Curve of an arc

Curve of a plane arc

Let us consider a arc parameterized of class $\ mathcal {C} ^2$ in the Euclidean plan directed $\ mathbb {R} ^2$ , supposed regular (of vector never derived no one). One can choose an origin and take for parameter the curvilinear Abscisse $s$ corresponding. It is about a normal parameter setting $\ vec {M} (S)$ (speed is of standard constant equal to 1), which makes it possible to easily define the Repère of Frenet:

origin: the point $\ vec {M} (S)$
first vector: the unit tangent vector $\ vec {T} (S) = \ frac {D \ vec {M}} {ds} (S)$ ,
second vector $\ vec {NR} (S)$ unit normal vector, supplementing the first in a direct orthonormal base

Curve introduced starting from acceleration

The unit tangent vector having a constant standard, one shows that its derivative is always orthogonal for him:

\ forall S \ in I, \ \ green| \ \ vec {T} (S) \ \ green|^2 \ = \ 1 \ quad \ Longrightarrow \ quad \ forall S \ in I, \ \ frac {D \ vec {T}} {ds} (S) \ \ cdot \ \ vec {T} (S) \ = \ 0

There thus exists a function γ, called algebraic curve , such as

\ frac {D \ vec {T} (S)}{ds} \ = \ \ gamma (S) \ \ vec {NR} (S) = \ frac {d^2 \ vec {M}} {ds^2} (S)

In normal parameter setting, the vector Accélération is thus normal (colinéaire with the unit normal vector) and the curve is its coordinate according to the unit normal vector.

In an unspecified parameter setting, the curve can also be obtained starting from speed and from acceleration by the formula

\ gamma (T) = \ frac {\ det \ left (\ frac {D \ vec {M}} {dt} (T), \ frac {d^2 \ vec {M}} {dt^2} (T) \ right)}{\ left \|\ frac {D \ vec {M}} {dt} (T) \ right \|^3}

Curve seen like number of revolutions of the reference mark of Frenet

To completely determine the vectors of the base of Frenet, it is enough to an angular parameter α, giving the angle between T and the first vector of the base fixes (vector of the X-coordinates)

\ alpha = \ widehat {(I, T)}\ qquad \ hbox {or} \ qquad T = \ begin {pmatrix} \ cos \ alpha \ \ \ sin \ alpha \ end {pmatrix} \ qquad NR = \ begin {pmatrix} - \ sin \ alpha \ \ \ cos \ alpha \ end {pmatrix}

This parameter α is thus interpreted as the angle which form the base of Frenet with a fixed direction.

It is then judicious to take the angle α for parameter and to derive the elements from the reference mark of Frénet compared to α. In mathematics at least, it is necessary to validate this operation. It is shown that if arc is birégulier (i.e. the vectors derived first and second are never colinéaires), it is possible to form angle α a regular function reparamétrant the arc, and thus to derive compared to α. One will find the discussion corresponding in the article on the Théorème of raising.

In this parameter setting, the vector NR is obtained either by carrying out a rotation of $\ frac \ pi {2}$ (quarter of turn in the direct direction), or by adding $\ frac \ pi {2}$ to α, or still while deriving compared to α

N = \ begin {pmatrix} - \ sin \ alpha \ \ \ cos \ alpha \ end {pmatrix} = \ frac {D T} {D \ alpha}

\ qquad T=- \ frac {D NR} {D \ alpha} The formulas of derivation of the vectors T and NR compared to α are rigorously identical to the coordinated formulas of derivation of the mobile base in polar, since it is exactly about the same situation.

The curve γ is worth then

\ gamma= \ frac {D \ alpha} {D S}

It is thus the number of revolutions of the base of Frenet compared to a fixed direction (once again, in normal parameter setting).

Geometrical introduction

The dimensional Analyze watch that for a problem of kinematics, γ is homogeneous contrary a length. One thus introduces frequently the radius of curvature (algebraic)

R= \ frac1 \ gamma= \ frac {ds} {D \ alpha}

To include/understand the significance of this size, it is interesting to examine the particular case of a circle

\ begin {boxes} X (T) =r \ cos T \ \ there (T) =r \ sin T \ end {boxes}

The application of the formula of computation of the curve gives R=r .

More generally, for any arc $\ mathcal {C} ^2$ birégulier at the point P of X-coordinate curvilinear S , one shows that there exists a single tangent circle with the curve in P and who “marries this curve the best possible one” in a vicinity of P . This circle is called osculatory Cercle with the curve in P . The ray of this osculatory circle, is equal to the absolute value of the radius of curvature.

Alternative convention: positive curve

Convention chosen previously made curve and radius of curvature of the algebraic quantities.

If the algebraic curve is positive, one falls down exactly on preceding conventions. If not, the curve and the radius of curvature are the absolute values of algebraic conventions, and the unit normal vector is the opposite of algebraic convention. The base of Frenet is not inevitably any more a direct base. One finds however the formula

\ frac {D \ vec T} {ds} = \ gamma \ vec N

With this convention the unit normal vector indicates the direction towards which the concavity of the curve is turned. A defect of convention is that if the curve is null, the unit normal vector is not defined.

Form

\ right) (Divergence of the Gradient standardized) |}

Curve of a left arc

Let us consider a arc parameterized of class $\ mathcal {C} ^2$ in Euclidean space directed $\ mathbb {R} ^3$ , supposed regular; one can again take for parameter the curvilinear Abscisse and define the Repère of Frenet and the curve. However the questions of orientation require to be examined more closely and one cannot define an algebraic coubure.

The unit tangent vector is always defined like the Flight Path Vector in normal parameter setting. Its derivative is nonnull and is orthogonal for him. One can thus pose

\ frac {D \ vec T} {ds} = \ frac {d^2 \ vec M} {ds^2} = \ gamma \ vec N

What defines the curve γ perfectly and the unit normal vector on the condition of imposing the positivity of γ.