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In mathematics, Watt's curve is a tricircular
plane algebraic curve In mathematics, an 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 cu ...
of degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)). A
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
of length 2''c'' 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 partially back and forth or completely around. It arose in connection with
James Watt James Watt (; 30 January 1736 (19 January 1736 OS) – 25 August 1819) 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 f ...
's pioneering work on the steam engine. The equation of the curve can be given in
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
as :r^2=b^2-\left \sin\theta\pm\sqrt\right2.


Derivation


Polar coordinates

The polar equation for the curve can be derived as follows: Working in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, let the centers of the circles be at ''a'' and ''−a'', and the connecting segment have endpoints at ''−a''+''be''''i'' λ and ''a''+''be''''i'' ρ. Let the angle of inclination of the segment be ψ with its midpoint at ''re''''i'' θ. Then the endpoints are also given by ''re''''i'' θ ± ''ce''''i'' ψ. Setting expressions for the same points equal to each other gives :a+be^=re^+ce^.\, :-a+be^=re^-ce^\, Add these and divide by two to get :re^=\tfrac(e^+e^)=b\cos(\tfrac)e^. Comparing radii and arguments gives :r=b\cos\alpha,\ \theta=\tfrac\ \mbox\ \alpha=\tfrac. Similarly, subtracting the first two equations and dividing by 2 gives :ce^-a=\tfrac(e^-e^)=i b\sin\alpha e^. Write :a=a\cos\theta\ e^ - i a\sin\theta\ e^.\, Then :ce^=i b\sin\alpha e^+a\cos\theta\ e^ - i a\sin\theta\ e^=(a\cos\theta\ +i(b\sin\alpha-a\sin\theta))e^, :c^2=a^2\cos^2\theta+(b\sin\alpha-a\sin\theta)^2,\, :b\sin\alpha=a\sin\theta\pm\sqrt,\, :r^2=b^2\cos^2\alpha=b^2-b^2\sin^2\alpha=b^2-\left \sin\theta\pm\sqrt\right2.,\,


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),\, :r^2-a^2-b^2+c^2+2a^2\sin^2\theta=\pm 2a\sin\theta\sqrt),\, :(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^2y^2(x^2+y^2-b^2)=0.\, Letting ''d'' 2=''a''2+''b''2–''c''2 simplifies this to :(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0.\,


Form of the curve

The construction requires a quadrilateral with sides 2''a'', ''b'', 2''c'', ''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 curve 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 In mathematics, particularly calculus, a vertical tangent is a tangent, tangent line that is Vertical direction, vertical. Because a vertical line has Infinity, infinite slope, a Function (mathematics), function whose graph of a function, graph ha ...
, 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 ''a''2+''b''2 <''c''2 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''=√2''c'' which produces the
lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...
. 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
acnode An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point. For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is ...
s. The curve again has a figure eight shape though the shape is distorted if ''d'' is close to ''b''. 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.Encyclopédie des Formes Mathématiques Remarquables page for section.


Watt's linkage

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 a^2=b^2+c^2 ( which is the case if the triangle with sides a, b and c is a
right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...
) 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.


See also

*
Four-bar linkage In the study of Mechanism (engineering), mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-Kinematic chain, chain movable linkage (mechanical), linkage. It consists of four Rigid body, bodies, called ''bars'' or ''link ...
*
Watt's linkage A Watt's linkage is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel a nearly straight path. Watt's described the linkage in his patent specification of 1784 for the ...


References


External links

* * * * {{DEFAULTSORT:Watt's Curve Sextic curves
Curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...