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| [[Image:Evolute_of_an_ellipse.gif||right|thumb|An [[ellipse]] (blue) and its evolute (green). The moving circle is the [[osculating circle]] to the ellipse, whose center is the center of curvature. It is also shown how the tangent line to the evolute is normal to the ellipse, i.e., the evolute is the [[envelope]] of the normals to the ellipse. The evolute of an ellipse is called an [[astroid]].]]
| | It is quite frustrating to struggle for getting pregnant without any results. * One of the best and proven tips for getting pregnant in a natural way is to have sex at least three times a week. To work out when you are ovulating, find days 12 – 16 in your cycle. For both women and men, the anatomical and hormonal factors are not the only ones to be monitored and corrected. The fertility diet program will show you exactly what they are and create one's body in a situation of balance. <br><br>Incorporate nutritious foods into your meal plan and begin exercising to get your body in shape. There is no "right" time to do so - and if it is causing you anxiety and worry, then you should consult a doctor. While toxoplasmosis is generally mild to the mother, it can cause serious complications in the fetus. Another option that is easier to have The Sims 3 ghost baby with, is to just make your own ghost for you to interact with. Antidepressant may helpful to calm the nervous system, long term intake may impair the reproductive organs in fertility function by causing damage to the DNA in their sperm, leading to abnormal sperm production including low sperm count and round head sperm. <br><br>It begins with the formation of the corpus luteum and ends in either pregnancy or luteolysis. ' It detects the LH surge which happens in the middle of Menstrual cycle, about 1 -1. Instead, go for doggy, missionary, spoon, as long as you. If you look at this "infertility" issue from a holistic point of view, as shown in Pregnancy Miracle by Lisa Olson, you'll find that it is nothing but a mere imbalance in your body and that you can still find ways to correct it and produce a child. Snooki's son Lorenzo is going to have a BFF in Baby Mathews, that's for sure. <br><br>The folic acid (vitamin Bc) should be taken by any future mother. All you have to do is enter your information and it will calculate when your fertile days are. Failing to conceive after so many attempts at ferility treatment can takes its toll physically and emotionally. I hope that you have found these tips on getting pregnant:now. In addition, whole grains, nuts, tofu, wheat germ, the consumption is also very useful for these products. <br><br>It is a shame that every woman cannot born a baby because of infertility problems. If a couple decides to go for consultation of medical experts then first thing is to fix up an appointment with infertility specialist. A male condom, on the other hand, is a thin rubber material worn over the penis, to prevent the sperm cells from reaching the woman's egg cells. Do I need to tell you about smoking and its effects on fertility. Having twins has something to do with your family history of having twins.<br><br>If you have any issues concerning the place and how to use pills to get pregnant with twins ([http://www.quebec-1759.info/sitemap/ read here]), you can get hold of us at our webpage. |
| [[Image:Evolute1.gif||right|thumb|The evolute of a curve (in this case, an ellipse) is the envelope of its normals.]]
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| In the [[differential geometry of curves]], the '''evolute''' of a [[curve]] is the [[locus (mathematics)|locus]] of all its [[Osculating circle|centers of curvature]]. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center.<ref>{{cite web|title=Circle Evolute - Wolfram Mathworld|url=http://mathworld.wolfram.com/CircleEvolute.html|accessdate=5 December 2012}}</ref>
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| Equivalently, an evolute is the [[envelope (mathematics)|envelope]] of the [[perpendicular|normals]] to a curve.
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| The evolute of a curve, a surface, or more generally a [[submanifold]], is the [[caustic (mathematics)|caustic]] of the normal map. Let ''M'' be a smooth, regular submanifold in '''R'''<sup>''n''</sup>. For each point ''p'' in ''M'' and each vector '''v''', based at ''p'' and normal to ''M'', we associate the point {{nowrap|1=''p'' + '''v'''}}. This defines a [[Lagrangian map]], called the normal map. The caustic of the normal map is the evolute of ''M''.<ref name="Arnold">{{Cite book|first=V. I.|last=Arnold|first2=A. N.|last2=Varchenko|first3=S. M.|last3=Gusein-Zade|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1|publisher=Birkhäuser|year=1985|isbn=0-8176-3187-9}}</ref> | |
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| ==History==
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| [[Apollonius of Perga|Apollonius]] (c. 200 BC) discussed evolutes in Book V of his ''Conics''. However, [[Christiaan Huygens|Huygens]] is sometimes credited with being the first to study them (1673).
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| ==Definition==
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| Let γ(''s'') be a plane curve, parameterized by its arclength ''s''. The unit tangent vector to the curve is, by virtue of the arclength parameterization,
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| :<math>\mathbf{T}(s) = \gamma'(s)</math>
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| and the unit normal to the curve is the unit vector '''N'''(''s'') perpendicular to '''T'''(''s'') chosen so that the pair ('''T''','''N''') is [[orientation (mathematics)|positively oriented]].
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| The '''[[curvature]]''' ''k'' of γ is defined by means of the equation
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| :<math>\mathbf{T}'(s) = k(s)\mathbf{N}(s)</math>
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| for each ''s'' in the domain of γ. The '''[[osculating circle|radius of curvature]]''' is the reciprocal of curvature:
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| :<math>R(s) = \frac{1}{k(s)}.</math>
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| The radius of curvature at γ(''s'') is, in magnitude, the radius of the circle which forms the best approximation of the curve to second order at the point: that is, it is the radius of the circle making second order [[contact (mathematics)|contact]] with the curve, the [[osculating circle]]. The sign of the radius of curvature indicates the direction in which the osculating circle moves if it is parameterized in the same direction as the curve at the point of contact: it is positive if the circle moves in a counterclockwise sense, and negative otherwise. | |
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| The '''center of curvature''' is the center of the osculating circle. It lies on the normal line through γ(''s'') at a distance of ''R'' from γ(''s''), in the direction determined by the sign of ''k''. In symbols, the center of curvature lies at the point:
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| :<math>E(s) = \gamma(s) + R(s)\mathbf{N}(s) = \gamma(s) + \frac{1}{k(s)}\mathbf{N}(s).</math>
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| As ''s'' varies, the center of curvature defined by this equation traces out a plane curve, the '''evolute''' of γ.
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| ===General parameterizations===
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| If γ(''t'') is given a general parameterization other than the parameterization by arclength, say
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| γ(''t'') = (''x''(''t''), ''y''(''t'')), then the parametric equation of the evolute can be expressed in terms of the radius of curvature ''R'' = 1/''k'' and the [[tangential angle]] φ, which is the angle the tangent to the curve makes with a fixed reference axis [the ''x''-axis]. In terms of ''R'' and φ, the evolute has the parametric equation
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| :<math>(X,Y) = (x,y) + R \mathbf{N} = (x-R\sin\varphi,y+R\cos\varphi)</math>
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| where the unit normal '''N''' = (−sinφ, cosφ) is obtained by rotating the unit tangent '''T''' = (cosφ, sinφ) through an angle of 90°.
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| The equation of the evolute may also be written entirely in terms of ''x'', ''y'' and their derivatives. Since
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| :<math>(\cos \varphi, \sin \varphi) = \frac{(x', y')}{(x'^2+y'^2)^{1/2}}</math> and <math>R = 1/k = \frac{(x'^2+y'^2)^{3/2}}{x'y''-x''y'},</math>
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| ''R'' and φ can be eliminated to obtain for a parametrically defined function:
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| :<math>X[x,y]= x-y'\frac{x'^2+y'^2}{x'y''-x''y'}</math>
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| :<math>Y[x,y]= y+x'\frac{x'^2+y'^2}{x'y''-x''y'}</math>
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| ==Properties==
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| ;Arclength
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| Suppose that the curve γ is parameterized with respect to its arclength ''s''. Then the arclength along the evolute ''E'' from ''s''<sub>1</sub> to ''s''<sub>2</sub> is given by
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| :<math>\int_{s_1}^{s_2}\left|\frac{dR}{ds}\right| ds.</math>
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| Thus, if the curvature of γ is [[monotonic function|strictly monotonic]], then
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| :<math>\int_{s_1}^{s_2}\left|\frac{dR}{ds}\right| ds = |R(s_2)-R(s_1)|.</math>
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| Equivalently, denoting the arclength parameter of the curve ''E'' by σ,
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| :<math>\frac{d\sigma}{ds} = \left|\frac{dR}{ds}\right|.</math>
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| This follows by differentiation of the formula
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| :<math>E(s) = \gamma(s) + R(s)\mathbf{N}(s)</math>
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| and employing the Frenet identity '''N'''′(''s'') = −''k''(''s'')'''T'''(''s''): | |
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| :<math>E'(s) = \gamma'(s) +R'(s)\mathbf{N}(s) - \mathbf{T}(s) = R'(s)\mathbf{N}(s)</math>
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| whence
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| {{NumBlk|:|<math>\frac{dE}{ds} = \frac{dR}{ds}\mathbf{N}\left ( s \right ) </math>|{{EquationRef|1}}}}
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| from which it follows that dσ/d''s'' = |d''R''/d''s''|, as claimed.
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| ;Unit tangent vector
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| Another consequence of ({{EquationNote|1}}) is that the tangent vector to the evolute ''E'' at ''E''(''s'') is normal to the curve γ at γ(''s'').
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| ;Curvature
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| The curvature of the evolute ''E'' is obtained by differentiating ''E'' twice with respect to its arclength parameter σ. Since dσ/d''s'' = |d''R''/d''s''|, it follows from ({{EquationNote|1}}) that
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| :<math> | |
| \frac{dE}{d\sigma} = \left.\frac{dE}{ds}\right/\frac{d\sigma}{ds} = \pm\mathbf{N}</math>
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| where the sign is that of d''R''/d''s''. Differentiating a second time, and using the Frenet equation ''N''′(''s'') = −''k''(''s'')'''T'''(''s'') gives
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| :<math>\frac{d^2E}{d\sigma^2} = \pm\left.\frac{d\mathbf{N}}{ds}\right/\frac{d\sigma}{ds} = -\frac{1}{RR'}\frac{dE}{d\sigma}.</math>
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| As a consequence, the curvature of ''E'' is
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| :<math>k_E = -\frac{1}{RR'}</math>
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| where ''R'' is the (signed) radius of curvature and the prime denotes the derivative with respect to ''s''.
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| ;Relation with involute
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| With an appropriate starting point, the involute of the evolute of a curve is the curve itself.
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| ;Intrinsic equation
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| If φ can be expressed as a function of ''R'', say φ = ''g''(''R''), then the [[Whewell equation]] for the evolute is Φ = ''g''(''R'') + π/2, where Φ is the tangential angle of the evolute and we take ''R'' as arclength along the evolute. From this we can derive the [[Cesàro equation]] as Κ = ''g''′(''R''), where Κ is the curvature of the evolute. | |
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| ===Relationship between a curve and its evolute===
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| [[Image:Evolute and parallel.gif|right|thumb|240px|An ellipse (red), its evolute (blue) and some parallel curves. Note how the parallel curves touch the evolute where they have cusps]]
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| By the above discussion, the derivative of (''X'', ''Y'') vanishes when d''R''/d''s'' = 0, so the evolute will have a [[cusp (singularity)|cusp]] when the curve has a [[Vertex (curve)|vertex]], that is when the curvature has a local maximum or minimum. At a point of inflection of the original curve the radius of curvature becomes infinite and so (''X'', ''Y'') will become infinite, often this will result in the evolute having an [[asymptote]]. Similarly, when the original curve has a cusp where the radius of curvature is 0 then the evolute will touch the original curve.
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| This can be seen in the figure to the right: the blue curve is the evolute of all the other curves. The cusp in the blue curve corresponds to a vertex in the other curves. The cusps in the green curve are on the evolute. Curves with the same evolute are [[Parallel curve|parallel]].
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| ==Radial curve==
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| A curve with a similar definition is the '''radial''' of a given curve. For each point on the curve take the vector from the point to the center of curvature and translate it so that it begins at the origin. Then the locus of points at the end of such vectors is called the radial of the curve. The equation for the radial is obtained by removing the ''x'' and ''y'' terms from the equation of the evolute. This produces (''X'', ''Y'') = (−''R'' sinφ, ''R'' cosφ) or
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| :<math>(X, Y)= \left(-y'\frac{x'^2+y'^2}{x'y''-x''y'}, x'\frac{x'^2+y'^2}{x'y''-x''y'}\right).</math>
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| ==Examples==
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| * The evolute of a [[parabola]] is a [[semicubical parabola]]. The cusp of the latter curve is the center of curvature of the parabola at its vertex.
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| * The evolute of a [[logarithmic spiral]] is a [[Congruence (geometry)|congruent]] spiral.
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| * The evolute of a [[cycloid]] is a congruent cycloid.
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| <!-- Animations don't work, need to reduce frame rate or resize original
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| {{Image gallery
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| |title=Examples of evolutes
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| |lines=3
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| |Evolute of tractrix.gif || The evolute of a [[tractrix]] is a [[catenary]].
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| |Evolute of parabola.gif || The evolute of a [[parabola]] is a [[semicubical parabola]] .
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| |Evolute of cycloid.gif || The evolute of a [[cycloid]] is an equal cycloid.
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| |Evolute of ellipse.gif || The evolute of an [[ellipse]] is a stretched [[astroid]] or a [[Lamé curve]].
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| |Evolute of epicycloid.gif || The evolute of an [[epicycloid]] is a smaller epicycloid.
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| |Evolute of astroid.gif || The evolute of an [[astroid]] is an enlarged astroid (twice as large).
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| }} -->
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| ==References==
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| {{reflist}}
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| * {{MathWorld|title=Evolute|urlname=Evolute}}
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| * {{springer|title=Evolute|id=E/e036670|last=Sokolov|first=D.D.}}
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| * Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Evolutes." pp. 86ff
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| * [http://www.2dcurves.com/derived/curvature.html#evolute Evolute on 2d curves.]
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| {{Differential transforms of plane curves}}
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| [[Category:Differential geometry]]
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| [[Category:Curves]]
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It is quite frustrating to struggle for getting pregnant without any results. * One of the best and proven tips for getting pregnant in a natural way is to have sex at least three times a week. To work out when you are ovulating, find days 12 – 16 in your cycle. For both women and men, the anatomical and hormonal factors are not the only ones to be monitored and corrected. The fertility diet program will show you exactly what they are and create one's body in a situation of balance.
Incorporate nutritious foods into your meal plan and begin exercising to get your body in shape. There is no "right" time to do so - and if it is causing you anxiety and worry, then you should consult a doctor. While toxoplasmosis is generally mild to the mother, it can cause serious complications in the fetus. Another option that is easier to have The Sims 3 ghost baby with, is to just make your own ghost for you to interact with. Antidepressant may helpful to calm the nervous system, long term intake may impair the reproductive organs in fertility function by causing damage to the DNA in their sperm, leading to abnormal sperm production including low sperm count and round head sperm.
It begins with the formation of the corpus luteum and ends in either pregnancy or luteolysis. ' It detects the LH surge which happens in the middle of Menstrual cycle, about 1 -1. Instead, go for doggy, missionary, spoon, as long as you. If you look at this "infertility" issue from a holistic point of view, as shown in Pregnancy Miracle by Lisa Olson, you'll find that it is nothing but a mere imbalance in your body and that you can still find ways to correct it and produce a child. Snooki's son Lorenzo is going to have a BFF in Baby Mathews, that's for sure.
The folic acid (vitamin Bc) should be taken by any future mother. All you have to do is enter your information and it will calculate when your fertile days are. Failing to conceive after so many attempts at ferility treatment can takes its toll physically and emotionally. I hope that you have found these tips on getting pregnant:now. In addition, whole grains, nuts, tofu, wheat germ, the consumption is also very useful for these products.
It is a shame that every woman cannot born a baby because of infertility problems. If a couple decides to go for consultation of medical experts then first thing is to fix up an appointment with infertility specialist. A male condom, on the other hand, is a thin rubber material worn over the penis, to prevent the sperm cells from reaching the woman's egg cells. Do I need to tell you about smoking and its effects on fertility. Having twins has something to do with your family history of having twins.
If you have any issues concerning the place and how to use pills to get pregnant with twins (read here), you can get hold of us at our webpage.