Introduction to Curvature

In the fourteenth century, philosopher and mathematician, Nicole Oresme introduced the concept of curvature, he defined it as the measurement of departure from the straight lines, and for circles, the curvature is inversely proportional to the radius and the extension of these as the varying magnitude to the other curves. Originally the curvature of the differentiable curves was defined with the help of osculating circles. In this attempt, another mathematician called Augustin-Louis Cauchy mentioned that a curve has normal lines that have two infinitely close points that are considered as the centre of curvature. The curvature meaning is as follows, in Mathematics it is one of the strongly related topics present in geometry, it is the amount to which the curve deviates from being a straight line or the surface deviates from a plane. 


Curvature for Curves, Surfaces, and Riemannian Manifolds

For curves, the best example is a circle, the curvature of the circle is equal to that of the reciprocal of the radius. The smaller circle has more curvature than the larger circle as it can bend sharply. At a point of a differentiable curve, the best approximation of the curvature at this point is the osculating circle. The curvature is normally a scalar point for the normal curve and it is expressed as a single real number. Whereas the curvature for the straight line is zero


For surfaces, generally for the higher-dimensional manifolds that are embedded in the Euclidean space, determining the curvature concept is complex as it depends on the direction to which the surface or manifold is formed. Thus the concepts mean curvature, minimal curvature, and maximal curvature. 


For Riemannian manifolds, these are not embedded in the Euclidean space without referring to the outer surface one can intrinsically define the curvature. The definition of these manifolds is done in terms of the lengths of the curve that are traced on the manifold. After tracing these are expressed with the help of linear algebra by Riemann curvature tensor.


Plane Curves:

Generally, curvature meaning for any part of the curve is the amount to which the curve changes its direction in the small distance it is travelled. Thus it is a measure of the rate of change of direction at a point that moves on the surface of the curve. The rate of change is larger for the larger curvature. It measures the speed of rotation of the unit tangent vector to the curve. Sometimes this instantaneous change is termed as curvature. If a point is moving on the curve at a position P(s) that is considered as the time of change or arc length from the origin, T(s) is the unit tangent vector of P(s) and a derivative of it then the derivative of T(s) is a vector. 


The definition of the curvature and the different characteristics are required while calculating curvature is continuously differentiable at that point. 

  1. Osculating Circle: The differentiable curve curvature was defined through the osculating circle that is the circle where it best approximates the curve at a point. A point P on a curve, every other point Q of the curve defines a circle passing through Q and tangent to the curve at P. Then the centre and the radius of curvature of the curve at P are the centres and the radius of the osculating circle. The curvature is generally the reciprocal of the radius of the circle. That is, the curvature is K = \[\frac{1}{R}\], where R is the radius of the circle. 

  2. Arc-length Parametrization: Every differentiable curve can be parametrized with the help of using the concept called arc length. In the case of a plane curve, this means the existence of a parametrization where x and y are real-valued differentiable functions whose derivatives satisfy ||γ¹(s)|| = \[\sqrt{x^{1}(s)^{2} + y^{1}(s)^{2}}\] = 1

  3. Polar Coordinates: If a curve is defined in polar coordinates by the radius expressed as a function of the polar angle, that is r is a function of , then its curvature is 

K(θ) = \[\frac{|r^{2} + 2(r')^{2}-rr''|}{(r^{2}+(r')^{2})^{\frac{3}{2}}}\]

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

x = r(θ) cos(θ)

y = r(θ) sin(θ)

  1. Graph of a Function: The graph of a function such as y = f(x), is a special case for the parametrized curve, that is in the form x = t, y = f(t). 

In the general case of a curve, the sign of the signed curvature is somehow arbitrary and it depends on the orientation of the curve. There is a natural orientation in the case of the graph of a function which is obtained by increasing the values of x. The sign of the signed curvature makes it significant. The sign of the signed curvature is the same as that of the sign of the second derivative of f. The graph has an upward concavity if this condition is positive then, and the graph has a downward concavity if it is negative. It is zero, then one has an undulation point or an inflection point.

  1. Implicit Curve: The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Also, changing F into –F does not change the curve but changes the sign of the numerator if the absolute value is omitted in the preceding formula. 

A point of the curve where F\[_{x}\] = F\[_{y}\] = 0 is a singular point, which means that the curve is not differentiable at this point, and thus that the curvature is not defined.

The above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has

\[\frac{dy}{dx}\] = - \[\frac{F_{x}}{F_{y}}\] 


Gaussian Curvature:

In contrast to curves that do have extrinsic curvature surfaces but do not have intrinsic curvature can have intrinsic curvature, independent of an embedding. The Gaussian curvature is named after the mathematician, Carl Friedrich Gauss where he defined it as equal to the product of the principal curvatures. It is an intrinsic property of the surface which means that it does not depend on the particular embedding of the surface. Generally, we can determine the Gaussian curvature on the surface on which the ants live.


For example, the curvature of the sphere on an ant living can be determined by the sum of the interior angles of a triangle and determine that it was greater than that of the 180 degrees, this is implying that the space inhabited by an ant had positive curvature. Whereas an ant that is found living on a cylinder would not detect any such departure from Euclidean geometry, this means that the ant could not detect that these two surfaces have different mean curvatures which is a purely extrinsic type of curvature. 


The Gaussian curvature only depends on the Riemannian metric of the surface. Where he found it while concerned with the mapmaking and geographic surveys. The discrete analog of curvature that is corresponding to the curvature being concentrated at a point and particularly useful for polyhedra is the defect, the analog for the Gauss-Bonnet theorem is Descartes' theorem on the total angular defect. Because Gaussian curvature can be defined without having the reference to an embedding space and it is not necessary that a surface that is being embedded in a higher-dimensional space is to be curved.  The simple example of a Riemannian manifold is such an intrinsically curved two-dimensional surface.


Sigmoid Curvature:

A sigmoid function is a mathematical function having a characteristic sigmoid curve or ‘S’ shaped curve. A logistic function is a common example of a sigmoid function. The curvature of this function is sigmoid curvature.

s(x) = \[\frac{e^{x}}{e^{x} + 1}\]

Special cases of the sigmoid function include the Gompertz curve which is used in modelling systems that saturate at large values of x and the ogee curve. Sigmoid functions have a domain of all real numbers, with return value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value on the y-axis is in the range of 0 to 1. −1 to 1 is another commonly used range.

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Fig: Sigmoid function.


Conclusion:

The curvature of a curve can be considered as a kinematic quantity naturally, that can be represented by the force felt by a certain observer moving along the curve. Another generalization of curvature comes from the study of parallel transport on a surface. Suppose for an instance, if a vector is moved around a loop on the surface of a sphere by keeping it parallel throughout the motion, then the obtained final position of the vector might not be the same as that of the initial position of the vector. The ability to compare a curved space with another space where both of them have constant curvature is another generalization of curvature. These generalizations are often done with triangles in the spaces.

FAQs (Frequently Asked Questions)

1. What is Curvature?

Ans: The curvature of a curve is defined as the amount to which the curve deviates from being a straight line or the surface deviates from a plane.

2. How to Measure the Curvature?

Ans: To measure the curvature at a point you have to find the circle of best fit at that point. This is called the osculating circle. The curvature of the curve at that point is generally defined as the reciprocal of the radius of the reference circle known as the osculating circle.