]
|
|
* ```
|
|
* @note the type of scalar should be equal to the dual quaternion.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> cv::operator-(const DualQuat<T>&, const T s);
|
|
|
|
/**
|
|
* @brief Subtraction operator of a scalar and a dual quaternions.
|
|
* Subtracts right hand operand from left hand operand.
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* double scalar = 2.0;
|
|
* std::cout << scalar - p << std::endl; //[1.0, -2, -3, -4, -5, -6, -7, -8]
|
|
* ```
|
|
* @note the type of scalar should be equal to the dual quaternion.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> cv::operator-(const T s, const DualQuat<T>&);
|
|
|
|
/**
|
|
* @brief Multiplication operator of a dual quaternions and a scalar.
|
|
* It multiplies right operand with the left operand and assign the result to left operand.
|
|
*
|
|
* Rule of dual quaternion multiplication with a scalar:
|
|
* \f[
|
|
* \begin{equation}
|
|
* \begin{split}
|
|
* p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\
|
|
* &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s].
|
|
* \end{split}
|
|
* \end{equation}
|
|
* \f]
|
|
*
|
|
* For example
|
|
* ```
|
|
* DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
|
|
* double s = 2.0;
|
|
* std::cout << p * s << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]
|
|
* ```
|
|
* @note the type of scalar should be equal to the dual quaternion.
|
|
*/
|
|
template <typename T>
|
|
friend DualQuat<T> cv::operator*(const DualQuat<T>&, const T s);
|
|
|
|
template <typename S>
|
|
friend std::ostream& cv::operator<<(std::ostream&, const DualQuat<S>&);
|
|
|
|
};
|
|
|
|
using DualQuatd = DualQuat<double>;
|
|
using DualQuatf = DualQuat<float>;
|
|
|
|
//! @} core
|
|
}//namespace
|
|
|
|
#include "dualquaternion.inl.hpp"
|
|
|
|
#endif /* OPENCV_CORE_QUATERNION_HPP */
|