Watt's curve

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In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart (taken to be at (±a, 0). A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate. It arose in connection with James Watt's pioneering work on the steam engine.

In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(xy) = 0, where F is a polynomial with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2. More precisely, if FFn + Fn−1 + ... + F1 + F0, where each Fi is homogeneous of degree i, then the curve F(xy) = 0 is circular if and only if Fn is divisible by x2 + y2. In mathematics, a affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve can be restricted to an affine algebraic plane curve by replacing by one some indeterminate of the defining homogeneous polynomial. As these two operations are each inverse to the other, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. James Watt was a Scottish inventor, mechanical engineer, and chemist who improved on Thomas Newcomen's 1712 Newcomen steam engine with his Watt steam engine in 1776, which was fundamental to the changes brought by the Industrial Revolution in both his native Great Britain and the rest of the world.

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The equation of the curve can be given in polar coordinates as

$r^{2}=b^{2}-\left[a\sin \theta \pm {\sqrt {c^{2}-a^{2}\cos ^{2}\theta }}\right]^{2}.$ Derivation

Polar coordinates

The polar equation for the curve can be derived as follows:  Working in the complex plane, let the centers of the circles be at a and −a, and the connecting segment have endpoints at −a+bei λ and a+bei ρ. Let the angle of inclination of the segment be ψ with its midpoint at rei θ. Then the endpoints are also given by rei θ ± cei ψ. Setting expressions for the same points equal to each other gives In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

$a+be^{i\rho }=re^{i\theta }+ce^{i\psi }.\,$ $-a+be^{i\lambda }=re^{i\theta }-ce^{i\psi }\,$ Add these and divide by two to get

$re^{i\theta }={\tfrac {b}{2}}(e^{i\rho }+e^{i\lambda })=b\cos({\tfrac {\rho -\lambda }{2}})e^{i{\tfrac {\rho +\lambda }{2}}}.$ Comparing radii and arguments gives

$r=b\cos \alpha ,\ \theta ={\tfrac {\rho +\lambda }{2}}\ {\mbox{where}}\ \alpha ={\tfrac {\rho -\lambda }{2}}.$ Similarly, subtracting the first two equations and dividing by 2 gives

$ce^{i\psi }-a={\tfrac {b}{2}}(e^{i\rho }-e^{i\lambda })=ib\sin \alpha e^{i\theta }.$ Write

$a=a\cos \theta \ e^{i\theta }-ia\sin \theta \ e^{i\theta }.\,$ Then

$ce^{i\psi }=ib\sin \alpha e^{i\theta }+a\cos \theta \ e^{i\theta }-ia\sin \theta \ e^{i\theta }=(a\cos \theta \ +i(b\sin \alpha -a\sin \theta ))e^{i\theta },$ $c^{2}=a^{2}\cos ^{2}\theta +(b\sin \alpha -a\sin \theta )^{2},\,$ $b\sin \alpha =a\sin \theta \pm {\sqrt {c^{2}-a^{2}\cos ^{2}\theta }},\,$ $r^{2}=b^{2}\cos ^{2}\alpha =b^{2}-b^{2}\sin ^{2}\alpha =b^{2}-\left[a\sin \theta \pm {\sqrt {c^{2}-a^{2}\cos ^{2}\theta }}\right]^{2}.,\,$ Cartesian coordinates

Expanding the polar equation gives

$r^{2}=b^{2}-(a^{2}\sin ^{2}\theta \ +c^{2}-a^{2}\cos ^{2}\theta \pm 2a\sin \theta {\sqrt {c^{2}-a^{2}\cos ^{2}\theta }}),\,$ $r^{2}-a^{2}-b^{2}+c^{2}+2a^{2}\sin ^{2}\theta =\pm 2a\sin \theta {\sqrt {c^{2}-a^{2}\cos ^{2}\theta }}),\,$ $(r^{2}-a^{2}-b^{2}+c^{2})^{2}+4a^{2}(r^{2}-a^{2}-b^{2}+c^{2})\sin ^{2}\theta +4a^{4}\sin ^{4}\theta =4a^{2}\sin ^{2}\theta (c^{2}-a^{2}\cos ^{2}\theta ),\,$ $(r^{2}-a^{2}-b^{2}+c^{2})^{2}+4a^{2}(r^{2}-b^{2})\sin ^{2}\theta =0,\,$ $(x^{2}+y^{2})(x^{2}+y^{2}-a^{2}-b^{2}+c^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0.\,$ Letting d2=a2+b2c2 simplifies this to

$(x^{2}+y^{2})(x^{2}+y^{2}-d^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0.\,$ Form of the curve

The construction requires a quadrilateral with sides 2a, b, 2c, b. Any side must be less than the sum of the remaining sides, so the curve is empty (at least in the real plane) unless a<b+c and c<b+a.

The has a crossing point at the origin if there is a triangle with sides a, b and c. Given the previous conditions, this means that the curve crosses the origin if and only if b<a+c. If b=a+c then two branches of the curve meet at the origin with a common vertical tangent, making it a quadruple point.

Given b<a+c, the shape of the curve is determined by the relative magnitude of b and d. If d is imaginary, that is if a2+b2 <c2 then the curve has the form of a figure eight. If d is 0 then the curve is a figure eight with two branches of the curve having a common horizontal tangent at the origin. If 0<d<b then the curve has two additional double points at ±d and the curve crosses itself at these points. The overall shape of the curve is pretzel-like in this case. If d=b then a=c and the curve decomposes into a circle of radius b and a lemniscate of Booth, a figure eight shaped curve. A special case of this is a=c, b=√2c which produces the lemniscate of Bernoulli. Finally, if d>b then the points ±d are still solutions to the Cartesian equation of the curve, but the curve does not cross these points and they are acnodes. The curve again has a figure eight shape though the shape is distorted if d is close to b. In geometry, a hippopede is a plane curve determined by an equation of the form In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4. An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are "isolated point or hermit point".

Given b>a+c, the shape of the curve is determined by the relative sizes of a and c. If a<c then the curve has the form of two loops that cross each other at ±d. If a=c then the curve decomposes into a circle of radius b and an oval of Booth. If a>c then the curve does not cross the x-axis at all and consists of two flattened ovals. 

When the curve crosses the origin, the origin is a point of inflection and therefore has contact of order 3 with a tangent. However, if a2=b2+<c2[ clarification needed ] then tangent has contact of order 5 with the tangent, in other words the curve is a close approximation of a straight line. This is the basis for Watt's linkage.

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