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Workmen's Compensation Act 1897
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The Workmen's Compensation Act 1897 was British law in operation from 1897 to 1946. Great Britain followed the German model. Joseph Chamberlain, leader of the Liberal Unionist party and in coalition with the Conservatives, designed a plan that was enacted under the Salisbury government in 1897. The Act was a key domestic achievement. It served its social purpose at no cost to the government, since compensation was paid for by insurance which employers were required to take out to pay for medical costs of injuries on the job. It replaced the Employers' Liability Act 1880, which gave the injured worker the right to sue the employer but put the burden of proof on the employee. After 1897, injured employees had only to show that they had been injured on the job.[1] These are roughly the same rights German workers were awarded in their 1884 law.[2] The Act was replaced by an expanded scheme under the Workmen's Compensation Act 1906.[3]
Contents
[hide]
1 Definitions
2 Interpretation by Courts
3 See also
4 Notes
5 References
Definitions[edit]
A "workman" was defined as,
"any person who is engaged in an employment to which this Act applies, whether by way of manual labour or otherwise."[4]
The employments to which the Act applied were stated to be railways, mining and quarrying, factory work and laundry work.
Interpretation by Courts[edit]
Courts took a restrictive interpretation of the Act in Simpson v. Ebbw Vale Steel, Iron & Coal Co. [5] A widow claimed for the death of a colliery manager who had been killed in an underground accident. Lord Collins MR held that her dead husband was outside the Act's scope, because though the act extended to non-manual workers[6] the victim "must still be a workman". He said the Act:
"presupposes a position of dependence; it treats the class of workmen as being in a sens inopes consilii, and the Legislature does for them what they cannot do for themselves: it gives them a sort of State insurance, it being assumed that they are either not sufficiently intelligent or not sufficiently in funds to insure themselves. In no sense can such a principle extend to those who are earning good salaries."[7]
See also[edit]
Workers' compensation
Workmen's Compensation Act 1906
Contracts of Employment Act 1963
UK labour law
English tort law
Notes[edit]
Jump up ^ History of Workers' Compensation, Florida AWC
Jump up ^ Burriesci, 2001.
Jump up ^ D. C. Hanes, The First British Workmen's Compensation Act of 1897 (1968).
Jump up ^ Workmen's Compensation Act 1897, s.7
Jump up ^ Simpson v. Ebbw Vale Steel, Iron & Coal Co. (1905)
Jump up ^ unlike the previous law under the Employers Liability Act 1880, from Morgan v. London General Omnibus Co. (1884) 13 QBD 832
Jump up ^ Simpson, 1905
References[edit]
Jack J. Burriesci, "Historical Summary Of Workers' Compensation Laws" (March 7, 2001) 2001-R-0261 online
D. C. Hanes, The First British Workmen's Compensation Act of 1897 (1968).
Simon Deakin, 'The historical process of wage formation', in Linda Clarke et al., The Dynamics of Wage Relations in the New Europe (2000) pp. 38–9
Simon Deakin, 'Welfare State and Contract of Employment', in Noel Whiteside et al., Governance, Industry and Labour Markets in Britain and France (1998) pp. 212 ff.
Frankel, Lee K., and Dawson, Miles, M., Workingmen's Insurance in Europe, New York: Charities Publication Committee, 1911.
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text:
Arc length
From Wikipedia, the free encyclopedia
Jump to: navigation, search
When rectified, the curve gives a straight line segment with the same length as the curve's arc length.
Example: arc length s of a logarithmic spiral as a function of its parameter θ.
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.
Contents
[hide]
1 General approach
2 Definition for a smooth curve
3 Finding arc lengths by integrating
3.1 Numerical integration
3.2 Curve on a surface
3.3 Other coordinate systems
4 Simple cases
4.1 Arcs of circles
4.1.1 Arcs of great circles on the Earth
4.2 Length of an arc of a parabola
5 Historical methods
5.1 Antiquity
5.2 17th century
5.3 Integral form
6 Curves with infinite length
7 Generalization to (pseudo-)Riemannian manifolds
8 See also
9 References and notes
10 External links
General approach[edit]
Approximation by multiple linear segments
A curve in the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance.[1]
If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small.
For some curves there is a smallest number L {\displaystyle L} that is an upper bound on the length of any polygonal approximation. These curves are called rectifiable and the number L {\displaystyle L} is defined as the arc length.
Definition for a smooth curve[edit]
See also: Length of a curve
Let f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} be a continuously differentiable function. The length of the curve defined by f {\displaystyle f} can be defined as the limit of the sum of line segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity. This means
L ( f ) = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | {\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}}
where t i = a + i ( b − a ) / N = a + i Δ t {\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t} for i = 0 , 1 , … , N . {\displaystyle i=0,1,\dotsc ,N.} This definition is equivalent to the standard definition of arc length as an integral:
lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) Δ t | Δ t = ∫ a b | f ′ ( t ) | d t . {\displaystyle \lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}=\lim _{N\to \infty }\sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt.}
The last equality above is true because the definition of the derivative as a limit implies that there is a positive real function δ ( ϵ ) {\displaystyle \delta (\epsilon )} of positive real ϵ {\displaystyle \epsilon } such that Δ t < δ ( ϵ ) {\displaystyle \Delta t<\delta (\epsilon )} implies | | f ( t i ) − f ( t i − 1 ) Δ t | − | f ′ ( t i ) | | < ϵ . {\displaystyle \left|{\bigg |}{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}{\bigg |}-{\Big |}f'(t_{i}){\Big |}\right|<\epsilon .} This means
∑ i = 1 N | f ( t i ) − f ( t i − 1 ) Δ t | Δ t − ∑ i = 1 N | f ′ ( t i ) | Δ t {\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t-\sum _{i=1}^{N}{\Big |}f'(t_{i}){\Big |}\Delta t}
has absolute value less than ϵ ( b − a ) {\displaystyle \epsilon (b-a)} for N > ( b − a ) / δ ( ϵ ) . {\displaystyle N>(b-a)/\delta (\epsilon ).} This means that in the limit N → ∞ , {\displaystyle N\rightarrow \infty ,} the left term above equals the right term, which is just the Riemann integral of | f ′ ( t ) | {\displaystyle |f'(t)|} on [ a , b ] . {\displaystyle [a,b].} This definition of arc length shows that the length of a curve f : [ a , b ] → R n {\displaystyle f:[a,b]\rightarrow \mathbb {R} ^{n}} continuously differentiable on [ a , b ] {\displaystyle [a,b]} is always finite. In other words, the curve is always rectifiable.
The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition
L ( f ) = sup ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | {\displaystyle L(f)=\sup \sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}}
where the supremum is taken over all possible partitions a = t 0 < t 1 < ⋯ < t N − 1 < t N = b {\displaystyle a=t_{0}<t_{1}<\dots <t_{N-1}<t_{N}=b} of [ a , b ] . {\displaystyle [a,b].} [2] This definition is also valid if f {\displaystyle f} is merely continuous, not differentiable.
A curve can be parameterized in infinitely many ways. Let φ : [ a , b ] → [ c , d ] {\displaystyle \varphi :[a,b]\to [c,d]} be any continuously differentiable bijection. Then g = f ∘ φ − 1 : [ c , d ] → R n {\displaystyle g=f\circ \varphi ^{-1}:[c,d]\to \mathbb {R} ^{n}} is another continuously differentiable parameterization of the curve originally defined by f . {\displaystyle f.} The arc length of the curve is the same regardless of the parameterization used to define the curve:
L ( f ) = ∫ a b | f ′ ( t ) | d t = ∫ a b | g ′ ( φ ( t ) ) φ ′ ( t ) | d t = ∫ a b | g ′ ( φ ( t ) ) | φ ′ ( t ) d t since φ must be non-decreasing = ∫ c d | g ′ ( u ) | d u using integration by substitution = L ( g ) . {\displaystyle {\begin{aligned}L(f)&=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt=\int _{a}^{b}{\Big |}g'(\varphi (t))\varphi '(t){\Big |}\ dt\\&=\int _{a}^{b}{\Big |}g'(\varphi (t)){\Big |}\varphi '(t)\ dt\quad {\textrm {since}}\ \varphi \ {\textrm {must}}\ {\textrm {be}}\ {\textrm {non-decreasing}}\\&=\int _{c}^{d}{\Big |}g'(u){\Big |}\ du\quad {\textrm {using}}\ {\textrm {integration}}\ {\textrm {by}}\ {\textrm {substitution}}\\&=L(g).\end{aligned}}}
Finding arc lengths by integrating[edit]
See also: Differential geometry of curves
Quarter circle
If a planar curve in R 2 {\displaystyle \mathbb {R} ^{2}} is defined by the equation y = f ( x ) , {\displaystyle y=f(x),} where f {\displaystyle f} is continuously differentiable, then it is simply a special case of a parametric equation where x = t {\displaystyle x=t} and y = f ( t ) , {\displaystyle y=f(t),} the arc length is given by:
s = ∫ a b 1 + ( d y d x ) 2 d x . {\displaystyle s=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx.}
Curves with closed-form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and straight line. The lack of a closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals.
Numerical integration[edit]
In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as y = 1 − x 2 . {\displaystyle y={\sqrt {1-x^{2}}}.} The interval x ∈ [ − 2 / 2 , 2 / 2 ] {\displaystyle x\in [-{\sqrt {2}}/2,{\sqrt {2}}/2]} corresponds to a quarter of the circle. Since d y / d x = − x / 1 − x 2 {\displaystyle dy/dx=-x/{\sqrt {1-x^{2}}}} and 1 + ( d y / d x ) 2 = 1 / ( 1 − x 2 ) , {\displaystyle 1+(dy/dx)^{2}=1/(1-x^{2}),} the length of a quarter of the unit circle is
∫ − 2 / 2 2 / 2 1 1 − x 2 d x . {\displaystyle \int _{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}{\frac {1}{\sqrt {1-x^{2}}}}dx.}
The 15-point Gauss-Kronrod rule estimate for this integral of 7000157079632680817♠1.570796326808177 differs from the true length of π / 2 {\displaystyle \pi /2} by 6989130000000000000♠1.3×10−11 and the 16-point Gaussian quadrature rule estimate of 7000157079632679472♠1.570796326794727 differs from the true length by only 6987170000000000000♠1.7×10−13. This means it is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations.
Curve on a surface[edit]
Let x ( u , v ) {\displaystyle \mathbf {x} (u,v)} be a surface mapping and let C ( t ) = ( u ( t ) , v ( t ) ) {\displaystyle \mathbf {C} (t)=(u(t),v(t))} be a curve on this surface. The integrand of the arc length integral is | ( x ∘ C ) ′ ( t ) | . {\displaystyle |(\mathbf {x} \circ \mathbf {C} )'(t)|.} Evaluating the derivative requires the chain rule for vector fields:
D ( x ∘ C ) = ( x u x v ) ( u ′ v ′ ) = x u u ′ + x v v ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=(\mathbf {x} _{u}\ \mathbf {x} _{v}){\binom {u'}{v'}}=\mathbf {x} _{u}u'+\mathbf {x} _{v}v'.}
The squared norm of this vector is ( x u u ′ + x v v ′ ) ⋅ ( x u u ′ + x v v ′ ) = g 11 ( u ′ ) 2 + 2 g 12 u ′ v ′ + g 22 ( v ′ ) 2 {\displaystyle (\mathbf {x} _{u}u'+\mathbf {x} _{v}v')\cdot (\mathbf {x} _{u}u'+\mathbf {x} _{v}v')=g_{11}(u')^{2}+2g_{12}u'v'+g_{22}(v')^{2}} (where g i j {\displaystyle g_{ij}} is the first fundamental form coefficient), so the integrand of the arc length integral can be written as g a b ( u a ) ′ ( u b ) ′ {\displaystyle {\sqrt {g_{ab}(u^{a})'(u^{b})'}}} (where u 1 = u {\displaystyle u^{1}=u} and u 2 = v {\displaystyle u^{2}=v} ).
Other coordinate systems[edit]
Let C ( t ) = ( r ( t ) , θ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t))} be a curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is
x ( r , θ ) = ( r cos  θ , r sin  θ ) . {\displaystyle \mathbf {x} (r,\theta )=(r\cos \theta ,r\sin \theta ).}
The integrand of the arc length integral is | ( x ∘ C ) ′ ( t ) | . {\displaystyle |(\mathbf {x} \circ \mathbf {C} )'(t)|.} The chain rule for vector fields shows that D ( x ∘ C ) = x r r ′ + x θ θ ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '.} So the squared integrand of the arc length integral is
( x r ⋅ x r ) ( r ′ ) 2 + 2 ( x r ⋅ x θ ) r ′ θ ′ + ( x θ ⋅ x θ ) ( θ ′ ) 2 = ( r ′ ) 2 + r 2 ( θ ′ ) 2 . {\displaystyle (\mathbf {x_{r}} \cdot \mathbf {x} _{r})(r')^{2}+2(\mathbf {x} _{r}\cdot \mathbf {x} _{\theta })r'\theta '+(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta })(\theta ')^{2}=(r')^{2}+r^{2}(\theta ')^{2}.}
So for a curve expressed in polar coordinates, the arc length is
∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 d t = ∫ θ ( t 1 ) θ ( t 2 ) ( d r d θ ) 2 + r 2 d θ . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}}}dt=\int _{\theta (t_{1})}^{\theta (t_{2})}{\sqrt {\left({\frac {dr}{d\theta }}\right)^{2}+r^{2}}}d\theta .}
Now let C ( t ) = ( r ( t ) , θ ( t ) , ϕ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t),\phi (t))} be a curve expressed in spherical coordinates where θ {\displaystyle \theta } is the polar angle measured from the positive z {\displaystyle z} -axis and ϕ {\displaystyle \phi } is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is
x ( r , θ , ϕ ) = ( r sin  θ cos  ϕ , r sin  θ sin  ϕ , r cos  θ ) . {\displaystyle \mathbf {x} (r,\theta ,\phi )=(r\sin \theta \cos \phi ,r\sin \theta \sin \phi ,r\cos \theta ).}
Using the chain rule again shows that D ( x ∘ C ) = x r r ′ + x θ θ ′ + x ϕ ϕ ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '+\mathbf {x} _{\phi }\phi '.} All dot products x i ⋅ x j {\displaystyle \mathbf {x} _{i}\cdot \mathbf {x} _{j}} where i {\displaystyle i} and j {\displaystyle j} differ are zero, so the squared norm of this vector is
( x r ⋅ x r ) ( r ′ 2 ) + ( x θ ⋅ x θ ) ( θ ′ ) 2 + ( x ϕ ⋅ x ϕ ) ( ϕ ′ ) 2 = ( r ′ ) 2 + r 2 ( θ ′ ) 2 + r 2 sin 2  θ ( ϕ ′ ) 2 . {\displaystyle (\mathbf {x} _{r}\cdot \mathbf {x} _{r})(r'^{2})+(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta })(\theta ')^{2}+(\mathbf {x} _{\phi }\cdot \mathbf {x} _{\phi })(\phi ')^{2}=(r')^{2}+r^{2}(\theta ')^{2}+r^{2}\sin ^{2}\theta (\phi ')^{2}.}
So for a curve expressed in spherical coordinates, the arc length is
∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 + r 2 sin 2  θ ( d ϕ d t ) 2 d t . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}+r^{2}\sin ^{2}\theta \left({\frac {d\phi }{dt}}\right)^{2}}}dt.}
A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is
∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 + ( d z d t ) 2 d t . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}}}dt.}
Simple cases[edit]
Arcs of circles[edit]
Arc lengths are denoted by s, since the Latin word for length (or size) is spatium.
In the following lines, r {\displaystyle r} represents the radius of a circle, d {\displaystyle d} is its diameter, C {\displaystyle C} is its circumference, s {\displaystyle s} is the length of an arc of the circle, and θ {\displaystyle \theta } is the angle which the arc subtends at the centre of the circle. The distances r , d , C , {\displaystyle r,d,C,} and s {\displaystyle s} are expressed in the same units.
C = 2 π r , {\displaystyle C=2\pi r,} which is the same as C = π d . {\displaystyle C=\pi d.} This equation is a definition of π . {\displaystyle \pi .}
If the arc is a semicircle, then s = π r . {\displaystyle s=\pi r.}
For an arbitrary circular arc:
If θ {\displaystyle \theta } is in radians then s = r θ . {\displaystyle s=r\theta .} This is a definition of the radian.
If θ {\displaystyle \theta } is in degrees, then s = π r θ 180 , {\displaystyle s={\frac {\pi r\theta }{180}},} which is the same as s = C θ 360 . {\displaystyle s={\frac {C\theta }{360}}.}
If θ {\displaystyle \theta } is in grads (100 grads, or grades, or gradians are one right-angle), then s = π r θ 200 , {\displaystyle s={\frac {\pi r\theta }{200}},} which is the same as s = C θ 400 . {\displaystyle s={\frac {C\theta }{400}}.}
If θ {\displaystyle \theta } is in turns (one turn is a complete rotation, or 360°, or 400 grads, or 2 π {\displaystyle 2\pi } radians), then s = C θ . {\displaystyle s=C\theta .}
Arcs of great circles on the Earth[edit]
Main article: Great-circle distance
Further information: Geodesics on an ellipsoid
Two units of length, the nautical mile and the metre (or kilometre), were originally defined so the lengths of arcs of great circles on the Earth's surface would be simply numerically related to the angles they subtend at its centre. The simple equation s = θ {\displaystyle s=\theta } applies in the following circumstances:
if s {\displaystyle s} is in nautical miles, and θ {\displaystyle \theta } is in arcminutes (1⁄60 degree), or
if s {\displaystyle s} is in kilometres, and θ {\displaystyle \theta } is in centigrades (1⁄100 grad).
The lengths of the distance units were chosen to make the circumference of the Earth equal 7004400000000000000♠40000 kilometres, or 7004216000000000000♠21600 nautical miles. These are the numbers of the corresponding angle units in one complete turn.
These definitions of the metre and nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes, and for some calculations. For example, they imply that one kilometre is exactly 0.54 nautical miles. Using official modern definitions, one nautical mile is exactly 1.852 kilometres,[3] which implies that 1 kilometre ≈ 6999539956800000000♠0.53995680 nautical miles.[4] This modern ratio differs from the one calculated from the original definitions by less than one part in ten thousand.
Length of an arc of a parabola[edit]
For a calculation of the length of a parabolic arc, see Parabola § Length of an arc of a parabola.
Historical methods[edit]
Antiquity[edit]
For much of the history of mathematics, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a way of finding the area beneath a curve with his "method of exhaustion", few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus, by approximation. People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. In particular, by inscribing a polygon of many sides in a circle, they were able to find approximate values of π.
17th century[edit]
In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: the logarithmic spiral by Evangelista Torricelli in 1645 (some sources say John Wallis in the 1650s), the cycloid by Christopher Wren in 1658, and the catenary by Gottfried Leibniz in 1691.
In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola.[5] The accompanying figures appear on page 145. On page 91, William Neile is mentioned as Gulielmus Nelius.
Integral form[edit]
Before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre de Fermat.
In 1659 van Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral). As an example of his method, he determined the arc length of a semicubical parabola, which required finding the area under a parabola.[6] In 1660, Fermat published a more general theory containing the same result in his De linearum curvarum cum lineis rectis comparatione dissertatio geometrica (Geometric dissertation on curved lines in comparison with straight lines).[7]
Fermat's method of determining arc length
Building on his previous work with tangents, Fermat used the curve
y = x 3 / 2 {\displaystyle y=x^{3/2}\,}
whose tangent at x = a had a slope of
3 2 a 1 / 2 {\displaystyle \textstyle {3 \over 2}a^{1/2}}
so the tangent line would have the equation
y = 3 2 a 1 / 2 ( x − a ) + f ( a ) . {\displaystyle y=\textstyle {3 \over 2}{a^{1/2}}(x-a)+f(a).}
Next, he increased a by a small amount to a + ε, making segment AC a relatively good approximation for the length of the curve from A to D. To find the length of the segment AC, he used the Pythagorean theorem:
A C 2 = A B 2 + B C 2 = ε 2 + 9 4 a ε 2 = ε 2 ( 1 + 9 4 a ) {\displaystyle {\begin{aligned}AC^{2}&{}=AB^{2}+BC^{2}\\&{}=\textstyle \varepsilon ^{2}+{9 \over 4}a\varepsilon ^{2}\\&{}=\textstyle \varepsilon ^{2}\left(1+{9 \over 4}a\right)\end{aligned}}}
which, when solved, yields
A C = ε 1 + 9 4 a . {\displaystyle AC=\textstyle \varepsilon {\sqrt {1+{9 \over 4}a\ }}.}
In order to approximate the length, Fermat would sum up a sequence of short segments.
Curves with infinite length[edit]
See also: Coastline paradox
The Koch curve.
The graph of xsin(1/x).
As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to quantify the size of such curves.
Generalization to (pseudo-)Riemannian manifolds[edit]
Let M {\displaystyle M} be a (pseudo-)Riemannian manifold, γ : [ 0 , 1 ] → M {\displaystyle \gamma :[0,1]\rightarrow M} a curve in M {\displaystyle M} and g {\displaystyle g} the (pseudo-) metric tensor.
The length of γ {\displaystyle \gamma } is defined to be
ℓ ( γ ) = ∫ 0 1 ± g ( γ ′ ( t ) , γ ′ ( t ) ) d t , {\displaystyle \ell (\gamma )=\int _{0}^{1}{\sqrt {\pm g(\gamma '(t),\gamma '(t))}}\,dt,}
where γ ′ ( t ) ∈ T γ ( t ) M {\displaystyle \gamma '(t)\in T_{\gamma (t)}M} is the tangent vector of γ {\displaystyle \gamma } at t . {\displaystyle t.} The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves . Thus the length of a curve in a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike.
In theory of relativity, arc length of timelike curves (world lines) is the proper time elapsed along the world line, and arc length of a spacelike curve the proper distance along the curve.
See also[edit]
Arc (geometry)
Circumference
Crofton formula
Elliptic integral
Geodesics
Intrinsic equation
Integral approximations
Meridian arc
Sinuosity
References and notes[edit]
Jump up ^ Ahlberg; Nilson (1967). The Theory of Splines and Their Applications. Academic Press. p. 51.
Jump up ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill, Inc. p. 137. ISBN 0-07-054235-X.
Jump up ^ physics.nist.gov/Pubs/SP811/appenB8.html
Jump up ^ CRC Handbook of Chemistry and Physics, page F-254
Jump up ^ Wallis, John (1659). Tractatus Duo. Prior, De Cycloide et de Corporibus inde Genitis…. Oxford: University Press. pp. 91–96.
Jump up ^ van Heuraet, Hendrik (1659). "Epistola de transmutatione curvarum linearum in rectas [Letter on the transformation of curved lines into right ones]". Renati Des-Cartes Geometria (2nd ed.). Amsterdam: Louis & Daniel Elzevir. pp. 517–520.
Jump up ^ M.P.E.A.S. (pseudonym of Fermat) (1660). De Linearum Curvarum cum Lineis Rectis Comparatione Dissertatio Geometrica. Toulouse: Arnaud Colomer.
Farouki, Rida T. (1999). "Curves from motion, motion from curves". In Laurent, P.-J.; Sablonniere, P.; Schumaker, L. L. Curve and Surface Design: Saint-Malo 1999. Vanderbilt Univ. Press. pp. 63–90. ISBN 0-8265-1356-5.
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Rectifiable curve", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
The History of Curvature
Weisstein, Eric W. "Arc Length". MathWorld.
Arc Length by Ed Pegg, Jr., The Wolfram Demonstrations Project, 2007.
[permanent dead link] Calculus Study Guide – Arc Length (Rectification)
Famous Curves Index The MacTutor History of Mathematics archive
Arc Length Approximation by Chad Pierson, Josh Fritz, and Angela Sharp, The Wolfram Demonstrations Project.
Length of a Curve Experiment Illustrates numerical solution of finding length of a curve.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Arc_length&oldid=799302264"
Categories:
Integral calculus
Curves
Length
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Johnny Western
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Johnny Western
Born
(1934-10-28) October 28, 1934 (age 83)
Two Harbors, Lake County
Minnesota, USA
Occupation(s)
Singer-songwriter, actor, radio show host (current)
Associated acts
Gene Autry, Johnny Cash
Johnny Western (born October 28, 1934) is an American country singer-songwriter, musician, actor, and radio show host. He is a member of the Western Music Association Hall of Fame and the Country Music Disc Jockey Hall of Fame.
Contents
[hide]
1 Early life
2 Musical career
2.1 Albums
2.2 Guest appearances
3 Acting career
4 Radio career
5 The Johnny Western Theatre
6 Awards and honors
7 External links
8 References
Early life[edit]
Johnny Western was born Johnny Westerlund in Two Harbors in Lake County in northeastern Minnesota but was primarily reared in Northfield in south central Minnesota.[1]
His father was an instructor and officer in several Civilian Conservation Corps camps, where Western spent some his earlier years. He also lived on Indian reservations along the Canada–United States border.[2]
When he was five years old, Western's parents took him to see the western film Guns and Guitars, which starred the actor/singer Gene Autry. The young boy decided he wanted to be a singing cowboy. At the age of twelve, he received a guitar. Within a year, he was performing professionally.[2]
Musical career[edit]
Johnny Western's professional career began as a young teenager, singing and playing rhythm guitar with a collegiate singing trio. He got a job on radio at the age of thirteen,[3] a feat publicized in Billboard as the youngest disc jockey and singer on American radio.[2] At age sixteen, Western began performing with the Sons of the Pioneers.
He made his first professional recordings in the summer of 1952 in the studio of WCAL Radio station of St. Olaf College of Northfield, Minnesota. The six songs which resulted from those sessions were released on three singles on the local J-O-C-O label.[4] After having played a supporting role in an episode of "Have Gun, Will Travel", Western wrote "The Ballad of Paladin" as a musical "thank-you-card" to Richard Boone. This landed him a deal with Columbia Records.
Between August 1958 and May 1963, Western recorded seven singles and one album (Have Gun, Will Travel, released in May 1962) for Columbia, until he was dropped from their roster. He then signed a contract with the Philips label, but only one single resulted ("Light The Fuse" b/w "Tender Years").
Have Gun Will Travel included the haunting Stan Jones song "Cowpoke", which members of the Western Writers of America chose as one of the Top 100 Western songs of all time.[5] The album was a mature work, redolent of Western lore. It also included Western's third version of "The Ballad of Paladin", a fine and introspective performance of "The Lonely Man", the gallows ballad "Hannah Lee", "The Streets of Laredo", "The Searchers" and "The Last Roundup".[6]
After this, Western was left without a recording contract. Over the years, he recorded various songs for small local labels such as Hep in Missouri or a promo-single for Dodge City. It was not until Johnny Cash invited him to record at his House Of Cash recording studio that Western was able to compile enough material for another LP, which was released under the simple title of Johnny Western in 1981. Apart from the House of Cash recordings, it contained some unreleased songs Western had recorded for the Hep label.
Western's third album was recorded September 17–20, 1984 at Jack Clement Studio in Nashville, with Art Sparer producing. It was released under the title Johnny Western Sings 20 Great Classics & Legends. Amongst others, it contained a new version of Western's own composition "The Gunfighter", featuring Harold Bradley on gut-string guitar, imitating the original "El Paso" sound, since Western had originally written that song with Marty Robbins in mind.
His scarce recording output notwithstanding, Johnny Western remained a consistent performer for over sixty years and so his name was never lost on the general public.
He performed with Gene Autry and was a part of the Johnny Cash road show from 1958 until 1997. He wrote and performed the theme song "The Ballad of Paladin" for the CBS television program Have Gun – Will Travel, with Richard Boone. In collaboration with Johnny Cash, he re-wrote the lyrics of NBC's Bonanza and the theme song, "The Rebel - Johnny Yuma", from the ABC series The Rebel, starring Nick Adams.[3]
In October 2013, Johnny Western announced that he would stop touring and giving concerts.[7]
Albums[edit]
Year
Album
Label
1962 Have Gun - Will Travel Columbia
1981 Johnny Western JRC Records
1981 The Gunfighter Bear Family
1984 Sings 20 Great Classics & Legends Americana Records
1989 Gunfight at O.K. Corral [re-issue of his 1984 album] Bear Family
1993 Heroes and Cowboys [3-CD career retrospective] Bear Family
2008 Country and Johnny Western [contains material from his Columbia & House of Cash catalog] Black Knight
[8]
Guest appearances[edit]
Year
Song
Artist
Album
1997 Tumbling Tumbleweeds The Home Rangers Bad Boots
1998 The Ballad of Palindrome Riders in the Sky A Great Big Western Howdy!
2007 Pancho and Lefty Rex Allen, Jr. The New West
Acting career[edit]
Western has appeared in several television programs, including the westerns, Have Gun - Will Travel, Tales of Wells Fargo, Pony Express (in the 1960 episode "The Story of Julesburg"), and Boots and Saddles. He has had roles in several movies including The Night Rider, Fort Bowie, and The Dalton Girls.[9]
Radio career[edit]
In February 1986, Western returned to radio with a daily country music show on KFDI (1070 AM, later KFTI) a country music station calling itself "The Radio Ranch" in Wichita, Kansas.[10] The station and sister station KFDI-FM were sold to Journal Broadcast Group in July 1999.[11][12] In early April 2010, Johnny announced that he would retire from broadcasting. His last day on the air was Saturday, April 10, 2010. Just over a month and a half later the station would switch to an oldies format.
The Johnny Western Theatre[edit]
The Johnny Western Theatre in Wild West World, a theme park located in Park City, Kansas near Wichita, had concerts scheduled to begin in October 2007. However, the park declared bankruptcy in July 2007.[13] Western was never able to perform in his namesake theater, which now sits among the ruins of the Wild West World amusement park, just outside Wichita, Kansas.
Awards and honors[edit]
TV Theme Hall of Fame (1993)
Country Music Disc Jockey Hall of Fame (2000)
Old Time Country Music Hall of Fame (2000)
Western Music Association Hall of Fame (2001)
Wichita Professional Broadcasters Hall of Fame (2002)
Kansas Cowboy Hall of Fame (2004)
Kansas Western Swing Hall of Fame (2004)
Biography portal
Radio portal
Television portal
Film portal
Music portal
Minnesota portal
California portal
External links[edit]
Johnny Western Tour Information
References[edit]
Jump up ^ "Country Music & Western Heroes". KFDI. Archived from the original on 2007-12-31. Retrieved 2008-01-05.
^ Jump up to: a b c "Artists: Biography, Johnny Western". Retrieved 2008-01-04.
^ Jump up to: a b "Johnny Western, Cowboy Entertainer". Boot Hill Museum & Front Street. Archived from the original on 2008-01-23. Retrieved 2008-01-04.
Jump up ^ All information pertaining to Johnny Western's discography is taken from a discography compiled by John L. Smith and Richard Weize, as it was published in the booklet accompanying the Heroes & Cowboys boxed set by Bear Family Records, Germany, 1991.
Jump up ^ Western Writers of America (2010). "The Top 100 Western Songs". American Cowboy. Archived from the original on 10 August 2014.
Jump up ^ Erlewine, Michael (1997). All Music Guide to Country: The Experts' Guide to the Best Recordings in Country Music. Hal Leonard Corporation. p. 501. Retrieved 2015-02-06.
Jump up ^ cf. information on: www.johnnywesterntouring.com, retrieved December 23, 2013.
Jump up ^ "Albums". Country Music Television, Inc. Retrieved 2008-01-10.
Jump up ^ "Johnny Western". Internet Movie Data Base. Retrieved 2008-01-04.
Jump up ^ "The Ranch Hands". KFDI. Archived from the original on 2007-12-01. Retrieved 2008-01-05.
Jump up ^ "Johnny Western". Retrieved 2008-01-04.
Jump up ^ "AM 1070 Morning Newsradio Debuts Monday". Journal Broadcast Group. 2008-01-14. Retrieved 2008-02-02.
Jump up ^ Arthur Levine. "Wild West World Theme Park". About.com. Retrieved 2008-02-09.
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My Two Dads
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My Two Dads
Genre
Family sitcom
Created by
Michael Jacobs
Danielle Alexandra
Starring
Paul Reiser
Greg Evigan
Staci Keanan
Florence Stanley
Dick Butkus
Giovanni Ribisi
Chad Allen
Theme music composer
Greg Evigan,
Lenny Macaluso &
Michael Jacobs
Opening theme
"You Can Count on Me,"
performed by Greg Evigan
Ending theme
"You Can Count on Me" (instrumental)
Composer(s)
Don Peake (1987–1988)
Ray Colcord (1988–1990)
Country of origin
United States
Original language(s)
English
No. of seasons
3
No. of episodes
60 (list of episodes)
Production
Executive producer(s)
Michael Jacobs
Camera setup
Videotape; Multi-camera
Running time
22 mins. (approx)
Production company(s)
Michael Jacobs Productions
Tri-Star Television
(1987–1988)
(season 1)
Columbia Pictures Television
(1988–1990)
(seasons 1-3)
Distributor
TeleVentures (1990)
Columbia Pictures Television (1991–1995)
Columbia TriStar Television (1995–2002)
Sony Pictures Television (2002–present)
Release
Original network
NBC
Picture format
480i (SDTV)
Original release
September 20, 1987 (1987-09-20) – April 30, 1990 (1990-04-30)
My Two Dads is an American sitcom that was produced by Michael Jacobs Productions in association with Tri-Star Television (later Columbia Pictures Television) and distributed by TeleVentures. It starred Paul Reiser, Greg Evigan, and Staci Keanan. The series premiered on NBC on September 20, 1987,[1] airing three seasons through April 30, 1990.
Contents
1 Synopsis
2 Cast and characters
2.1 Cast
2.2 Characters
2.3 Night Court crossover
3 Episodes
4 Broadcast
5 Home media
6 Awards
7 See also
8 References
9 External links
Synopsis[edit]
The series begins after Marcy Bradford (played by Emma Samms in the second-season episode "In Her Dreams"), the mother of 12-year-old Nicole Bradford (Keanan), has died. The two men who had competed for Marcy's affections in Key West during the summer of 1974 - Michael Taylor (Reiser), a successful financial advisor; and struggling artist, Joey Harris (Evigan), former friends who grew to hate one another over their mutual interest in the woman - are awarded joint custody of Nicole.[2]
Each week, the mix-ups and trials of two single men raising a teenage daughter provide the stories. Judge Margaret W. Wilbur (Florence Stanley), a family court judge who gave custody of Nicole to Michael and Joey, frequently visits the new family. She bought the building in which Joey lived, so is now the live-in landlord. Michael originally has his own condo uptown, but in Episode 2, Nicole stages a sit in at school because she feels she had no home, and the men decide it is better to all live in one home. They choose Joey's loft.
Nicole's actual paternity is never revealed on the show, but in the episode "Pop, the Question", Michael and Joey — after a falling out — have a DNA test to determine which of them is Nicole's biological father. The test is conducted against Nicole's wishes, she is happier not knowing who her true father is, and she destroys the results before opening them. Michael and Joey later resolve their differences and reconcile. Judge Wilbur looks at the results, but throws them away without revealing them to the audience.
The series comes to an end (in the episode called "See You in September?") when Joey reconnects with a former girlfriend named Sarah, and eventually moves to San Francisco to live with her and her daughter, Grace. He keeps in contact with Nicole, Michael and Judge Wilbur, all of whom remain in New York. Nicole writes a letter to Joey, in which she makes reference to going out to San Francisco to visit, and she ends by saying that no matter what, or where he is, or who he is with, she would always be happy with him as one of her two dads.
Cast and characters[edit]
Cast[edit]
The lead cast of My Two Dads: Paul Reiser and Greg Evigan (top); Staci Keanan (bottom)
Paul Reiser – Michael Taylor
Greg Evigan – Joey Harris
Staci Keanan – Nicole Bradford
Florence Stanley – Judge Margaret W. Wilbur
Giovanni Ribisi (credited as Vonni Ribisi) – Cory Kupkus
Dick Butkus – Ed Klawicki (seasons 1–2)
Chad Allen – Zach Nichols (seasons 2–3)
Characters[edit]
The series regularly featured Giovanni Ribisi and Chad Allen as two boys (Cory Kupkus and Zach Nichols, respectively) who competed for Nicole's affections, just as her two dads had done for her mother's. Florence Stanley appeared as Judge Margaret Wilbur, who was responsible for assigning Nicole's custody, and who regularly looked over the family; she was also their landlady, being the resident owner of the apartment building where the family lived. Amy Hathaway played Nicole's worldly best friend, Shelby Haskell. The cast was rounded out by former football player Dick Butkus, who managed the cafe on the building's first floor. The cafe (Klawicki's) was the second spot in the show around which a plot usually revolved; the first being the family's apartment. In the third season, when Dick Butkus left the series, the diner was then run by cook Julian (Don Yesso), but there was no explanation as to what happened to Ed Klawicki. Ownership of the diner was explained to have been taken over by Judge Wilbur at this point, and it was renamed The Judge's Court Cafe.
Night Court crossover[edit]
My Two Dads had crossovers with another NBC show, Night Court. Judge Margaret W. Wilbur, played by Florence Stanley, appeared on Night Court. In turn, Richard Moll guest starred as his Night Court character Bull Shannon in the My Two Dads episode "Playing with Fire", protecting Judge Wilbur from a recently released criminal that she had sent to prison years before.[3]
Episodes[edit]
Main article: List of My Two Dads episodes
Season
Episodes
Originally aired
First aired
Last aired
1 22 September 20, 1987 (1987-09-20) August 14, 1988 (1988-08-14)
2 16 January 11, 1989 (1989-01-11) July 29, 1989 (1989-07-29)
3 22 September 24, 1989 (1989-09-24) April 30, 1990 (1990-04-30)
Broadcast[edit]
In August 1990 USA Network paid Columbia Pictures Television Distribution about $250,000 an episode, for as many as 15 runs over a three-year period, a price that was the syndication record until Major Dad surpassed that price two years later.[4] Reruns of the show aired on the USA Network through the 1990s and again in the early 2000s.
The show returned in January 2017, airing on both Antenna TV[5] and Up.[citation needed]
Home media[edit]
Shout! Factory (under license from Sony Pictures Home Entertainment) has released the first two seasons of My Two Dads on DVD in Region 1. Season 2 was released as a Shout! Factory Exclusives title, available exclusively through their online store.[6]
Mill Creek Entertainment released a 10 episode best-of set entitled My Two Dads- You Can Count on Me on March 22, 2011. The single disc release features episodes from the first 2 seasons.[7]
DVD name
Ep #
Release date
Special Features
The Complete First Season 22 March 3, 2009 A look back at My Two Dads with Greg Evigan and Staci Keanan.
The Complete Second Season 16 March 16, 2010 N/A
Awards[edit]
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2017) (Learn how and when to remove this template message)
Year
Award
Category
Recipient
Result
Refs
1988 Young Artist Awards The Most Promising New Fall Television Series My Two Dads Won
Best Young Actress Featured, Co-starring, Supporting, Recurring Role in a Comedy or Drama Series or Special Staci Keanan Won
People's Choice Awards Favorite New TV Comedy Program My Two Dads Won
1989 Young Artist Awards Best Young Actor Guest Starring in a Drama or Comedy Series Chad Allen Won
See also[edit]
My Sister Sam (1986)
Three Men and a Baby (1987)
References[edit]
Jump up ^ Howard Rosenberg (September 19, 1987). "The New Fall TV Season : Television Reviews : 'My Two Dads'". Los Angeles Times. Retrieved June 13, 2017.
Jump up ^ "BBC – Comedy Guide – My Two Dads". BBC. January 29, 2005. Archived from the original on January 29, 2005. Retrieved July 6, 2016. CS1 maint: Unfit url (link)
Jump up ^ "Good, "Bad," Hilarious: TV's Best Judges". NBC 7 San Diego. Retrieved 2018-09-20.
Jump up ^ Jim Benson; John Dempsey (October 28, 1992). "'Dad' shipped off to cable". Variety. Retrieved June 13, 2017.
Jump up ^ "Episode". My Two Dads. November 12, 2017. Event occurs at 20:00. Antenna TV. WLUK digital subchannel 11.2. |access-date= requires |url= (help)
Jump up ^ "My Two Dads – Retailers May Spurn Stocking a 2nd Season, but Shout! Keeps Going Anyway". TVShowsonDVD.com. Archived from the original on 2015-02-04.
Jump up ^ "My Two Dads – You Can Count on Me!". 18 January 2011. Retrieved July 6, 2016 – via Amazon.
External links[edit]
My Two Dads on IMDb
My Two Dads at TV.com
My Two Dads at epguides.com
show
v
t
e
People's Choice Awards for Favorite New TV Comedy
The Love Boat (1978)
Mork & Mindy (1979)
Too Close for Comfort (1981)
Private Benjamin (1982)
Cheers (1983)
Webster (1984)
The Cosby Show (1985)
The Golden Girls (1986)
ALF (1987)
A Different World / My Two Dads (1988)
Roseanne (1989)
Doogie Howser, M.D. (1990)
In Living Color / The Simpsons (1991)
Home Improvement (1992)
Mad About You (1993)
Frasier / Grace Under Fire (1994)
Ellen / Friends (1995)
Caroline in the City (1996)
Cosby (1997)
Veronica's Closet / Dharma & Greg (1998)
Jesse / Will & Grace (1999)
Stark Raving Mad (2000)
Ed (2001)
My Wife and Kids (2002)
8 Simple Rules (2003)
Two and a Half Men (2004)
Joey (2005)
My Name Is Earl (2006)
The Class (2007)
Samantha Who? (2008)
Gary Unmarried (2009)
Glee (2010)
$h*! My Dad Says (2011)
2 Broke Girls (2012)
The New Normal (2013)
Super Fun Night (2014)
Jane the Virgin (2015)
Scream Queens (2016)
Man with a Plan (2017)
Retrieved from "https://en.wikipedia.org/w/index.php?title=My_Two_Dads&oldid=860445072"
Categories:
1987 American television series debuts
1990 American television series endings
1980s American sitcoms
1990s American sitcoms
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NBC network shows
Television series by Sony Pictures Television
Television shows set in New York City
Television series created by Michael Jacobs
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