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63-472: A helix is a spiral-like space curve. Helix may also refer to: Helix (mythology) Helix The pitch of a helix is the height of one complete helix turn , measured parallel to the axis of the helix. A double helix consists of two (typically congruent ) helices with the same axis, differing by a translation along the axis. A circular helix (i.e. one with constant radius) has constant band curvature and constant torsion . The slope of

126-584: A a 2 + b 2 cos ⁡ s a 2 + b 2 i + − a a 2 + b 2 sin ⁡ s a 2 + b 2 j + 0 k {\displaystyle {\frac {d\mathbf {T} }{ds}}=\kappa \mathbf {N} ={\frac {-a}{a^{2}+b^{2}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {-a}{a^{2}+b^{2}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} } Its curvature

189-407: A b | f ′ ( t ) |   d t . {\displaystyle L(f)=\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 is proved by the following steps: With

252-899: A b | g ′ ( φ ( t ) ) φ ′ ( t ) |   d t = ∫ a b | g ′ ( φ ( t ) ) | φ ′ ( t )   d t in the case  φ  is 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 {\text{in

315-976: A 2 + b 2 | a | = ( − a sin ⁡ t ) 2 + ( a cos ⁡ t ) 2 = a s ( t ) = ∫ 0 t a 2 + b 2 d τ = a 2 + b 2 t {\displaystyle {\begin{aligned}\mathbf {r} &=a\cos t\mathbf {i} +a\sin t\mathbf {j} +bt\mathbf {k} \\[6px]\mathbf {v} &=-a\sin t\mathbf {i} +a\cos t\mathbf {j} +b\mathbf {k} \\[6px]\mathbf {a} &=-a\cos t\mathbf {i} -a\sin t\mathbf {j} +0\mathbf {k} \\[6px]|\mathbf {v} |&={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}+b^{2}}}={\sqrt {a^{2}+b^{2}}}\\[6px]|\mathbf {a} |&={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}}}=a\\[6px]s(t)&=\int _{0}^{t}{\sqrt {a^{2}+b^{2}}}d\tau ={\sqrt {a^{2}+b^{2}}}t\end{aligned}}} So

378-442: A 2 + b 2 i − b cos ⁡ s a 2 + b 2 j + a k ) d B d s = 1 a 2 + b 2 ( b cos ⁡ s a 2 + b 2 i + b sin ⁡ s

441-430: A 2 + b 2 j + 0 k {\displaystyle \mathbf {N} =-\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} } The binormal vector is B = T × N = 1 a 2 + b 2 ( b sin ⁡ s

504-565: A 2 + b 2 j + 0 k ) {\displaystyle {\begin{aligned}\mathbf {B} =\mathbf {T} \times \mathbf {N} &={\frac {1}{\sqrt {a^{2}+b^{2}}}}\left(b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +a\mathbf {k} \right)\\[12px]{\frac {d\mathbf {B} }{ds}}&={\frac {1}{a^{2}+b^{2}}}\left(b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} \right)\end{aligned}}} Its torsion

567-455: A 2 + b 2 k {\displaystyle {\frac {d\mathbf {r} }{ds}}=\mathbf {T} ={\frac {-a}{\sqrt {a^{2}+b^{2}}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {a}{\sqrt {a^{2}+b^{2}}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {b}{\sqrt {a^{2}+b^{2}}}}\mathbf {k} } The normal vector is d T d s = κ N = −

630-564: A ) {\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)} , and N > ( b − a ) / δ ( ε ) {\displaystyle N>(b-a)/\delta (\varepsilon )} . In the limit N → ∞ , {\displaystyle N\to \infty ,} δ ( ε ) → 0 {\displaystyle \delta (\varepsilon )\to 0} so ε → 0 {\displaystyle \varepsilon \to 0} thus

693-1301: A ) / δ ( ε ) {\textstyle N>(b-a)/\delta (\varepsilon )} so that Δ t < δ ( ε ) {\displaystyle \Delta t<\delta (\varepsilon )} , it becomes Δ t ∑ i = 1 N ( | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | − | f ′ ( t i ) | ) < ε N Δ t {\displaystyle \Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\left|f'(t_{i})\right|\right)<\varepsilon N\Delta t} with | f ′ ( t i ) | = ∫ 0 1 | f ′ ( t i ) | d θ {\displaystyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta } , ε N Δ t = ε ( b −

SECTION 10

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756-509: A + i ( b − a ) / N = a + i Δ t {\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t} with Δ t = b − a N = t i − t i − 1 {\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}} for i = 0 , 1 , … , N . {\displaystyle i=0,1,\dotsc ,N.} This definition

819-508: 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 =

882-661: A , b ] → R n {\displaystyle f:[a,b]\to \mathbb {R} ^{n}} on [ a , b ] {\displaystyle [a,b]} is always finite, i.e., 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

945-414: A , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve defined by f {\displaystyle f} can be defined as the limit of the sum of linear segment lengths for a regular partition of [

1008-459: A sin ⁡ t i + a cos ⁡ t j + b k a = − a cos ⁡ t i − a sin ⁡ t j + 0 k | v | = ( − a sin ⁡ t ) 2 + ( a cos ⁡ t ) 2 + b 2 =

1071-510: A / b ⁠ (or pitch 2 πb ) expressed in Cartesian coordinates as the parametric equation has an arc length of a curvature of and a torsion of A helix has constant non-zero curvature and torsion. A helix is the vector-valued function r = a cos ⁡ t i + a sin ⁡ t j + b t k v = −

1134-566: A general helix or cylindrical helix if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant. A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the moving frame of a general helix. For more general helix-like space curves can be found, see space spiral ; e.g., spherical spiral . Helices can be either right-handed or left-handed. With

1197-406: A rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} , then

1260-399: A circular helix is commonly defined as the ratio of the circumference of the circular cylinder that it spirals around, and its pitch (the height of one complete helix turn). A conic helix , also known as a conic spiral , may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis. A curve is called

1323-407: A closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals . 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

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1386-817: 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 .} The second expression

1449-403: 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

1512-788: 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

1575-696: 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

1638-440: A curve length determination by approximating the curve as connected (straight) line segments is called rectification of a curve. 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

1701-608: A helix can be reparameterized as a function of s , which must be unit-speed: r ( s ) = a cos ⁡ s a 2 + b 2 i + a sin ⁡ s a 2 + b 2 j + b s a 2 + b 2 k {\displaystyle \mathbf {r} (s)=a\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +a\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {bs}{\sqrt {a^{2}+b^{2}}}}\mathbf {k} } The unit tangent vector

1764-423: A helix having an angle equal to that between the chord of the element and a plane perpendicular to the propeller axis; see also: pitch angle (aviation) . Arc length Arc length is the distance between two points along a section of a curve . Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification . For

1827-490: A helix is a curve in 3- dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; perhaps the simplest equations for one is As the parameter t increases, the point ( x ( t ) , y ( t ) , z ( t ) ) {\displaystyle (x(t),y(t),z(t))} traces a right-handed helix of pitch 2 π (or slope 1) and radius 1 about

1890-434: A quarter of the circle. Since d y / d x = − x / 1 − x 2 {\textstyle dy/dx=-x{\big /}{\sqrt {1-x^{2}}}} and 1 + ( d y / d x ) 2 = 1 / ( 1 − x 2 ) , {\displaystyle 1+(dy/dx)^{2}=1{\big /}\left(1-x^{2}\right),}

1953-481: 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 \left[-{\sqrt {2}}/2,{\sqrt {2}}/2\right]} corresponds to

Helix (disambiguation) - Misplaced Pages Continue

2016-397: Is d r d s = T = − a a 2 + b 2 sin ⁡ s a 2 + b 2 i + a a 2 + b 2 cos ⁡ s a 2 + b 2 j + b

2079-711: 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 \left(\mathbf {x} _{u}u'+\mathbf {x} _{v}v'\right)\cdot (\mathbf {x} _{u}u'+\mathbf {x} _{v}v')=g_{11}\left(u'\right)^{2}+2g_{12}u'v'+g_{22}\left(v'\right)^{2}} (where g i j {\displaystyle g_{ij}}

2142-543: Is | ( x ∘ C ) ′ ( t ) | . {\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.} 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

2205-471: Is κ = | d T d s | = | a | a 2 + b 2 {\displaystyle \kappa =\left|{\frac {d\mathbf {T} }{ds}}\right|={\frac {|a|}{a^{2}+b^{2}}}} . The unit normal vector is N = − cos ⁡ s a 2 + b 2 i − sin ⁡ s

2268-735: Is τ = | d B d s | = b a 2 + b 2 . {\displaystyle \tau =\left|{\frac {d\mathbf {B} }{ds}}\right|={\frac {b}{a^{2}+b^{2}}}.} An example of a double helix in molecular biology is the nucleic acid double helix . An example of a conic helix is the Corkscrew roller coaster at Cedar Point amusement park. Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions . Most hardware screw threads are right-handed helices. The alpha helix in biology as well as

2331-620: 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 Euclidean distance of each infinitesimal segment of the arc can be given by: d x 2 + d y 2 = 1 + ( d y d x ) 2 d x . {\displaystyle {\sqrt {dx^{2}+dy^{2}}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.} The arc length

2394-463: Is an upper bound on the length of all polygonal approximations (rectification). These curves are called rectifiable and the arc length is defined as the number L {\displaystyle L} . A signed arc length can be defined to convey a sense of orientation or "direction" with respect to a reference point taken as origin in the curve (see also: curve orientation and signed distance ). Let f : [

2457-430: 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 = ∫

2520-597: Is equivalent to the standard definition of arc length as an integral: L ( f ) = 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 = ∫

2583-420: Is for a polar graph r = r ( θ ) {\displaystyle r=r(\theta )} parameterized by t = θ {\displaystyle t=\theta } . Now let C ( t ) = ( r ( t ) , θ ( t ) , ϕ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t),\phi (t))} be

Helix (disambiguation) - Misplaced Pages Continue

2646-426: Is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations. 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

2709-669: 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}\left(u^{a}\right)'\left(u^{b}\right)'\,}}} (where u 1 = u {\displaystyle u^{1}=u} and u 2 = v {\displaystyle u^{2}=v} ). Let C ( t ) = ( r ( t ) , θ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t))} be

2772-455: Is then 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

2835-423: The A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed. In music , pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths , so as to represent octave equivalency . In aviation, geometric pitch is the distance an element of an airplane propeller would advance in one revolution if it were moving along

2898-533: The Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summation of the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance . If the curve is not already a polygonal path, then using a progressively larger number of line segments of smaller lengths will result in better curve length approximations. Such

2961-410: 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].} This definition as the supremum of

3024-458: The z -axis, in a right-handed coordinate system. In cylindrical coordinates ( r , θ , h ) , the same helix is parametrised by: A circular helix of radius a and slope ⁠ a / b ⁠ (or pitch 2 πb ) is described by the following parametrisation: Another way of mathematically constructing a helix is to plot the complex-valued function e as a function of the real number x (see Euler's formula ). The value of x and

3087-2618: The above step result, it becomes ∑ i = 1 N | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | Δ t − ∑ i = 1 N | f ′ ( t i ) | Δ t . {\displaystyle \sum _{i=1}^{N}\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.} Terms are rearranged so that it becomes Δ t ∑ i = 1 N ( | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | − ∫ 0 1 | f ′ ( t i ) | d θ ) ≦ Δ t ∑ i = 1 N ( ∫ 0 1 | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) |   d θ − ∫ 0 1 | f ′ ( t i ) | d θ ) = Δ t ∑ i = 1 N ∫ 0 1 | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) | − | f ′ ( t i ) |   d θ {\displaystyle {\begin{aligned}&\Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad \leqq \Delta t\sum _{i=1}^{N}\left(\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|\ d\theta -\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad =\Delta t\sum _{i=1}^{N}\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\ d\theta \end{aligned}}} where in

3150-605: The all possible partition sums 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}}

3213-792: The arc length integral is | ( x ∘ C ) ′ ( t ) | . {\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.} 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

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3276-432: The case }}\varphi {\text{ is non-decreasing}}\\&=\int _{c}^{d}{\Big |}g'(u){\Big |}\ du\quad {\text{using integration by substitution}}\\&=L(g).\end{aligned}}} 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}

3339-686: 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

3402-426: The curve is rectifiable (i.e., it has a finite length). The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a finite number of points on the curve using (straight) line segments to create a polygonal path . Since it is straightforward to calculate the length of each linear segment (using

3465-665: The left side of < {\displaystyle <} approaches 0 {\displaystyle 0} . In other words, ∑ 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}\left|f'(t_{i})\right|\Delta t} in this limit, and

3528-806: The leftmost side | f ′ ( t i ) | = ∫ 0 1 | f ′ ( t i ) | d θ {\textstyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta } is used. By | | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) | − | f ′ ( t i ) | | < ε {\textstyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon } for N > ( b −

3591-425: The length of a quarter of the unit circle is ∫ − 2 / 2 2 / 2 d x 1 − x 2 . {\displaystyle \int _{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}{\frac {dx}{\sqrt {1-x^{2}}}}\,.} The 15-point Gauss–Kronrod rule estimate for this integral of 1.570 796 326 808 177 differs from

3654-427: The line of sight along the helix's axis, if a clockwise screwing motion moves the helix away from the observer, then it is called a right-handed helix; if towards the observer, then it is a left-handed helix. Handedness (or chirality ) is a property of the helix, not of the perspective: a right-handed helix cannot be turned to look like a left-handed one unless it is viewed in a mirror, and vice versa. In mathematics ,

3717-419: The real and imaginary parts of the function value give this plot three real dimensions. Except for rotations , translations , and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the x , y or z components. A circular helix of radius a and slope ⁠

3780-452: The right side of this equality is just the Riemann integral of | f ′ ( t ) | {\displaystyle \left|f'(t)\right|} on [ a , b ] . {\displaystyle [a,b].} This definition of arc length shows that the length of a curve represented by a continuously differentiable function f : [

3843-937: 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 \left(\mathbf {x_{r}} \cdot \mathbf {x_{r}} \right)\left(r'\right)^{2}+2\left(\mathbf {x} _{r}\cdot \mathbf {x} _{\theta }\right)r'\theta '+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}.} So for

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3906-1110: 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 \left(\mathbf {x} _{r}\cdot \mathbf {x} _{r}\right)\left(r'^{2}\right)+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}+\left(\mathbf {x} _{\phi }\cdot \mathbf {x} _{\phi }\right)\left(\phi '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}+r^{2}\sin ^{2}\theta \left(\phi '\right)^{2}.} So for

3969-446: The true length of arcsin ⁡ x | − 2 / 2 2 / 2 = π 2 {\displaystyle \arcsin x{\bigg |}_{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}={\frac {\pi }{2}}} by 1.3 × 10 and the 16-point Gaussian quadrature rule estimate of 1.570 796 326 794 727 differs from the true length by only 1.7 × 10 . This means it

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