183 (number)

This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection.

Special affine curvature, also known as the equi-affine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that preserves area). The curves of constant equi-affine curvature k are precisely all non-singular plane conics. Those with k > 0 are ellipses, those with k = 0 are parabolas, and those with k < 0 are hyperbolas.

The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle, the unique circle making second order contact (having three point contact) with the curve at the point. In the same way, the special affine curvature of a curve at a point P is the special affine curvature of its hyperosculating conic, which is the unique conic making fourth order contact (having five point contact) with the curve at P. In other words it is the limiting position of the (unique) conic through P and four points P₁, P₂, P₃, P₄ on the curve, as each of the points approaches P:

$P_1,P_2,P_3,P_4\to P.$

In some contexts, the affine curvature refers to a differential invariant κ of the general affine group, which may readily obtained from the special affine curvature k by κ = k^−3/2dk/ds, where s is the special affine arc length. Where the general affine group is not used, the special affine curvature k is sometimes also called the affine curvature Template:Harv.

Formal definition

Special affine arclength

To define the special affine curvature, it is necessary first to define the special affine arclength (also called the equi-affine arclength). Consider an affine plane curve $\beta (t)$. Choose co-ordinates for the affine plane such that the area of the parallelogram spanned by two vectors $a=(a_1,a_2)$ and $b=(b_1,b_2)$ is given by the determinant

$\det \left[ab\right]=a_1{b}_2-a_2{b}_1.$

In particular, the determinant

$\det \left[\begin{array}{cc}\frac{d\beta }{dt}& \frac{d^2\beta }{d{t}^2}\end{array}\right]$

is a well-defined invariant of the special affine group, and gives the signed area of the parallelogram spanned by the velocity and acceleration of the curve β. Consider a reparameterization of the curve β, say with a new parameter s related to t by means of a regular reparameterization s = s(t). This determinant undergoes then a transformation of the following sort, by the chain rule:

$\begin{array}{rl}\det \left[\begin{array}{cc}\frac{d\beta }{dt}& \frac{d^2\beta }{d{t}^2}\end{array}\right]& =\det \left[\begin{array}{cc}\frac{d\beta }{ds}\frac{ds}{dt}& (\frac{d^2\beta }{d{s}^2}{\left(\frac{ds}{dt}\right)}^2+\frac{d\beta }{ds}\frac{d^{2}s}{d{t}^2})\end{array}\right]\\ & ={\left(\frac{ds}{dt}\right)}^3\det \left[\begin{array}{cc}\frac{d\beta }{ds}& \frac{d^2\beta }{d{s}^2}\end{array}\right].\end{array}$

The reparameterization can be chosen so that

$\det \left[\begin{array}{cc}\frac{d\beta }{ds}& \frac{d^2\beta }{d{s}^2}\end{array}\right]=1$

provided the velocity and acceleration, dβ/dt and d²β/dt² are linearly independent. Existence and uniqueness of such a parameterization follows by integration:

$s(t)=\underset{a}{\overset{t}{\int }}\sqrt[3]{\det \left[\begin{array}{cc}\frac{d\beta }{dt}& \frac{d^2\beta }{d{t}^2}\end{array}\right]}dt.$

This integral is called the special affine arclength, and a curve carrying this parameterization is said to be parameterized with respect to its special affine arclength.

Special affine curvature

Suppose that β(s) is a curve parameterized with its special affine arclength. Then the special affine curvature (or equi-affine curvature) is given by

$k(s)=\det \left[\begin{array}{cc}{\beta }^{″}(s)& {\beta }^{‴}(s)\end{array}\right].$

Here β′ denotes the derivative of β with respect to s.

More generally (Template:Harvnb; Template:Harvnb), for a plane curve with arbitrary parameterization

$t\mapsto (x(t),y(t)),$

the special affine curvature is:

$\begin{array}{rl}k(t)& =\frac{x^{″}{y}^{‴}-x^{‴}{y}^{″}}{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime }{)}^{5/3}}-\frac{1}{2}{\left[{\frac{1}{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime }{)}^{2/3}}}^{″}\right]}^{″}\\ & =\frac{4(x^{″}{y}^{‴}-x^{‴}{y}^{″})+(x^{\prime }{y}^{⁗}-x^{⁗}{y}^{\prime })}{3(x^{\prime }{y}^{″}-x^{″}{y}^{\prime }{)}^{5/3}}-\frac{5}{9}\frac{(x^{\prime }{y}^{‴}-x^{‴}{y}^{\prime }{)}^2}{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime }{)}^{8/3}}\end{array}$

provided the first and second derivatives of the curve are linearly independent. In the special case of a graph y = y(x), these formulas reduce to

$k=-\frac{1}{2}{\left({\frac{1}{(y^{″}{)}^{2/3}}}^{″}\right)}^{″}=\frac{1}{3}\frac{y^{⁗}}{(y^{″}{)}^{5/3}}-\frac{5}{9}\frac{(y^{‴}{)}^2}{(y^{″}{)}^{8/3}}$

where the prime denotes differentiation with respect to x (Template:Harvnb; Template:Harvnb).

Affine curvature

Suppose as above that β(s) is a curve parameterized by special affine arclength. There are a pair of invariants of the curve that are invariant under the full general affine group Template:Harv — the group of all affine motions of the plane, not just those that are area-preserving. The first of these is

$\sigma =\int \sqrt{k(s)}ds,$

sometimes called the affine arclength (although this risks confusion with the special affine arclength described above). The second is referred to as the affine curvature:

$\kappa =\frac{1}{k^{3/2}}\frac{dk}{ds}.$

Conics

Suppose that β(s) is a curve parameterized by special affine arclength with constant affine curvature k. Let

$C_{\beta }(s)=\left[\begin{array}{cc}{\beta }^{\prime }(s)& {\beta }^{″}(s)\end{array}\right].$

Note that det C_β, since β is assumed to carry the special affine arclength parameterization, and that

$k=\det ({C_{\beta }}^{\prime }).$

It follows from the form of C_β that

${C_{\beta }}^{\prime }=C_{\beta }\left[\begin{array}{cc}0& -k\\ 1& 0\end{array}\right].$

By applying a suitable special affine transformation, we can arrange that C_β(0) = I is the identity matrix. Since k is constant, it follows that C_β is given by the matrix exponential

$\begin{array}{rl}{C}_{\beta }(s)& =\exp \{s\cdot \left[\begin{array}{cc}0& -k\\ 1& 0\end{array}\right]\}\\ & =\left[\begin{array}{cc}\cos \sqrt{k}s& \sqrt{k}\sin \sqrt{k}s\\ -\frac{1}{\sqrt{k}}\sin \sqrt{k}s& \cos \sqrt{k}s\end{array}\right].\end{array}$

The three cases are now as follows.

k = 0

If the curvature vanishes identically, then upon passing to a limit,

$C_{\beta }(s)=\left[\begin{array}{cc}1& 0\\ s& 1\end{array}\right]$

so β'(s) = (1,s), and so integration gives

$\beta (s)=(s,s^2/2)$

up to an overall constant translation, which is the special affine parameterization of the parabola y = x²/2.

k > 0

If the special affine curvature is positive, then it follows that

${\beta }^{\prime }(s)=(\cos \sqrt{k}s,\frac{1}{\sqrt{k}}\sin \sqrt{k}s)$

so that

$\beta (s)=(\frac{1}{\sqrt{k}}\sin \sqrt{k}s,-\frac{1}{k}\cos \sqrt{k}s)$

up to a translation, which is the special affine parameterization of the ellipse kx² + k²y² = 1.

k < 0

If k is negative, then the trigonometric functions in C_β give way to hyperbolic functions:

$C_{\beta }(s)=\left[\begin{array}{cc}\cosh \sqrt{\left|k\right|}s& \sqrt{\left|k\right|}\sinh \sqrt{\left|k\right|}s\\ \frac{1}{\sqrt{\left|k\right|}}\sinh \sqrt{\left|k\right|}s& \cosh \sqrt{\left|k\right|}s\end{array}\right].$

Thus

$\beta (s)=(\frac{1}{\sqrt{\left|k\right|}}\sinh \sqrt{\left|k\right|}s,\frac{1}{\left|k\right|}\cosh \sqrt{\left|k\right|}s)$

up to a translation, which is the special affine parameterization of the hyperbola

$-\left|k\right|x^2+|k{|}^2{y}^2=1.$

Characterization up to affine congruence

The special affine curvature of an immersed curve is the only (local) invariant of the curve in the following sense:

• If two curves have the same special affine curvature at every point, then one curve is obtained from the other by means of a special affine transformation.

In fact, a slightly stronger statement holds:

• Given any continuous function k : [a,b] → R, there exists a curve β whose first and second derivatives are linearly independent, such that the special affine curvature of β relative to the special affine parameterization is equal to the given function k. The curve β is uniquely determined up to a special affine transformation.

This is analogous to the fundamental theorem of curves in the classical Euclidean differential geometry of curves, in which the complete classification of plane curves up to Euclidean motion depends on a single function κ, the curvature of the curve. It follows essentially by applying the Picard–Lindelöf theorem to the system

${C_{\beta }}^{\prime }=C_{\beta }\left[\begin{array}{cc}0& -k\\ 1& 0\end{array}\right]$

where C_β = [β′ β′′]. An alternative approach, rooted in the theory of moving frames, is to apply the existence of a primitive for the Darboux derivative.

Derivation of the curvature by affine invariance

The special affine curvature can be derived explicitly by techniques of invariant theory. For simplicity, suppose that an affine plane curve is given in the form of a graph y = y(x). The special affine group acts on the Cartesian plane via transformations of the form

$\begin{array}{rl}x& \mapsto ax+by+\alpha \\ y& \mapsto cx+dy+\beta ,\end{array}$

with ad − bc = 1. The following vector fields span the Lie algebra of infinitesimal generators of the special affine group:

$T_1={\partial }_x,T_2=\partial y$

$X_1=x{\partial }_y,X_2=y{\partial }_x,H=x{\partial }_x-y{\partial }_y.$

An affine transformation not only acts on points, but also on the tangent lines to graphs of the form y = y(x). That is, there is an action of the special affine group on triples of coordinates

$(x,y,y^{\prime }).$

The group action is generated by vector fields

$T_1^{(1)},T_2^{(1)},X_1^{(1)},X_2^{(1)},H^{(1)}$

defined on the space of three variables (x,y,y′). These vector fields can be determined by the following two requirements:

• Under the projection onto the xy-plane, they must to project to the corresponding original generators of the action $T_1,T_2,X_1,X_2,H$, respectively.
• The vectors must preserve up to scale the contact structure of the jet space

${\theta }_1=dy-y^{\prime }dx.$

Concretely, this means that the generators X⁽¹⁾ must satisfy

$L_{X^{(1)}}{\theta }_1\equiv 0(\mathrm{mod}{\theta }_1)$

where L is the Lie derivative.

Similarly, the action of the group can be extended to the space of any number of derivatives

$(x,y,y^{\prime },y^{″},\dots ,y^{(k)}).$

The prolonged vector fields generating the action of the special affine group must then inductively satisfy, for each generator X ∈ {T₁,T₂,X₁,X₂,H}:

• The projection of X^(k) onto the space of variables (x,y,y′,…,y^(k−1)) is X^(k−1).
• X^(k) preserves the contact ideal:

$L_{X^{(k)}}{\theta }_k\equiv 0(\mathrm{mod}{\theta }_1,\dots ,{\theta }_k)$

where

${\theta }_i=d{y}^{(i-1)}-y^{(i)}dx.$

Carrying out the inductive construction up to order 4 gives

$T_1^{(4)}={\partial }_x,T_2^{(4)}={\partial }_y$

$X_1^{(4)}=x{\partial }_y+{\partial }_{y^{\prime }}$

$\begin{array}{rl}{X}_2^{(4)}=y{\partial }_x& -y{\text{'}}^2{\partial }_{y^{\prime }}-3y^{\prime }{y}^{″}{\partial }_{y^{″}}-(3y^{\prime }{\text{'}}^2+4y^{\prime }{y}^{‴}){\partial }_{y^{‴}}\\ & -(10y^{″}{y}^{‴}+5y^{\prime }{y}^{⁗}){\partial }_{y^{⁗}}\end{array}$

$H^{(4)}=x{\partial }_x-y{\partial }_y-2y^{\prime }{\partial }_{y^{\prime }}-3y^{″}{\partial }_{y^{″}}-4y^{‴}{\partial }_{y^{‴}}-5y^{⁗}{\partial }_{y^{⁗}}.$

The special affine curvature

$k=\frac{1}{3}\frac{y^{⁗}}{(y^{″}{)}^{5/3}}-\frac{5}{9}\frac{(y^{‴}{)}^2}{(y^{″}{)}^{8/3}}$

does not depend explicitly on x, y, or y′, and so satisfies

$T_1^{(4)}k=T_2^{(4)}k=X_1^{(4)}k=0.$

The vector field H acts diagonally as a modified homogeneity operator, and it is readily verified that H⁽⁴⁾k = 0. Finally,

$X_2^{(4)}k=\frac{1}{2}[H,X1{]}^{(4)}k=\frac{1}{2}[H^{(4)},X{1}^{(4)}]k=0.$

The five vector fields

$T_1^{(4)},T_2^{(4)},X_1^{(4)},X_2^{(4)},H^{(4)}$

form an involutive distribution on (an open subset of) R⁶ so that, by the Frobenius integration theorem, they integrate locally to give a foliation of R⁶ by five-dimensional leaves. Concretely, each leaf is a local orbit of the special affine group. The function k parameterizes these leaves.

Human Motor System

Human curvilinear 2-dimensional drawing movements tend to follow the equi-affine parametrization. This is more commonly known as the two thirds power law, according to which the hand's speed is proportional to the Euclidean curvature raised to the minus third power. Namely,

$v=\gamma {\kappa }^{-\frac{1}{3}},$

where $v$ is the speed of the hand, $\kappa$ is the Euclidean curvature and $\gamma$ is a constant termed the velocity gain factor.

See also

• Affine geometry of curves
• Affine sphere

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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  my web-site http://himerka.com/
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.