Polar equation

The plan is provided with an orthonormal reference mark (O, I, J) . If $f\; \backslash ,$ is a function of $\backslash \; mathbb\; \{R\}$ in $;\; +\; \backslash \; infty\; [\backslash ,$, one can consider the whole of the points M whose them [[coordinated polar] $(\backslash \; rho,\; \backslash \; theta)\; \backslash ,$ checks the following equation

$\backslash \; rho\; =\; F\; (\backslash \; theta)\; \backslash ,$

It is said that the curve in question has for polar equation :

$\backslash \; rho\; =\; F\; (\backslash \; theta)\; \backslash ,$

rem: if $\backslash \; rho\; =\; 0\; \backslash ,$, one will then place the point M at the origin of the reference mark although in any theory, one cannot define the angle $more\; (\backslash \; vec\; \{I\},\; \backslash \; vec\; \{OM\})$.

If a curve has a polar equation and if the interval $\backslash ,$ is included in the field of field of definition, the restriction of the curve with this interval can be traversed while turning in the trigonometrical direction of the angle $\backslash \; theta\_1\; \backslash ,$ with the angle $\backslash \; theta\_2\; \backslash ,$.

Base mobile

One introduces for each value of θ a direct orthonormal base (U (θ), v (θ)) , obtained by rotation of θ starting from the base (I, J) . Thus

$\backslash \; vec\; U\; (\backslash \; theta)\; =\; \backslash \; begin\; \{pmatrix\}\; \backslash \; cos\; \backslash \; theta\; \backslash \; \backslash \; \backslash \; sin\; \backslash \; theta\; \backslash \; end\; \{pmatrix\}$

\ qquad \ vec v (\ theta) = \ begin {pmatrix} - \ sin \ theta \ \ \ cos \ theta \ end {pmatrix} = \ vec U (\ theta+ \ frac \ pi 2)

One will endeavor to express all the geometrical concepts using this base. However as these two vectors depend on θ, one should not forget to derive them too them.

$\backslash \; frac\; \{D\; \backslash \; vec\; U\}\; \{D\; \backslash \; theta\}\; =\; \backslash \; vec\; v\; \backslash \; qquad\; \backslash \; frac\; \{D\; \backslash \; vec\; v\}\; \{D\; \backslash \; theta\}\; =\; \backslash \; vec\; u$

Note: to derive these vectors amounts making them undergo a rotation of π/2.

Vector position

By definition even of the polar Coordinated , $\backslash \; vec\; \{U\}$ is an unit vector colinéaire and of the same direction than $\backslash \; vec\; \{OM\}$ and thus

$\backslash \; vec\; \{OM\}\; =f\; (\backslash \; theta)\; \backslash \; vec\; u$

Coupled with the formulas of derivation of the vectors U and v Ci above, this formula makes it possible to calculate all the objects of differential geometry usual.

Tangent with the curve

If $f\; \backslash ,$ is a derivable function then

$\backslash \; frac\; \{D\; \backslash \; vec\; \{OM\}\}\; \{D\; \backslash \; theta\}\; =\; f\text{'}\; (\backslash \; theta)\; \backslash \; vec\; \{U\}\; (\backslash \; theta)\; +\; F\; (\backslash \; theta)\; \backslash \; vec\; \{v\}\; (\backslash \; theta)$

This vector is a directing vector of the tangent (T) to the curve at the point associated with $\backslash \; theta$. In any rigor there is a particular case, which is treated in the article tangent.

If $\backslash \; alpha$ is the angle which form (T) and (OM), one obtains then the following relation:

$\backslash \; tan\; (\backslash \; alpha)\; =\; |\backslash \; frac\; \{F\; (\backslash \; theta)\}\{f\text{'}\; (\backslash \; theta)\}|$ if $f\text{'}\; (\backslash \; theta)\; \backslash ,$ is nonnull

$\backslash \; alpha\; =\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}$ if not

Curvilinear X-coordinate

If the origin is taken in $\backslash \; theta\_0\; \backslash ,$ then the curvilinear Abscisse, i.e. the algebraic length of the curve between the point $M\; (\backslash \; theta\_0)\; \backslash ,$ and $M\; (\backslash \; theta\_1)\; \backslash ,$ is:

$\backslash \; int\_\; \{\backslash \; theta\_0\}\; ^\; \{\backslash \; theta\_1\}\; \backslash \; sqrt\; \{f\text{'}^2\; (\backslash \; theta)\; +f^2\; (\backslash \; theta)\}D\; \backslash \; theta$

Radius of curvature

The radius of curvature is the ray of the tangent circle with (T) and which approaches “as well as possible” the curve.

If the function $f\; \backslash ,$ is twice derivable, and if $2f\text{'}^2\; (\backslash \; theta)\; +\; f^2\; (\backslash \; theta)\; -\; F\; (\backslash \; theta)\; F$ (\ theta) \, is nonnull, the radius of curvature is:

$\backslash \; frac\; \{(f\text{'}^2\; (\backslash \; theta)\; +\; f^2\; (\backslash \; theta))^\; \{3/2\}\}\; \{2f\text{'}^2\; (\backslash \; theta)\; +\; f^2\; (\backslash \; theta)\; -\; F\; (\backslash \; theta)\; F$ (\ theta)}

Infinite branches

To study the infinite branches one returns in Cartesian coordinates.

Parametric polar equations

If the curve is given by a parametric polar equation R (T), θ (T) , the Flight Path Vectors and acceleration can be calculated in the mobile base. One notes by a point derivation compared to the parameter T

$\backslash \; vec\; V\; =\; \backslash \; dowry\; \{R\}\; \backslash \; vec\; u+\; R\; \backslash \; dowry\; \{\backslash \; theta\}\; \backslash \; vec\; v$

$\backslash \; vec\; has\; =\; (\backslash \; ddot\; \{R\}\; -\; R\; \backslash \; dowry\; \{\backslash \; theta\}\; ^2)\; \backslash \; vec\; u+\; (R\; \backslash \; ddot\; \{\backslash \; theta\}\; +2\; \backslash \; dowry\; \{R\}\; \backslash \; dowry\; \{\backslash \; theta\})\; \backslash \; vec\; v$

See too

• Curved plane
• Coordinated polar

Random links:

Dubious Time | Gordon Brown | Hajimé Saïto | Glen Osborne | Xylokastro | Claude Verret | Newburg,_comté_de_Clearfield,_Pennsylvanie

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