Ingeometry, anantiparallelogram is a type ofself-crossingquadrilateral. Like aparallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in generalparallel. Instead, each pair of sides isantiparallel with respect to the other, with sides in the longer pair crossing each other as in ascissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also calledcontraparallelograms[1] orcrossed parallelograms.[2]
Antiparallelograms occur as thevertex figures of certainnonconvex uniform polyhedra. In the theory offour-bar linkages, the linkages with the form of an antiparallelogram are also calledbutterfly linkages orbow-tie linkages, and are used in the design ofnon-circular gears. Incelestial mechanics, they occur in certain families of solutions to the4-body problem.
Every antiparallelogram has anaxis of symmetry, with all four vertices on a circle. It can be formed from anisosceles trapezoid by adding the two diagonals and removing two parallel sides. Thesigned area of every antiparallelogram is zero.
An antiparallelogram is a special case of acrossed quadrilateral, with two pairs of equal-length edges.[3] In general, crossed quadrilaterals can have unequal edges.[3] Special forms of the antiparallelogram are thecrossed rectangles and crossed squares, obtained by replacing two opposite sides of a rectangle or square by the two diagonals. In these shapes, the twouncrossed sides are parallel and equal.[4] Every antiparallelogram is acyclic quadrilateral, meaning that its four vertices all lie on a singlecircle.[3] Additionally, the four extended sides of any antiparallelogram are thebitangents of two circles, making antiparallelograms closely related to thetangential quadrilaterals,ex-tangential quadrilaterals, andkites (which are both tangential and ex-tangential).[5]
Every antiparallelogram has anaxis of symmetry through its crossing point. Because of this symmetry, it has two pairs of equal angles and two pairs of equal sides.[2] The fourmidpoints of its sides lie on a line perpendicular to the axis of symmetry; that is, for this kind of quadrilateral, theVarignon parallelogram is adegenerate quadrilateral of area zero, consisting of four collinear points.[6][7] Theconvex hull of an antiparallelogram is anisosceles trapezoid, and every antiparallelogram may be formed from an isosceles trapezoid (or its special cases, the rectangles and squares) by replacing two parallel sides by the two diagonals of the trapezoid.[4]
Because an antiparallelogram forms two congruent triangular regions of the plane, but loops around those two regions in opposite directions, itssigned area is the difference between the regions' areas and is therefore zero.[7] The polygon's unsigned area (the total area it surrounds) is the sum, rather than the difference, of these areas. For an antiparallelogram with two parallel diagonals of lengths and, separated by height, this sum is.[4] It follows from applying thetriangle inequality to these two triangular regions that the crossing pair of edges in an antiparallelogram must always be longer than the two uncrossed edges.[8]
Severalnonconvex uniform polyhedra, including thetetrahemihexahedron,cubohemioctahedron,octahemioctahedron,small rhombihexahedron,small icosihemidodecahedron, andsmall dodecahemidodecahedron, have antiparallelograms as theirvertex figures, the cross-sections formed by slicing the polyhedron by a plane that passes near a vertex, perpendicularly to the axis between the vertex and the center.[9]
One form of a non-uniform butflexible polyhedron, theBricard octahedron, can be constructed as abipyramid over an antiparallelogram.[10]
The antiparallelogram has been used as a form offour-bar linkage, in which four rigid beams of fixed length (the four sides of the antiparallelogram) may rotate with respect to each other at joints placed at the four vertices of the antiparallelogram. In this context it is also called abutterfly orbow-tie linkage. As a linkage, it has a point of instability in which it can be converted into a parallelogram and vice versa, but either of these linkages can be braced to prevent this instability.[12][11]
For both the parallelogram and antiparallelogram linkages, if one of the long (crossed) edges of the linkage is fixed as a base, the free joints move on equal circles, but in a parallelogram they move in the same direction with equal velocities while in the antiparallelogram they move in opposite directions with unequal velocities.[13] AsJames Watt discovered, if an antiparallelogram has its long side fixed in this way, the midpoint of the unfixed long edge will trace out a lemniscate or figure eight curve. For the antiparallelogram formed by the sides and diagonals of a square, it is thelemniscate of Bernoulli.[14][15]
The antiparallelogram with its long side fixed is a variant ofWatt's linkage.[14] An antiparallelogram is an important feature in the design ofHart's inversor, a linkage that (like thePeaucellier–Lipkin linkage) can convert rotary motion to straight-line motion.[16] An antiparallelogram-shaped linkage can also be used to connect the twoaxles of a four-wheeled vehicle, decreasing theturning radius of the vehicle relative to a suspension that only allows one axle to turn.[2] A pair of nested antiparallelograms was used in a linkage defined byAlfred Kempe as part ofKempe's universality theorem, stating that any algebraic curve may be traced out by the joints of a suitably defined linkage. Kempe called the nested-antiparallelogram linkage a "multiplicator", as it could be used to multiply an angle by an integer.[1] Used in the other direction, to divide angles, it can be used forangle trisection (although not as astraightedge and compass construction).[17] Kempe's original constructions using this linkage overlooked the parallelogram-antiparallelogram instability, but bracing the linkages fixes his proof of the universality theorem.[12]
Suppose that one of the uncrossed edges of an antiparallelogram linkage is fixed in place, and the remaining linkage moves freely. As the linkage moves, each antiparallelogram formed can be divided into two congruent triangles meeting at the crossing point. In the triangle based on the fixed edge, the lengths of the two moving sides sum to the constant length of one of the antiparallelogram's crossed edges, and therefore the moving crossing point traces out anellipse with the fixed points as its foci. Symmetrically, the second (moving) uncrossed edge of the antiparallelogram has as its endpoints the foci of a second ellipse, formed from the first one by reflection across atangent line through the crossing point.[2][18] Because the second ellipserolls around the first, this construction of ellipses from the motion of an antiparallelogram can be used in the design ofelliptical gears that convert uniform rotation into non-uniform rotation or vice versa.[19]
In then-body problem, the study of the motions of point masses underNewton's law of universal gravitation, an important role is played bycentral configurations, solutions to then-body problem in which all of the bodies rotate around some central point as if they were rigidly connected to each other. For instance, for three bodies, there are five solutions of this type, given by the fiveLagrangian points. For four bodies, with two pairs of the bodies having equal masses (but with the ratio between the masses of the two pairs varying continuously), numerical evidence indicates that there exists a continuous family of central configurations, related to each other by the motion of an antiparallelogram linkage.[20]
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