Curvature

Dictyostelium discoideum cell whose boundary is colored by curvature. Scale bar: 5 µm. A migrate wild-typecell whose boundary is colored by curvature. scale bar : 5 µm. In mathematics, curvature is any of several strongly related concepts in geometry. intuitively, the curvature is the sum by which a curve deviates from being a straight line, or a come on deviates from being a plane.

For curves, the canonic exemplar is that of a set, which has a curvature peer to the reciprocal of its radius. Smaller circles bend more sharply, and therefore have higher curvature. The curvature at a point of a differentiable curl is the curvature of its osculating set, that is the r-2 that best approximates the curve near this point. The curvature of a uncoiled occupation is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real issue. For surfaces ( and, more generally for higher-dimensional manifolds ), that are embedded in a euclidian quad, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold paper. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature. For riemannian manifolds ( of dimension at least two ) that are not necessarily embedded in a euclidian space, one can define the curvature intrinsically, that is without referring to an external quad. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor .

history

In Tractatus de configurationibus qualitatum et motuum, [ 1 ] the 14th-century philosopher and mathematician Nicole Oresme introduces the concept of curvature as a bill of passing from straightness ; for circles he has the curvature as being inversely proportional to the radius ; and he attempts to extend this idea to other curves as a endlessly varying magnitude. [ 2 ] The curvature of a differentiable bend was primitively defined through osculating circles. In this set, Augustin-Louis Cauchy showed that the center of curvature is the intersection detail of two infinitely cheeseparing normal lines to the bend. [ 3 ]

plane curves

intuitively, the curvature describes for any separate of a bend how much the crook guidance changes over a small distance travelled ( e.g. fish in rad/m ), so it is a measure of the instantaneous rate of exchange of direction of a point that moves on the curve : the larger the curvature, the larger this rate of switch. In other words, the curvature measures how firm the whole tangent vector to the curl rotates [ 4 ] ( fast in terms of bend position ). In fact, it can be proved that this instantaneous rate of change is precisely the curvature. More precisely, suppose that the decimal point is moving on the curl at a changeless travel rapidly of one unit of measurement, that is, the place of the bespeak P ( s ) is a affair of the parameter sulfur, which may be thought as the prison term or as the bow duration from a given beginning. Let T ( s ) be a unit tangent vector of the curl at P ( s ), which is besides the derivative of P ( s ) with obedience to s. then, the derived function of T ( s ) with esteem to s is a vector that is convention to the curvature and whose length is the curvature. For being meaningful, the definition of the curvature and its different characterizations require that the curve is endlessly differentiable near P, for having a tangent that varies endlessly ; it requires besides that the arch is twice differentiable at P, for insuring the being of the involve limits, and of the derivative of T ( s ). The word picture of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating traffic circle, but formulas for computing the curvature are easier to deduce. Therefore, and besides because of its use in kinematics, this characterization is much given as a definition of the curvature .

Osculating set

historically, the curvature of a differentiable crook was defined through the snog circle, which is the r-2 that best approximates the curve at a point. More precisely, given a orient P on a curvature, every other target Q of the swerve defines a circle ( or sometimes a line ) passing through Q and tangent to the arch at P. The osculate lap is the terminus ad quem, if it exists, of this r-2 when Q tends to P. then the center and the radius of curvature of the curve at P are the kernel and the radius of the osculate circle. The curvature is the reciprocal cross of radius of curvature. That is, the curvature is

κ = 1 R, { \displaystyle \kappa = { \frac { 1 } { R } }, }

where R is the radius of curvature [ 5 ] ( the unharmed encircle has this curvature, it can be read as sour 2π over the distance 2πR ). This definition is difficult to manipulate and to express in formula. therefore, early equivalent definitions have been introduced .

In terms of arc-length parametrization

Every differentiable curve can be parametrized with deference to arc length. [ 6 ] [ dead link ] In the case of a plane arch, this means the being of a parametrization γ ( s ) = ( x ( s ), y ( s ) ), where ten and y are real-valued differentiable functions whose derivatives satisfy

‖ γ ′ ‖ = x ′ ( mho ) 2 + yttrium ′ ( randomness ) 2 = 1. { \displaystyle \| { \boldsymbol { \gamma } } ‘\|= { \sqrt { x ‘ ( s ) ^ { 2 } +y ‘ ( s ) ^ { 2 } } } =1. }

This means that the tangent vector

T ( s ) = ( x ′ ( sulfur ), yttrium ′ ( s ) ) { \displaystyle \mathbf { T } ( mho ) = { \bigl ( } x ‘ ( second ), y ‘ ( mho ) { \bigr ) } }

has a average equal to one and is thus a whole tangent vector. If the curvature is twice differentiable, that is, if the second derivatives of ten and yttrium exist, then the derivative of T ( s ) exists. This vector is normal to the curl, its norm is the curvature κ ( s ), and it is oriented toward the center of curvature. That is ,

T ( s ) = γ ′ ( mho ), T 2 ( s ) = 1 ( constant ) ⟹ T ′ ( s ) ⋅ T ( s ) = 0 κ ( s ) = ‖ T ′ ( s ) ‖ = ‖ γ ″ ( s ) ‖ = x ″ ( second ) 2 + yttrium ″ ( s ) 2 { \displaystyle { \begin { aligned } & \mathbf { T } ( sulfur ) = { \boldsymbol { \gamma } } ‘ ( s ), \\ & \mathbf { T } ^ { 2 } ( sulfur ) =1\ { \text { ( constant ) } } \implies \mathbf { T } ‘ ( sulfur ) \cdot \mathbf { T } ( sulfur ) =0\\ & \kappa ( s ) =\|\mathbf { T } ‘ ( south ) \|=\| { \boldsymbol { \gamma } } ” ( s ) \|= { \sqrt { ten ” ( s ) ^ { 2 } +y ” ( s ) ^ { 2 } } } \\\end { align } } }

furthermore, as the spoke of curvature is

R ( s ) = 1 κ ( south ), { \displaystyle R ( s ) = { \frac { 1 } { \kappa ( second ) } }, }

and the center of curvature is on the convention to the curvature, the center of curvature is the point

C ( s ) = γ ( s ) + 1 κ ( s ) 2 T ′ ( second ). { \displaystyle \mathbf { C } ( south ) = { \boldsymbol { \gamma } } ( south ) + { \frac { 1 } { \kappa ( s ) ^ { 2 } } } \mathbf { T } ‘ ( sulfur ). }

If N ( s ) is the whole normal vector obtained from T ( s ) by a counterclockwise rotation of π/2, then

T ′ ( s ) = k ( s ) N ( randomness ), { \displaystyle \mathbf { T } ‘ ( s ) =k ( s ) \mathbf { N } ( sulfur ), }

with k ( s ) = ± κ ( s ). The real number phone number k ( s ) is called the oriented curvature or signed curvature. It depends on both the predilection of the plane ( definition of counterclockwise ), and the orientation of the curve provided by the parametrization. In fact, the change of variable star s → – s provides another arc-length parametrization, and changes the sign of k ( s ) .

In terms of a general parametrization

Let γ ( t ) = ( x ( t ), y ( t ) ) be a proper parametric representation of a doubly differentiable plane curl. here proper means that on the sphere of definition of the parametrization, the derivative d γ / dt is defined, differentiable and nowhere adequate to the nothing vector. With such a parametrization, the sign curvature is

thousand = x ′ y ″ − y ′ x ″ ( x ′ 2 + y ′ 2 ) 3 2, { \displaystyle k= { \frac { x’y ” -y’x ” } { \left ( { x ‘ } ^ { 2 } + { y ‘ } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }, }

where primes refer to derivatives with esteem to t. The curvature κ is frankincense

κ = | x ′ y ″ − y ′ x ″ | ( x ′ 2 + y ′ 2 ) 3 2. { \displaystyle \kappa = { \frac { |x’y ” -y’x ” | } { \left ( { ten ‘ } ^ { 2 } + { yttrium ‘ } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }. }

These can be expressed in a coordinate-free way as

potassium = det ( γ ′, γ ″ ) ‖ γ ′ ‖ 3, κ = | det ( γ ′, γ ″ ) | ‖ γ ′ ‖ 3. { \displaystyle k= { \frac { \det ( { \boldsymbol { \gamma } } ‘, { \boldsymbol { \gamma } } ” ) } { \| { \boldsymbol { \gamma } } ‘\|^ { 3 } } }, \qquad \kappa = { \frac { |\det ( { \boldsymbol { \gamma } } ‘, { \boldsymbol { \gamma } } ” ) | } { \| { \boldsymbol { \gamma } } ‘\|^ { 3 } } }. }

These formulas can be derived from the special case of arc-length parametrization in the trace way. The above condition on the parametrisation entail that the discharge length mho is a differentiable flat function of the parameter metric ton, and conversely that deoxythymidine monophosphate is a flat function of s. furthermore, by changing, if needed, s to – s, one may suppose that these functions are increasing and have a positive derivative. Using notation of the preceding section and the chain rule, one has

five hundred γ vitamin d triiodothyronine = five hundred s five hundred metric ton T, { \displaystyle { \frac { d { \boldsymbol { \gamma } } } { dt } } = { \frac { darmstadtium } { dt } } \mathbf { T }, }

and thus, by taking the average of both sides

five hundred thymine vitamin d s = 1 ‖ γ ′ ‖, { \displaystyle { \frac { dt } { vitamin d } } = { \frac { 1 } { \| { \boldsymbol { \gamma } } ‘\| } }, }

where the choice denotes the derivation with obedience to t. The curvature is the norm of the derivative of T with respect to s. By using the above convention and the chain rule this derivative and its average can be expressed in terms of γ ′ and γ ″ only, with the arc-length argument second wholly eliminated, giving the above rule for the curvature .

Graph of a serve

The graph of a function y = f ( x ), is a particular lawsuit of a parametrized curl, of the human body

ten = metric ton yttrium = fluorine ( thyroxine ). { \displaystyle { \begin { aligned } x & =t\\y & =f ( thymine ) .\end { align } } }

As the first and second derivatives of x are 1 and 0, previous formulas simplify to

κ = | y ″ | ( 1 + y ′ 2 ) 3 2, { \displaystyle \kappa = { \frac { |y ” | } { \left ( 1+ { y ‘ } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }, }

for the curvature, and to

potassium = yttrium ″ ( 1 + y ′ 2 ) 3 2, { \displaystyle k= { \frac { yttrium ” } { \left ( 1+ { yttrium ‘ } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }, }

for the sign curvature. In the general case of a arch, the sign of the sign curvature is somehow arbitrary, as depending on an orientation course of the wind. In the case of the graph of a function, there is a natural orientation course by increasing values of adam. This makes significant the sign of the zodiac of the sign curvature. The sign of the gestural curvature is the like as the sign of the second derivative of f. If it is positive then the graph has an up concavity, and, if it is negative the graph has a down concave shape. It is zero, then one has an inflection point or an wave point. When the slope of the graph ( that is the derivative of the function ) is minor, the sign curvature is well approximated by the second derived function. More precisely, using big O notation, one has

k ( x ) = y ″ ( 1 + O ( yttrium ′ 2 ) ). { \displaystyle k ( x ) =y ” \left ( 1+O\left ( { yttrium ‘ } ^ { 2 } \right ) \right ). }

It is common in physics and engineering to approximate the curvature with the moment derivative instrument, for case, in beam theory or for deriving wave equation of a tense string, and other applications where humble slopes are involved. This much allows systems that are otherwise nonlinear to be considered as linear .

polar coordinates

If a curvature is defined in diametric coordinates by the radius expressed as a function of the pivotal angle, that is gas constant is a function of θ, then its curvature is

κ ( θ ) = | r 2 + 2 radius ′ 2 − gas constant r ″ | ( r 2 + r ′ 2 ) 3 2 { \displaystyle \kappa ( \theta ) = { \frac { \left|r^ { 2 } +2 { r ‘ } ^ { 2 } -r\, r ” \right| } { \left ( r^ { 2 } + { gas constant ‘ } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } } }

where the choice refers to specialization with respect to θ. This results from the formula for general parametrizations, by considering the parametrization

ten = gas constant ( θ ) carbon monoxide ⁡ θ y = radius ( θ ) sin ⁡ θ { \displaystyle { \begin { aligned } x & =r ( \theta ) \cos \theta \\y & =r ( \theta ) \sin \theta \end { aligned } } }

Implicit crook

For a bend defined by an implicit equation F ( x, y ) = 0 with fond derivatives denoted Fx, Fy, Fxx, Fxy, Fyy, the curvature is given by [ 7 ]

κ = | F y 2 F x x − 2 F x F y F x y + F x 2 F y yttrium | ( F x 2 + F y 2 ) 3 2. { \displaystyle \kappa = { \frac { \left|F_ { y } ^ { 2 } F_ { twenty } -2F_ { adam } F_ { y } F_ { xy } +F_ { x } ^ { 2 } F_ { yy } \right| } { \left ( F_ { adam } ^ { 2 } +F_ { yttrium } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }. }

The sign curvature is not defined, as it depends on an orientation course of the crook that is not provided by the implicit equality. besides, changing F into – F does not change the swerve, but changes the signal of the numerator if the absolute value is omitted in the preceding formula. A degree of the swerve where Fx = Fy = 0 is a singular point, which means that the arch is not differentiable at this point, and thus that the curvature is not defined ( most frequently, the point is either a intersect luff or a cusp ). Above formula for the curvature can be derived from the construction of the curvature of the graph of a function by using the implicit serve theorem and the fact that, on such a curvature, one has

five hundred y five hundred x = − F x F y. { \displaystyle { \frac { dy } { dx } } =- { \frac { F_ { ten } } { F_ { y } } }. }

Examples

It can be useful to verify on childlike examples that the unlike formulas given in the precede sections give the lapp leave .

circle

A common parametrization of a traffic circle of radius r is γ ( t ) = ( r carbon monoxide t, r sin t ). The formula for the curvature gives

thousand ( thyroxine ) = r 2 sin 2 ⁡ thyroxine + r 2 colorado 2 ⁡ thyroxine ( roentgen 2 carbon monoxide 2 ⁡ thyroxine + r 2 sin 2 ⁡ metric ton ) 3 2 = 1 radius. { \displaystyle k ( t ) = { \frac { r^ { 2 } \sin ^ { 2 } t+r^ { 2 } \cos ^ { 2 } thyroxine } { ( r^ { 2 } \cos ^ { 2 } t+r^ { 2 } \sin ^ { 2 } deoxythymidine monophosphate ) ^ { \frac { 3 } { 2 } } } } = { \frac { 1 } { roentgen } }. }

It follows, as expected, that the radius of curvature is the spoke of the lap, and that the center of curvature is the center of the circle. The circle is a rare character where the arc-length parametrization is easy to compute, as it is

γ ( s ) = ( roentgen conscientious objector ⁡ second r, gas constant sin ⁡ s gas constant ). { \displaystyle { \boldsymbol { \gamma } } ( mho ) =\left ( r\cos { \frac { s } { roentgen } }, r\sin { \frac { s } { gas constant } } \right ). }

It is an arc-length parametrization, since the average of

γ ′ ( s ) = ( − sin ⁡ second roentgen, cosine ⁡ s radius ) { \displaystyle { \boldsymbol { \gamma } } ‘ ( s ) =\left ( -\sin { \frac { s } { roentgen } }, \cos { \frac { s } { roentgen } } \right ) }

is adequate to one. This parametrization gives the same rate for the curvature, as it amounts to division by r 3 in both the numerator and the denominator in the preceding formula. The lapp circle can besides be defined by the implicit equation F ( x, y ) = 0 with F ( x, y ) = x 2 + y 2 – r 2. then, the convention for the curvature in this lawsuit gives

κ = | F y 2 F x x − 2 F x F y F x y + F x 2 F yttrium y | ( F x 2 + F y 2 ) 3 2 = 8 yttrium 2 + 8 ten 2 ( 4 adam 2 + 4 yttrium 2 ) 3 2 = 8 gas constant 2 ( 4 gas constant 2 ) 3 2 = 1 r. { \displaystyle { \begin { aligned } \kappa & = { \frac { \left|F_ { y } ^ { 2 } F_ { twenty } -2F_ { adam } F_ { yttrium } F_ { xy } +F_ { ten } ^ { 2 } F_ { yy } \right| } { \left ( F_ { ten } ^ { 2 } +F_ { yttrium } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } } \\ & = { \frac { 8y^ { 2 } +8x^ { 2 } } { \left ( 4x^ { 2 } +4y^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } } \\ & = { \frac { 8r^ { 2 } } { \left ( 4r^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } } = { \frac { 1 } { r } } .\end { align } } }

parabola

Consider the parabola y = ax 2 + bx + c. It is the graph of a officiate, with derivative 2 ax + b, and second derived function 2 a. sol, the bless curvature is

potassium ( x ) = 2 a ( 1 + ( 2 a ten + b ) 2 ) 3 2. { \displaystyle kilobyte ( x ) = { \frac { 2a } { \left ( 1+\left ( 2ax+b\right ) ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }. }

It has the sign of a for all values of ten. This means that, if a > 0, the concavity is up directed everywhere ; if a < 0, the concave shape is downward directed ; for a = 0, the curvature is zero everywhere, confirming that the parabola degenerates into a line in this sheath. The ( unsigned ) curvature is maximal for x = – b /2 a, that is at the stationary point ( zero derivative ) of the function, which is the vertex of the parabola. Consider the parametrization γ ( t ) = ( t, at 2 + bt + c ) = ( x, y ). The first gear derivative of x is 1, and the second derived function is zero. Substituting into the recipe for general parametrizations gives precisely the same result as above, with x replaced by t. If we use primes for derivatives with respect to the parameter metric ton. The same parabola can besides be defined by the implicit equation F ( x, y ) = 0 with F ( x, y ) = ax 2 + bx + cy. As Fy = –1, and Fyy = Fxy = 0, one obtains precisely the same prize for the ( unsigned ) curvature. however, the sign curvature is meaningless here, as – F ( x, y ) = 0 is a valid implicit equation for the lapp parabola, which gives the opposite sign for the curvature .

Frenet–Serret formula for plane curves

T

and N at two points on a plane curve, a translated version of the second frame (dotted), and δ T the change in T. Here δs is the distance between the points. In the limit

dT

/

ds

will be in the direction N. The curvature describes the rate of rotation of the frame. The vectorsandat two points on a plane swerve, a translate adaptation of the second frame ( dotted ), andthe variety in. Hereis the outdistance between the points. In the limitwill be in the direction. The curvature describes the pace of rotation of the skeleton. The saying of the curvature In terms of arc-length parametrization is basically the beginning Frenet–Serret formula

T ′ ( s ) = κ ( s ) N ( sulfur ), { \displaystyle \mathbf { T } ‘ ( second ) =\kappa ( s ) \mathbf { N } ( randomness ), }

where the primes refer to the derivatives with deference to the arch distance second, and N ( s ) is the convention unit vector in the focus of T ′ ( randomness ). As planar curves have zero torsion, the second Frenet–Serret formula provides the relation

d N vitamin d second = − κ T, = − κ five hundred γ vitamin d sulfur. { \displaystyle { \begin { aligned } { \frac { d\mathbf { N } } { d } } & =-\kappa \mathbf { T }, \\ & =-\kappa { \frac { d { \boldsymbol { \gamma } } } { bureau of diplomatic security } } .\end { align } } }

For a general parametrization by a parameter deoxythymidine monophosphate, one needs expressions involving derivatives with regard to t. As these are obtained by multiplying by ds / dt the derivatives with regard to s, one has, for any proper parametrization

N ′ ( t ) = − κ ( thyroxine ) γ ′ ( thyroxine ). { \displaystyle \mathbf { N } ‘ ( deoxythymidine monophosphate ) =-\kappa ( thymine ) { \boldsymbol { \gamma } } ‘ ( t ). }

space curves

T ′ ( s ) animation of the curvature and the acceleration vector As in the character of curves in two dimensions, the curvature of a regular space curve C in three dimensions ( and higher ) is the order of magnitude of the acceleration of a atom moving with unit speed along a arch. frankincense if γ ( s ) is the arc-length parametrization of C then the unit tangent vector T ( s ) is given by

T ( s ) = γ ′ ( s ) { \displaystyle \mathbf { T } ( s ) = { \boldsymbol { \gamma } } ‘ ( mho ) }

and the curvature is the magnitude of the acceleration :

κ ( s ) = ‖ T ′ ( s ) ‖ = ‖ γ ″ ( s ) ‖. { \displaystyle \kappa ( s ) =\|\mathbf { T } ‘ ( mho ) \|=\| { \boldsymbol { \gamma } } ” ( s ) \|. }

The focus of the acceleration is the unit normal vector N ( s ), which is defined by

N ( s ) = T ′ ( s ) ‖ T ′ ( s ) ‖. { \displaystyle \mathbf { N } ( sulfur ) = { \frac { \mathbf { T } ‘ ( mho ) } { \|\mathbf { T } ‘ ( second ) \| } }. }

The plane containing the two vectors T ( s ) and N ( s ) is the osculating flat to the swerve at γ ( s ). The curvature has the follow geometric interpretation. There exists a r-2 in the osculating airplane tangent to γ ( s ) whose Taylor series to second ordering at the charge of contact agrees with that of γ ( s ). This is the osculating set to the curl. The radius of the circle R ( s ) is called the radius of curvature, and the curvature is the multiplicative inverse of the spoke of curvature :

κ ( s ) = 1 R ( randomness ). { \displaystyle \kappa ( s ) = { \frac { 1 } { R ( mho ) } }. }

The tangent, curvature, and normal vector together describe the second-order behavior of a bend near a point. In three dimensions, the third-order behavior of a wind is described by a refer notion of torsion, which measures the extent to which a swerve tends to move as a coiling path in space. The torsion and curvature are related by the Frenet–Serret formula ( in three dimensions ) and their generalization ( in higher dimensions ) .

cosmopolitan expressions

For a parametrically-defined quad arch in three dimensions given in cartesian coordinates by γ ( t ) = ( x ( t ), y ( t ), z ( t ) ), the curvature is

κ = ( z ″ yttrium ′ − y ″ z ′ ) 2 + ( ten ″ z ′ − z ″ x ′ ) 2 + ( y ″ x ′ − x ″ y ′ ) 2 ( adam ′ 2 + y ′ 2 + z ′ 2 ) 3 2, { \displaystyle \kappa = { \frac { \sqrt { \left ( omega ” y’-y ” z’\right ) ^ { 2 } +\left ( ten ” z’-z ” x’\right ) ^ { 2 } +\left ( yttrium ” x’-x ” y’\right ) ^ { 2 } } } { \left ( { adam ‘ } ^ { 2 } + { y ‘ } ^ { 2 } + { omega ‘ } ^ { 2 } \right ) ^ { \frac { 3 } { 2 } } } }, }

where the flower denotes differentiation with respect to the parameter t. This can be expressed independently of the coordinate organization by means of the formula

κ = ‖ γ ′ × γ ″ ‖ ‖ γ ′ ‖ 3 { \displaystyle \kappa = { \frac { \| { \boldsymbol { \gamma } } ‘\times { \boldsymbol { \gamma } } ”\| } { \| { \boldsymbol { \gamma } } ‘\|^ { 3 } } } }

where × denotes the vector cross product. Equivalently ,

κ = det ( ( γ ′ × γ ″ ) T ( γ ′ × γ ″ ) ) ‖ γ ′ ‖ 3 = ‖ γ ′ ‖ 2 ‖ γ ″ ‖ 2 − ( γ ′ ⋅ γ ″ ) 2 ‖ γ ′ ‖ 3. { \displaystyle \kappa = { \frac { \sqrt { \det \left ( \left ( { \boldsymbol { \gamma } } ‘\times { \boldsymbol { \gamma } } ”\right ) ^ { \mathsf { T } } ( \gamma ‘\times \gamma ” ) \right ) } } { \| { \boldsymbol { \gamma } } ‘\|^ { 3 } } } = { \frac { \sqrt { \| { \boldsymbol { \gamma } } ‘\|^ { 2 } \| { \boldsymbol { \gamma } } ”\|^ { 2 } -\left ( { \boldsymbol { \gamma } } ‘\cdot { \boldsymbol { \gamma } } ”\right ) ^ { 2 } } } { \| { \boldsymbol { \gamma } } ‘\|^ { 3 } } }. }

hera the T denotes the matrix permute of the vector. This death formula ( without crabbed merchandise ) is besides valid for the curvature of curves in a euclidian space of any dimension .

curvature from arch and harmonize length

Given two points P and Q on C, let s ( P, Q ) be the bow length of the part of the curl between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q. The curvature of C at P is given by the limit [ citation needed ]

κ ( P ) = lim Q → P 24 ( mho ( P, Q ) − five hundred ( P, Q ) ) sulfur ( P, Q ) 3 { \displaystyle \kappa ( P ) =\lim _ { Q\to P } { \sqrt { \frac { 24 { \bigl ( } randomness ( P, Q ) -d ( P, Q ) { \bigr ) } } { s ( P, Q ) ^ { 3 } } } } }

where the specify is taken as the period Q approaches P on C. The denominator can evenly well be taken to be d ( P, Q ) 3. The formula is valid in any proportion. furthermore, by considering the limit independently on either side of P, this definition of the curvature can sometimes accommodate a singularity at P. The formula follows by verifying it for the osculate circle .

Surfaces

For broader coverage of this topic, see differential geometry of surfaces The curvature of curves drawn on a coat is the chief cock for the define and studying the curvature of the surface .

Curves on surfaces

For a curve withdraw on a airfoil ( embedded in three-dimensional Euclidean quad ), respective curvatures are defined, which relates the direction of curvature to the surface ‘s whole normal vector, including the :
Any non-singular curve on a legato surface has its tangent vector T contained in the tangent plane of the surface. The normal curvature, k north, is the curvature of the curve projected onto the plane containing the curve ‘s tangent T and the surface normal u ; the geodesic curvature, k thousand, is the curvature of the wind projected onto the come on ‘s tangent flat ; and the geodesic torsion ( or relative torsion ), τ roentgen, measures the rate of change of the surface normal around the curl ‘s tangent. Let the crook be arc-length parametrized, and let t = u × T so that T, t, u form an orthonormal footing, called the Darboux frame. The above quantities are related by :

( T ′ deoxythymidine monophosphate ′ u ′ ) = ( 0 κ gigabyte κ n − κ g 0 τ roentgen − κ n − τ r 0 ) ( T t uranium ) { \displaystyle { \begin { pmatrix } \mathbf { T } ‘\\\mathbf { triiodothyronine } ‘\\\mathbf { uranium } ‘\end { pmatrix } } = { \begin { pmatrix } 0 & \kappa _ { \mathrm { gigabyte } } & \kappa _ { \mathrm { n } } \\-\kappa _ { \mathrm { gravitational constant } } & 0 & \tau _ { \mathrm { radius } } \\-\kappa _ { \mathrm { normality } } & -\tau _ { \mathrm { radius } } & 0\end { pmatrix } } { \begin { pmatrix } \mathbf { T } \\\mathbf { triiodothyronine } \\\mathbf { u } \end { pmatrix } } }

star curvature

All curves on the airfoil with the like tangent vector at a given bespeak will have the lapp normal curvature, which is the same as the curvature of the wind obtained by intersecting the surface with the plane containing T and u. Taking all possible tangent vectors, the maximum and minimal values of the normal curvature at a point are called the principal curvatures, k 1 and k 2, and the directions of the match tangent vectors are called principal normal directions .

normal sections

curvature can be evaluated along surface normal sections, alike to § Curves on surfaces above ( see for case the Earth radius of curvature ) .

gaussian curvature

In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature ( they only have a curvature given an embed ), surfaces can have intrinsic curvature, mugwump of an implant. The gaussian curvature, named after Carl Friedrich Gauss, is adequate to the intersection of the principal curvatures, k 1 k 2. It has a property of length−2 and is positive for spheres, negative for one-sheet hyperboloids and zero for planes and cylinders. It determines whether a surface is locally convex ( when it is positive ) or locally saddle-shaped ( when it is negative ). gaussian curvature is an intrinsic property of the surface, meaning it does not depend on the particular embed of the coat ; intuitively, this means that ants living on the open could determine the gaussian curvature. For example, an ant live on a sector could measure the sum of the home angles of a triangulum and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other bridge player, an ant living on a cylinder would not detect any such departure from Euclidean geometry ; in particular the ant could not detect that the two surfaces have different hateful curvatures ( see below ), which is a strictly extrinsic type of curvature. formally, gaussian curvature lone depends on the riemannian metric of the open. This is Gauss ‘s observe Theorema Egregium, which he found while concerned with geographic surveys and mapmaking. An intrinsic definition of the gaussian curvature at a point P is the be : imagine an ant which is tied to P with a short thread of length r. It runs around P while the weave is wholly stretched and measures the length C ( r ) of one complete stumble around P. If the surface were flat, the ant would find C ( r ) = 2π r. On wind surfaces, the formula for C ( r ) will be different, and the gaussian curvature K at the compass point P can be computed by the Bertrand–Diguet–Puiseux theorem as

K = lim gas constant → 0 + 3 ( 2 π gas constant − C ( gas constant ) π r 3 ). { \displaystyle K=\lim _ { r\to 0^ { + } } 3\left ( { \frac { 2\pi r-C ( gas constant ) } { \pi r^ { 3 } } } \right ). }

The integral of the gaussian curvature over the whole surface is closely related to the surface ‘s Euler characteristic ; see the Gauss–Bonnet theorem. The discrete analogue of curvature, corresponding to curvature being concentrated at a degree and particularly useful for polyhedron, is the ( angular ) blemish ; the analogue for the Gauss–Bonnet theorem is Descartes ‘ theorem on full angular blemish. Because ( gaussian ) curvature can be defined without character to an embedding space, it is not necessary that a open be embedded in a higher-dimensional space in order to be curved. such an intrinsically curved two-dimensional airfoil is a dim-witted exercise of a riemannian manifold .

mean curvature

The mean curvature is an extrinsic measure of curvature equal to half the sum of the chief curvatures, k 1 + k 2/2. It has a dimension of length−1. beggarly curvature is close related to the first variation of coat area. In finical, a minimal surface such as a soap film has mean curvature zero and a soap ripple has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the implant, for exemplify, a cylinder and a plane are locally isometric but the entail curvature of a plane is zero while that of a cylinder is nonzero .

second fundamental shape

The intrinsic and extrinsic curvature of a open can be combined in the second fundamental shape. This is a quadratic shape in the tangent plane to the open at a point whose value at a particular tangent vector X to the airfoil is the convention part of the acceleration of a curvature along the surface tangent to X ; that is, it is the normal curvature to a bend tangent to X ( see above ). symbolically ,

I I ⁡ ( X, X ) = N ⋅ ( ∇ X X ) { \displaystyle \operatorname { I\ ! I } ( \mathbf { ten }, \mathbf { X } ) =\mathbf { N } \cdot ( \nabla _ { \mathbf { x } } \mathbf { X } ) }

where N is the whole normal to the surface. For whole tangent vectors X, the moment fundamental form assumes the maximum value k 1 and minimal value k 2, which occur in the principal directions u 1 and u 2, respectively. therefore, by the principal axis theorem, the second fundamental form is

I I ⁡ ( X, X ) = k 1 ( X ⋅ u 1 ) 2 + potassium 2 ( X ⋅ u 2 ) 2. { \displaystyle \operatorname { I\ ! I } ( \mathbf { ten }, \mathbf { X } ) =k_ { 1 } \left ( \mathbf { X } \cdot \mathbf { uranium } _ { 1 } \right ) ^ { 2 } +k_ { 2 } \left ( \mathbf { X } \cdot \mathbf { u } _ { 2 } \right ) ^ { 2 }. }

therefore the irregular fundamental shape encodes both the intrinsic and extrinsic curvatures .

form hustler

An encapsulation of surface curvature can be found in the supreme headquarters allied powers europe operator, S, which is a self-adjoint linear operator from the tangent plane to itself ( specifically, the derived function of the Gauss map ). For a surface with tangent vectors X and normal N, the form operator can be expressed compactly in index summation notation as

∂ a N = − S b a adam bel. { \displaystyle \partial _ { a } \mathbf { N } =-S_ { bachelor of arts } \mathbf { X } _ { bacillus }. }

( Compare the option saying of curvature for a plane curl. ) The Weingarten equations give the value of S in terms of the coefficients of the first gear and second fundamental forms as

S = ( E G − F 2 ) − 1 ( vitamin e G − f F degree fahrenheit G − gram F degree fahrenheit E − e F g E − f F ). { \displaystyle S=\left ( EG-F^ { 2 } \right ) ^ { -1 } { \begin { pmatrix } eG-fF & fG-gF\\fE-eF & gE-fF\end { pmatrix } }. }

The principal curvatures are the eigenvalues of the shape operator, the principal curvature directions are its eigenvectors, the Gauss curvature is its deciding, and the base curvature is half its trace .

curvature of space

“ curvature of space ” redirects here. It is not to be confused with Curvature of space-time By extension of the erstwhile controversy, a outer space of three or more dimensions can be intrinsically curved. The curvature is intrinsic in the sense that it is a place defined at every point in the quad, rather than a property defined with respect to a larger space that contains it. In general, a curved space may or may not be conceived as being embedded in a higher-dimensional ambient quad ; if not then its curvature can entirely be defined intrinsically. After the discovery of the intrinsic definition of curvature, which is close connected with non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically big. In the hypothesis of general relativity, which describes gravity and cosmology, the estimate is slenderly generalised to the “ curvature of spacetime “ ; in relativity hypothesis spacetime is a pseudo-Riemannian manifold. Once a time coordinate is defined, the cubic space corresponding to a detail time is generally a arch riemannian manifold ; but since the time organize choice is largely arbitrary, it is the underlie spacetime curvature that is physically significant. Although an randomly curved distance is identical building complex to describe, the curvature of a space which is locally isotropic and homogeneous is described by a individual gaussian curvature, as for a surface ; mathematically these are strong conditions, but they correspond to reasonable physical assumptions ( all points and all directions are indistinguishable ). A cocksure curvature corresponds to the inverse squarely radius of curvature ; an exemplar is a celestial sphere or hypersphere. An case of negatively curved space is hyperbolic geometry. A space or space-time with zero curvature is called flat. For model, Euclidean space is an example of a bland outer space, and Minkowski quad is an exemplar of a flat spacetime. There are early examples of flat geometries in both settings, though. A torus or a cylinder can both be given flat metrics, but differ in their topology. other topologies are besides potential for wind distance. See besides shape of the population .

Generalizations

ANBA yields a different vector. This failure to return to the initial vector is measured by the holonomy of the surface. Parallel transporting a vector fromyields a unlike vector. This failure to return to the initial vector is measured by the holonomy of the airfoil. The mathematical notion of curvature is besides defined in much more general context. [ 8 ] Many of these generalizations emphasize different aspects of the curvature as it is sympathize in lower dimensions. One such abstraction is kinematic. The curvature of a curve can naturally be considered as a kinematic quantity, representing the military unit felt by a certain perceiver moving along the curve ; analogously, curvature in higher dimensions can be regarded as a kind of tidal force ( this is one manner of think of the sectional curvature ). This generalization of curvature depends on how nearby test particles diverge or converge when they are allowed to move freely in the quad ; see Jacobi sphere. Another across-the-board generalization of curvature comes from the report of parallel transport on a come on. For example, if a vector is moved around a loop on the surface of a sector keeping analogue throughout the gesticulate, then the final position of the vector may not be the lapp as the initial stead of the vector. This phenomenon is known as holonomy. [ 9 ] Various generalizations capture in an pilfer form this mind of curvature as a measure of holonomy ; see curvature class. A closely relate notion of curvature comes from bore theory in physics, where the curvature represents a field and a vector electric potential for the field is a quantity that is in general path-dependent : it may change if an observer moves around a coil. Two more generalizations of curvature are the scalar curvature and Ricci curvature. In a swerve airfoil such as the sphere, the area of a phonograph record on the surface differs from the sphere of a magnetic disk of the same radius in directly outer space. This remainder ( in a suitable limit ) is measured by the scalar curvature. The difference in area of a sector of the disk is measured by the Ricci curvature. Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are peculiarly significant in relativity theory, where they both appear on the side of Einstein ‘s field equations that represents the geometry of spacetime ( the other side of which represents the presence of matter and energy ). These generalizations of curvature underlie, for example, the notion that curvature can be a property of a measurement ; see curvature of a standard. Another generalization of curvature relies on the ability to compare a wind space with another space that has constant curvature. Often this is done with triangles in the spaces. The notion of a triangle makes senses in metric unit spaces, and this gives rise to CAT ( k ) spaces .