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Screw axis

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A screw axis (helical axis or twist axis) is a line that is simultaneously the <a href="Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a> and the line along which <a href="Translation_(geometry)" title="Translation (geometry)">translation</a> of a body occurs. <a href="Chasles'_theorem_(kinematics)" title="Chasles' theorem (kinematics)">Chasles' theorem</a> shows that each <a href="Euclidean_group" title="Euclidean group">Euclidean displacement</a> in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

<a href="Pl%C3%BCcker_coordinates" title="Plücker coordinates">Plücker coordinates</a> are used to locate a screw axis in <a href="Cartesian_space" class="mw-redirect" title="Cartesian space">space</a>, and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position. The special case when the first vector is zero is interpreted as a pure translation in the direction of the second vector. A screw axis is associated with each pair of vectors in the algebra of screws, also known as <a href="Screw_theory" title="Screw theory">screw theory</a>.<a href="#cite_note-3">[3]</a>

The spatial movement of a body can be represented by a continuous set of displacements. Because each of these displacements has a screw axis, the movement has an associated ruled surface known as a screw surface. This surface is not the same as the axode, which is traced by the instantaneous screw axes of the movement of a body. The instantaneous screw axis, or 'instantaneous helical axis' (IHA), is the axis of the helicoidal field generated by the velocities of every point in a moving body.

When a spatial displacement specializes to a planar displacement, the screw axis becomes the displacement pole, and the instantaneous screw axis becomes the velocity pole, or <a href="Instantaneous_center_of_rotation" class="mw-redirect" title="Instantaneous center of rotation">instantaneous center of rotation</a>, also called an instant center. The term centro is also used for a velocity pole, and the locus of these points for a planar movement is called a <a href="Centrode" title="Centrode">centrode</a>.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The proof that a spatial displacement can be decomposed into a rotation around, and translation along, a line in space is attributed to <a href="Michel_Chasles" title="Michel Chasles">Michel Chasles</a> in 1830.<a href="#cite_note-5">[5]</a> Recently the work of Giulio Mozzi has been identified as presenting a similar result in 1763.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Screw_axis_symmetry">Screw axis symmetry</h2></div>

A screw displacement (also screw operation or rotary translation) is the composition of a rotation by an angle φ about an axis (called the screw axis) with a translation by a distance d along this axis. A positive rotation direction usually means one that corresponds to the translation direction by the <a href="Right-hand_rule" title="Right-hand rule">right-hand rule</a>. This means that if the rotation is clockwise, the displacement is away from the viewer. Except for φ = 180°, we have to distinguish a screw displacement from its <a href="Mirror_image" title="Mirror image">mirror image</a>. Unlike for rotations, a righthand and lefthand screw operation generate different groups.

The combination of a rotation about an axis and a translation in a direction perpendicular to that axis is a rotation about a parallel axis. However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that. Thus the effect of a rotation combined with any translation is a screw operation in the general sense, with as special cases a pure translation, a pure rotation and the identity. Together these are all the direct <a href="Euclidean_group#Overview_of_isometries_in_up_to_three_dimensions" title="Euclidean group">isometries in 3D</a>.

In <a href="Crystallography" title="Crystallography">crystallography</a>, a screw axis symmetry is a combination of rotation about an axis and a translation parallel to that axis which leaves a crystal unchanged. If φ = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠360°/n⁠ for some positive integer n, then screw axis symmetry implies <a href="Translational_symmetry" title="Translational symmetry">translational symmetry</a> with a translation vector which is n times that of the screw displacement.

Applicable for <a href="Space_group" title="Space group">space groups</a> is a rotation by ⁠360°/n⁠ about an axis, combined with a translation along the axis by a multiple of the distance of the translational symmetry, divided by n. This multiple is indicated by a subscript. So, 63 is a rotation of 60° combined with a translation of one half of the lattice vector, implying that there is also 3-fold <a href="Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> about this axis. The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the <a href="Enantiomorphous" class="mw-redirect" title="Enantiomorphous">enantiomorphous</a> 32, 43, 64, and 65.<a href="#cite_note-8">[8]</a>

Considering a screw axis nm, if g is the <a href="Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of n and m, then there is also a g-fold rotation axis. When ⁠n/g⁠ screw operations have been performed, the displacement will be ⁠m/g⁠, which since it is a whole number means one has moved to an equivalent point in the lattice, while carrying out a rotation by ⁠360°/g⁠. So 42, 62 and 64 create two-fold rotation axes, while 63 creates a three-fold axis.

A non-discrete screw axis <a href="Isometry_group" title="Isometry group">isometry group</a> contains all combinations of a rotation about some axis and a proportional translation along the axis (in <a href="Rifling" title="Rifling">rifling</a>, the constant of proportionality is called the <a href="Twist_rate" class="mw-redirect" title="Twist rate">twist rate</a>); in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold <a href="Helix" title="Helix">helix</a>; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.

<div class="mw-heading mw-heading2"><h2 id="Screw_axis_of_a_spatial_displacement">Screw axis of a spatial displacement</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Geometric_argument">Geometric argument</h3></div>

Let D : R3 → R3 be an orientation-preserving rigid motion of R3. The set of these transformations is a subgroup of <a href="Euclidean_motion" class="mw-redirect" title="Euclidean motion">Euclidean motions</a> known as the special Euclidean group SE(3). These rigid motions are defined by transformations of x in R3 given by

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<annotation encoding="application/x-tex">{\displaystyle D(\mathbf {x} )=A(\mathbf {x} )+\mathbf {d} }</annotation>

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</math><img src="./a02abe0b49bceeabe02feb547176a1a3638e2bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.532ex; height:2.843ex;" alt="{\displaystyle D(\mathbf {x} )=A(\mathbf {x} )+\mathbf {d} }" loading="lazy"></dd></dl>

consisting of a three-dimensional rotation A followed by a translation by the vector d.

A three-dimensional <a href="Rotation_matrix" title="Rotation matrix">rotation</a> A has a unique axis that defines a line L. Let the unit vector along this line be S so that the translation vector d can be resolved into a sum of two vectors, one parallel and one perpendicular to the axis L, that is,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} =\mathbf {d} _{L}+\mathbf {d} _{\perp },\quad \mathbf {d} _{L}=(\mathbf {d} \cdot \mathbf {S} )\mathbf {S} ,\quad \mathbf {d} _{\perp }=\mathbf {d} -\mathbf {d} _{L}.}</annotation>

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</math><img src="./1970cf1019373a9895f9c0330d77d8f42051f682.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.754ex; height:2.843ex;" alt="{\displaystyle \mathbf {d} =\mathbf {d} _{L}+\mathbf {d} _{\perp },\quad \mathbf {d} _{L}=(\mathbf {d} \cdot \mathbf {S} )\mathbf {S} ,\quad \mathbf {d} _{\perp }=\mathbf {d} -\mathbf {d} _{L}.}" loading="lazy"></dd></dl>

In this case, the rigid motion takes the form

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<annotation encoding="application/x-tex">{\displaystyle D(\mathbf {x} )=(A(\mathbf {x} )+\mathbf {d} _{\perp })+\mathbf {d} _{L}.}</annotation>

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</math><img src="./5dc47e8d564e04db340a60130c22a88f8ba662b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.176ex; height:2.843ex;" alt="{\displaystyle D(\mathbf {x} )=(A(\mathbf {x} )+\mathbf {d} _{\perp })+\mathbf {d} _{L}.}" loading="lazy"></dd></dl>

Now, the orientation preserving rigid motion D* = A(x) + d⊥ transforms all the points of R3 so that they remain in planes perpendicular to L. For a rigid motion of this type there is a unique point c in the plane P perpendicular to L through 0, such that

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<annotation encoding="application/x-tex">{\displaystyle D^{*}(\mathbf {C} )=A(\mathbf {C} )+\mathbf {d} _{\perp }=\mathbf {C} .}</annotation>

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</math><img src="./619af15d6115797b4dc92dcdb93d5109e1ac8628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.814ex; height:2.843ex;" alt="{\displaystyle D^{*}(\mathbf {C} )=A(\mathbf {C} )+\mathbf {d} _{\perp }=\mathbf {C} .}" loading="lazy"></dd></dl>

The point C can be calculated as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =[I-A]^{-1}\mathbf {d} _{\perp },}</annotation>

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</math><img src="./a33d32e8b0d70ad2d4b0f88334a304ac9ede96f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.054ex; height:3.176ex;" alt="{\displaystyle \mathbf {C} =[I-A]^{-1}\mathbf {d} _{\perp },}" loading="lazy"></dd></dl>

because d⊥ does not have a component in the direction of the axis of A.

A rigid motion D* with a fixed point must be a rotation of around the axis Lc through the point c. Therefore, the rigid motion

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<annotation encoding="application/x-tex">{\displaystyle D(\mathbf {x} )=D^{*}(\mathbf {x} )+\mathbf {d} _{L},}</annotation>

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</math><img src="./76b10c5fc9c7594cfcee8b9dd422540a5fc086ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.766ex; height:2.843ex;" alt="{\displaystyle D(\mathbf {x} )=D^{*}(\mathbf {x} )+\mathbf {d} _{L},}" loading="lazy"></dd></dl>

consists of a rotation about the line Lc followed by a translation by the vector dL in the direction of the line Lc.

Conclusion: every rigid motion of R3 is the result of a rotation of R3 about a line Lc followed by a translation in the direction of the line. The combination of a rotation about a line and translation along the line is called a screw motion.

<div class="mw-heading mw-heading3"><h3 id="Computing_a_point_on_the_screw_axis">Computing a point on the screw axis</h3></div>

A point C on the screw axis satisfies the equation:<a href="#cite_note-McCarthy-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle D^{*}(\mathbf {C} )=A(\mathbf {C} )+\mathbf {d} _{\perp }=\mathbf {C} .}</annotation>

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</math><img src="./619af15d6115797b4dc92dcdb93d5109e1ac8628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.814ex; height:2.843ex;" alt="{\displaystyle D^{*}(\mathbf {C} )=A(\mathbf {C} )+\mathbf {d} _{\perp }=\mathbf {C} .}" loading="lazy"></dd></dl>

Solve this equation for C using <a href="Rotation_matrix#Skew_parameters_via_Cayley's_formula" title="Rotation matrix">Cayley's formula</a> for a rotation matrix

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<annotation encoding="application/x-tex">{\displaystyle [A]=[I-B]^{-1}[I+B],}</annotation>

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</math><img src="./8f1fc95267a00ede6614b99b00fc650a081ee604.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.255ex; height:3.176ex;" alt="{\displaystyle [A]=[I-B]^{-1}[I+B],}" loading="lazy"></dd></dl>

where [B] is the skew-symmetric matrix constructed from <a href="Rotation_formalisms_in_three_dimensions#Rodrigues_parameters_and_Gibbs_representation" title="Rotation formalisms in three dimensions">Rodrigues' vector</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\tan {\frac {\phi }{2}}\mathbf {S} ,}</annotation>

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</math><img src="./1434812f9d6ee93083330222109c8ac52b34f652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.684ex; height:5.343ex;" alt="{\displaystyle \mathbf {b} =\tan {\frac {\phi }{2}}\mathbf {S} ,}" loading="lazy"></dd></dl>

such that

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<annotation encoding="application/x-tex">{\displaystyle [B]\mathbf {y} =\mathbf {b} \times \mathbf {y} .}</annotation>

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</math><img src="./c59017f8f8d1d59331b37d816b249816d8b115ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.951ex; height:2.843ex;" alt="{\displaystyle [B]\mathbf {y} =\mathbf {b} \times \mathbf {y} .}" loading="lazy"></dd></dl>

Use this form of the rotation A to obtain

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} =[I-B]^{-1}[I+B]\mathbf {C} +\mathbf {d} _{\perp },\quad [I-B]\mathbf {C} =[I+B]\mathbf {C} +[I-B]\mathbf {d} _{\perp },}</annotation>

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</math><img src="./2d9c22a02bf4d9b0e0ae4d9d5f774faa55b4e2ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.28ex; height:3.176ex;" alt="{\displaystyle \mathbf {C} =[I-B]^{-1}[I+B]\mathbf {C} +\mathbf {d} _{\perp },\quad [I-B]\mathbf {C} =[I+B]\mathbf {C} +[I-B]\mathbf {d} _{\perp },}" loading="lazy"></dd></dl>

which becomes

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<annotation encoding="application/x-tex">{\displaystyle -2[B]\mathbf {C} =[I-B]\mathbf {d} _{\perp }.}</annotation>

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</math><img src="./0f591373bd9a7fedb9f7d33edc8c386f8aedc21a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.771ex; height:2.843ex;" alt="{\displaystyle -2[B]\mathbf {C} =[I-B]\mathbf {d} _{\perp }.}" loading="lazy"></dd></dl>

This equation can be solved for C on the screw axis P(t) to obtain,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ={\frac {\mathbf {b} \times \mathbf {d} -\mathbf {b} \times (\mathbf {b} \times \mathbf {d} )}{2\mathbf {b} \cdot \mathbf {b} }}.}</annotation>

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</math><img src="./17908ead1c19b1f4767da9530d623f4eb0f7658b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.11ex; height:5.843ex;" alt="{\displaystyle \mathbf {C} ={\frac {\mathbf {b} \times \mathbf {d} -\mathbf {b} \times (\mathbf {b} \times \mathbf {d} )}{2\mathbf {b} \cdot \mathbf {b} }}.}" loading="lazy"></dd></dl>

The screw axis P(t) = C + tS of this spatial displacement has the <a href="Pl%C3%BCcker_coordinates" title="Plücker coordinates">Plücker coordinates</a> S = (S, C × S).<a href="#cite_note-McCarthy-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Dual_quaternion">Dual quaternion</h2></div>

The screw axis appears in the <a href="Dual_quaternion" title="Dual quaternion">dual quaternion</a> formulation of a spatial displacement D = ([A], d). The dual quaternion is constructed from the <a href="Screw_theory" title="Screw theory">dual vector</a> S = (S, V) defining the screw axis and the dual angle (φ, d), where φ is the rotation about and d the slide along this axis, which defines the displacement D to obtain,

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<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}=\cos {\frac {\hat {\varphi }}{2}}+\sin {\frac {\hat {\varphi }}{2}}{\mathsf {S}}.}</annotation>

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</math><img src="./6860a9c3380c2775c85f18c4f6b7a8074aa43f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.934ex; height:5.509ex;" alt="{\displaystyle {\hat {S}}=\cos {\frac {\hat {\varphi }}{2}}+\sin {\frac {\hat {\varphi }}{2}}{\mathsf {S}}.}" loading="lazy"></dd></dl>

A spatial displacement of points q represented as a vector quaternion can be defined using <a href="Quaternion" title="Quaternion">quaternions</a> as the mapping

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} \mapsto S\mathbf {q} S^{-1}+\mathbf {d} }</annotation>

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</math><img src="./ca818e22fd043aa7244eaa9b1c2cf26134a54da6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.125ex; height:3.009ex;" alt="{\displaystyle \mathbf {q} \mapsto S\mathbf {q} S^{-1}+\mathbf {d} }" loading="lazy"></dd></dl>

where d is translation vector quaternion and S is a unit quaternion, also called a <a href="Versor" title="Versor">versor</a>, given by

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<annotation encoding="application/x-tex">{\displaystyle S=\cos \theta +\mathbf {S} \sin \theta ,\ \ \mathbf {S} ^{2}=-1,}</annotation>

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</math><img src="./25797d8c47ce1f2a63bc92b21d312ccc98201c8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.683ex; height:3.009ex;" alt="{\displaystyle S=\cos \theta +\mathbf {S} \sin \theta ,\ \ \mathbf {S} ^{2}=-1,}" loading="lazy"></dd></dl>

that defines a rotation by 2θ around an axis S.

In the proper <a href="Euclidean_group" title="Euclidean group">Euclidean group</a> E+(3) a rotation may be <a href="Conjugacy_class" title="Conjugacy class">conjugated</a> with a translation to move it to a parallel rotation axis. Such a conjugation, using <a href="Quaternion_analysis" class="mw-redirect" title="Quaternion analysis">quaternion homographies</a>, produces the appropriate screw axis to express the given spatial displacement as a screw displacement, in accord with <a href="Chasles'_theorem_(kinematics)" title="Chasles' theorem (kinematics)">Chasles’ theorem</a>.

<div class="mw-heading mw-heading2"><h2 id="Mechanics">Mechanics</h2></div>

The instantaneous motion of a <a href="Rigid_body" title="Rigid body">rigid body</a> may be the combination of rotation about an axis (the screw axis) and a translation along that axis. This screw move is characterized by the velocity vector for the translation and the <a href="Angular_velocity" title="Angular velocity">angular velocity</a> vector in the same or opposite direction. If these two vectors are constant and along one of the <a href="Moment_of_inertia#Inertia_tensor" title="Moment of inertia">principal axes</a> of the body, no external forces are needed for this motion (moving and spinning]]). As an example, if gravity and drag are ignored, this is the motion of a <a href="Bullet" title="Bullet">bullet</a> fired from a <a href="Rifling" title="Rifling">rifled</a> <a href="Gun" title="Gun">gun</a>.

<div class="mw-heading mw-heading3"><h3 id="Biomechanics">Biomechanics</h3></div>

This parameter is often used in <a href="Biomechanics" title="Biomechanics">biomechanics</a>, when describing the motion of <a href="Joint" title="Joint">joints</a> of the body. For any period of time, joint motion can be seen as the movement of a single point on one articulating surface with respect to the adjacent surface (usually <a href="Anatomical_terms_of_location#Proximal_and_distal" title="Anatomical terms of location">distal</a> with respect to <a href="Anatomical_terms_of_location#Proximal_and_distal" title="Anatomical terms of location">proximal</a>). The total translation and rotations along the path of motion can be defined as the time integrals of the instantaneous translation and rotation velocities at the IHA for a given reference time.<a href="#cite_note-10">[10]</a>

In any single <a href="Plane_(mathematics)" title="Plane (mathematics)">plane</a>, the path formed by the locations of the moving instantaneous axis of rotation (IAR) is known as the 'centroid', and is used in the description of joint motion.

<ul><li><a href="Corkscrew_(roller_coaster_element)" class="mw-redirect" title="Corkscrew (roller coaster element)">Corkscrew (roller coaster element)</a></li>

<li><a href="Euler's_rotation_theorem" title="Euler's rotation theorem">Euler's rotation theorem</a> – rotations without translation</li>

<li><a href="Glide_reflection" title="Glide reflection">Glide reflection</a></li>

<li><a href="Helical_symmetry" class="mw-redirect" title="Helical symmetry">Helical symmetry</a></li>

<li><a href="Line_group" title="Line group">Line group</a></li>

<li><a href="Screw_theory" title="Screw theory">Screw theory</a></li>

<li><a href="Space_group" title="Space group">Space group</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Bottema, O., and B. Roth, Theoretical Kinematics, Dover Publications (September 1990), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f8I4yGVi9ocC">link to Google books</a>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> Hunt, K. H., Kinematic Geometry of Mechanism, Oxford University Press, 1990

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> R.S. Ball, A Treatise on the Theory of Screws, Hodges, Dublin, 1876, Appendix 1, University Press, Cambridge, 1900, p. 510

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> Homer D. Eckhardt, Kinematic Design of Machines and Mechanisms, McGraw-Hill (1998) p. 63 <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-018953-6</bdi> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GGBv_Gcy5g8C">on-line at Google books</a>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> M. Chasles, Note sur les Propriétés Generales du Système de Deux Corps Semblables entr'eux, Bullettin de Sciences Mathématiques, Astronomiques Physiques et Chimiques, Baron de Ferussac, Paris, 1830, pp. 321±326

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> G. Mozzi, Discorso matematico sopra il rotamento momentaneo dei corpi, Stamperia di Donato Campo, Naples, 1763

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> M. Ceccarelli, Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion, Mechanism and Machine Theory 35 (2000) 761-770

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFWalter_Borchardt-Ott1995" class="citation book cs1">Walter Borchardt-Ott (1995). Crystallography. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-59478-7</bdi>.</cite>

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<li id="cite_note-McCarthy-9">^ <a href="#cite_ref-McCarthy_9-0">a</a> <a href="#cite_ref-McCarthy_9-1">b</a> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jv9mQyjRIw4C&q=geometric+design+of+linkages">J. M. McCarthy and G. S. Soh, Geometric Design of Linkages, 2nd Edition, Springer 2010</a>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> Woltring HJ, de Lange A, Kauer JMG, Huiskes R. 1987 Instantaneous helical axes estimation via natural, cross-validated splines. In: Bergmann G, Kölbel R, Rohlmann A (Editors). Biomechanics: Basic and Applied Research. Springer, pp 121-128. <a rel="nofollow" class="external text" href="http://library.tue.nl/csp/dare/LinkToRepository.csp?recordnumber=587036">full text</a>

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