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| | | Before a person decides to invest in stocks, they need to decide if their goals are short-term or [http://en.wiktionary.org/wiki/long-term long-term]. They also need to know how much money they can afford to invest and what kind of investments that they can choose from. Account fees and commissions on active trading should also be considered. If you do not what to pay a lot of fees, do not sell and buy a lot of stock. To avoid extra fees invest in some solid performing stocks and hold them because many brokers earn their money from the transaction fees.<br><br>As said before, one way to trade cotton is to trade cotton futures that get traded on the various futures exchanges. Now, futures trading is somewhat different than stock trading. Futures contracts get marked to the market every day. What this means is that if your positions get worse, you can get a margin call from your broker to either close the position or put more funds in the account. Futures market is highly volatile and an inexperienced trader can get wiped out in matter of minutes.<br><br><br><br>However, when trading futures, you get the [http://www.alexa.com/search?q=benefit&r=topsites_index&p=bigtop benefit] of using gearing or what we call leverage as high as 10:1 as compared to 2:1 in the stock market. This means you can trade cotton futures with a much lower deposit in your trading account as compared to trading stocks.<br><br>Obviously there are exceptions. Most investors are not professionals. However, if you know the market like the palm of your hand, go on and keep investing in stocks. Warren Buffett has already passed the age at which normal people would be advised to stay away from equities but still remains firm and strong in investing. But Buffett knows what he's doing, he knows how to invest in the stock market.<br><br>You see after the tech boom, the market fell apart - the Russell 2000 and Dow alike. But when it recovered in 2003, the Russell spiked up much faster and much higher than the Dow. That's what [http://news.goldgrey.org/category/silver/ http://news.goldgrey.org/category/silver/] small caps do. If you can time when those spikes are about to occur, you can make big money. That's the real point of staring at indexes day in and day out.<br><br>Creating cash within the stock market is one of the most satisfying feelings you can get. With the correct method and attitude, you can find yourself making a substantial income from trading! |
| In [[kinematics]], the motion of a [[rigid body]] is defined as a continuous set of displacements. One-parameter motions can be defined
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| as a continuous displacement of moving object with respect to a fixed frame in Euclidean three-space (''E''<sup>3</sup>), where the displacement depends on one parameter, mostly identified as time.
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| '''Rational motions''' are defined by [[rational function]]s (ratio of two [[polynomial function]]s) of time. They produce rational [[trajectories]], and therefore they integrate well with the existing [[NURBS]] (Non-Uniform Rational B-Spline) based industry standard [[CAD/CAM]] systems. They are readily amenable to the applications of existing [[Computational geometry|computer-aided geometric design]] (CAGD) algorithms. By combining kinematics of rigid body motions with NURBS geometry of [[curves]] and [[surfaces]], methods have been developed for [[computer-aided design]] of rational motions.
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| These CAD methods for motion design find applications in [[animation]] in computer graphics (key frame [[interpolation]]), trajectory planning in [[robotics]] (taught-position interpolation), spatial navigation in [[virtual reality]], computer-aided geometric design of motion via interactive interpolation, [[CNC]] [[tool path planning]], and task specification in [[mechanism synthesis]].
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| ==Background==
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| There has been a great deal of research in applying the principles of computer-aided geometric design (CAGD) to the problem of computer-aided motion design.
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| In recent years, it has been well established that [[Bézier curve|rational Bézier]] and [[Nonuniform rational B-spline|rational B-spline]] based curve representation schemes can be combined with [[dual quaternion]] representation <ref name=McCarthy1990>{{Cite document
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| | author = McCarthy, J. M.
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| | year = 1990
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| | publisher = MIT Press Cambridge, MA, USA
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| | postscript = <!--None-->
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| }}</ref> of [[spatial displacements]] to obtain rational Bézier and B-spline
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| motions. Ge and Ravani,<ref name=Ge1994a>{{cite journal
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| | author = Ge, Q. J.; Ravani, B.
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| | year = 1994
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| | title = Computer-Aided Geometric Design of Motion Interpolants
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| | journal = Journal of mechanical design(1990)
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| | volume = 116
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| | issue = 3
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| | pages = 756–762
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| | doi = 10.1115/1.2919447
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| | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMDEDB000116000003000756000001&idtype=cvips&gifs=yes
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| }}</ref><ref name=Ge1994b>{{cite journal
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| | author = Ge, Q. J.; Ravani, B.
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| | year = 1994
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| | title = Geometric Construction of Bézier Motions
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| | journal = Journal of mechanical design(1990)
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| | volume = 116
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| | issue = 3
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| | pages = 749–755
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| | doi = 10.1115/1.2919446
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| | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMDEDB000116000003000749000001&idtype=cvips&gifs=yes
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| }}</ref> developed a new framework for geometric constructions
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| of spatial motions by combining the concepts from kinematics and CAGD. Their work was built upon the seminal paper of Shoemake,<ref name=Shoemake1985>{{cite journal
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| | author = Shoemake, K.
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| | year = 1985
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| | title = Animating rotation with quaternion curves
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| | journal = Proceedings of the 12th annual conference on Computer graphics and interactive techniques
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| | pages = 245–254
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| | doi = 10.1145/325334.325242
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| | url = http://portal.acm.org/citation.cfm?id=325242
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| }}</ref> in which he
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| used the concept of a [[quaternion]] <ref name=Bottema1990>
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| {{Cite book
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| | author = Bottema, O.; Roth, B.
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| | year = 1990
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| | url = http://books.google.com/?id=f8I4yGVi9ocC
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| | publisher = [[Dover Publications]]
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| | isbn =0-486-66346-9
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| | title = Theoretical kinematics
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| | type= Theoretical kinematics
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| }}</ref> for [[rotation]] interpolation. A detailed list of references on this topic can be found in <ref name=Roschel1998>{{cite journal
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| | author = Röschel, O.
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| | year = 1998
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| | title = Rational motion design—a survey
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| | journal = Computer-Aided Design
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| | volume = 30
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| | issue = 3
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| | pages = 169–178
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| | doi = 10.1016/S0010-4485(97)00056-0
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| | url = http://www.ingentaconnect.com/content/els/00104485/1998/00000030/00000003/art00056
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| }}</ref> and.<ref name=Purwar2005>{{cite journal
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| | author = Purwar, A.; Ge, Q. J.
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| | year = 2005
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| | title = On the effect of dual weights in computer-aided design of rational motions
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| | journal = ASME Journal of Mechanical Design
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| | volume = 127
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| | issue = 5
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| | pages = 967–972
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| | doi = 10.1115/1.1906263
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| | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMDEDB000127000005000967000001&idtype=cvips&gifs=yes88
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| }}</ref>
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| ==Rational Bézier and B-spline motions==
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| Let <math>\hat {\textbf{q}} = \textbf{q} + \varepsilon \textbf{q}^0</math>
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| denote a unit dual quaternion. A homogeneous dual quaternion may be
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| written as a pair of quaternions, <math>\hat {\textbf{Q}}= \textbf{Q} +
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| \varepsilon \textbf{Q}^0</math>; where <math>\textbf{Q} = w\textbf{q},
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| \textbf{Q}^0 = w\textbf{q}^0 + w^0\textbf{q}</math>. This is obtained by
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| expanding <math>\hat {\textbf{Q}} = \hat {w} \hat {\textbf{q}}</math> using
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| [[dual number]] algebra (here, <math>\hat{w}=w+\varepsilon w^0</math>).
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| In terms of dual quaternions and the [[homogeneous coordinates]] of a point <math>\textbf{P}:(P_1, P_2, P_3, P_4)</math> of the object, the transformation equation in terms of quaternions is given by (see <ref name=Purwar2005>{{Who|date=July 2008}}</ref> for details)
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| <math>
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| \tilde {\textbf{P}} = \textbf{Q}\textbf{P}\textbf{Q}^\ast + P_4 [(\textbf{Q}^0)\textbf{Q}^\ast - \textbf{Q}(\textbf{Q}^0)^\ast], </math>
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| where <math>\textbf{Q}^\ast</math> and <math>(\textbf{Q}^0)^\ast</math> are
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| conjugates of <math>\textbf{Q}</math> and <math>\textbf{Q}^0</math>, respectively and
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| <math>\tilde {\textbf{P}}</math> denotes homogeneous coordinates of the point
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| after the displacement.
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| Given a set of unit dual quaternions and dual weights <math>\hat
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| {\textbf{q}}_i, \hat {w}_i; i = 0...n</math> respectively, the
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| following represents a rational Bézier curve in the space of
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| dual quaternions.
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| <math> \hat{\textbf{Q}}(t) = \sum\limits_{i = 0}^n {B_i^n (t)\hat {\textbf{Q}}_i} =
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| \sum\limits_{i = 0}^n {B_i^n (t)\hat {w}_i \hat{\textbf{q}}_i}
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| </math> | |
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| where <math>B_i^n(t)</math> are the Bernstein polynomials. The Bézier dual quaternion curve given by above equation defines a rational Bézier motion of
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| degree <math>2n</math>.
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| Similarly, a B-spline dual quaternion curve, which defines a NURBS
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| motion of degree 2''p'', is given by,
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| : <math> \hat {\textbf{Q}}(t) =
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| \sum\limits_{i = 0}^n {N_{i,p}(t) \hat {\textbf{Q}}_i } =
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| \sum\limits_{i = 0}^n {N_{i,p}(t) \hat {w}_i \hat {\textbf{q}}_i }
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| </math>
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| where <math>N_{i,p}(t)</math> are the ''p''th-degree B-spline basis functions.
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| A representation for the rational Bézier motion and rational B-spline motion in the Cartesian space can be obtained by substituting either of the above two preceding expressions for <math> \hat {\textbf{Q}}(t)</math> in the equation for point transform. In what follows, we deal with the case of rational Bézier motion. The, the trajectory of a point undergoing rational Bézier motion is given by,
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| : <math> \tilde {\textbf{P}}^{2n}(t) =
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| [H^{2n}(t)]\textbf{P}, </math> | |
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| : <math> H^{2n}(t)] = \sum\limits_{k = 0}^{2n} | |
| {B_k^{2n}(t)[H_k]}, </math>
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| where <math>[H^{2n}(t)]</math> is the matrix
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| representation of the rational Bézier motion of degree
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| <math>2n</math> in Cartesian space. The following matrices
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| <math>[H_k ]</math> (also referred to as Bézier Control
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| Matrices) define the ''affine control structure'' of the motion:
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| : <math> [H_k] = \frac{1}{C_k^{2n}}
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| \sum\limits_{i+j=k}{C_i^n C_j^n w_i w_j [H_{ij}^\ast]}, </math>
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| where <math>[H_{ij}^\ast] = [H_i^+][H_j^-] +
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| [H_j^-][H_i^{0+}] - [H_i^+][H_j^{0-} ] + (\alpha_i - \alpha_j
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| )[H_j^-][Q_i^+]</math>.
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| In the above equations, <math>C_i^n</math> and <math>C_j^n</math>
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| are binomial coefficients and <math>\alpha_i = w_i^0/w_i, \alpha_j =
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| w_j^0/w_j</math> are the weight ratios and
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| : <math> [H_j^-] = \left[ \begin{array}{rrrr}
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| q_{j,4} & -q_{j,3} & q_{j,2} & -q_{j,1} \\
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| q_{j,3} & q_{j,4} & -q_{j,1} & -q_{j,2} \\
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| -q_{j,2} & q_{j,1} & q_{j,4} & -q_{j,3} \\
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| q_{j,1} & q_{j,2} & q_{j,3} & q_{j,4} \\
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| \end{array} \right],
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| </math>
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| : <math> [Q_i^+] = \left[ \begin{array}{rrrr}
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| 0 & 0 & 0 & q_{i,1} \\
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| 0 & 0 & 0 & q_{i,2} \\
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| 0 & 0 & 0 & q_{i,3} \\
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| 0 & 0 & 0 & q_{i,4} \\
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| \end{array} \right],
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| </math>
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| : <math> [H_i^{0+}] = \left[
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| \begin{array}{rrrr}
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| 0 & 0 & 0 & q_{i,1}^0 \\
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| 0 & 0 & 0 & q_{i,2}^0 \\
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| 0 & 0 & 0 & q_{i,3}^0 \\
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| 0 & 0 & 0 & q_{i,4}^0 \\
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| \end{array} \right],
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| </math>
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| : <math> [H_j^{0-}] = \left[
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| \begin{array}{rrrr}
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| 0 & 0 & 0 & -q_{j,1}^0 \\
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| 0 & 0 & 0 & -q_{j,2}^0 \\
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| 0 & 0 & 0 & -q_{j,3}^0 \\
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| 0 & 0 & 0 & q_{j,4}^0 \\
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| \end{array} \right],
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| </math>
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| : <math> [H_i^+] = \left[ \begin{array}{rrrr}
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| q_{i,4} & -q_{i,3} & q_{i,2} & q_{i,1} \\
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| q_{i,3} & q_{i,4} & -q_{i,1} & q_{i,2} \\
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| -q_{i,2} & q_{i,1} & q_{i,4} & q_{i,3} \\
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| -q_{i,1} & -q_{i,2} & -q_{i,3} & q_{i,4} \\
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| \end{array} \right].
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| </math>
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| In above matrices, <math>(q_{i,1}, q_{i,2}, q_{i,3}, q_{i,4})</math>
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| are four components of the real part <math>(\textbf{q}_i)</math> and
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| <math>(q_{i,1}^0, q_{i,2}^0, q_{i,3}^0, q_{i,4}^0)</math> are four | |
| components of the dual part<math>(\textbf{q}_i^0)</math> of the unit
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| dual quaternion <math>(\hat {\textbf{q}}_i)</math>.
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| ==Example==
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| [[File:Rational Bezier motion of degree-6.jpg|thumb|600px|none|A teapot under Rational Bézier motion of degree 6 with (on the left) unit real weights (<math>\hat{w}_i = 1 + \epsilon 0; i = 0..3</math>) (on the right) non-unit real weights (<math>\hat{w}_i = 1 + \epsilon 0; i = 0,3</math> and <math>\hat{w}_i = 4 + \epsilon 0; i = 1,2</math>); also shown are affine positions (distorted) as well as the given control positions (in blue color).]]
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| *[http://cadcam.eng.sunysb.edu/ Computational Design Kinematics Lab]
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| *[http://my.fit.edu/~pierrel/ Robotics and Spatial Systems Laboratory (RASSL)]
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| *[http://synthetica.eng.uci.edu:16080/~mccarthy/ Robotics and Automation Laboratory]
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| ==See also==
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| * [[Quaternion]] and [[Dual quaternion]]
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| * [[NURBS]]
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| * [[Computer animation]]
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| * [[Robotics]]
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| * [[Robot kinematics]]
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| * [[Computational geometry]]
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| * [[CNC]] machining
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| * [[Mechanism design]]
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| [[Category:Kinematics]]
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