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Curvature

In differential geometry, curvature quantifies the extent to which a geometric object, such as a curve or surface, deviates from being a straight line or flat plane, respectively, providing a measure of its bending or warping at a given point.^[1] For a curve parameterized by arc length, the curvature \kappa is formally defined as the magnitude of the derivative of the unit tangent vector with respect to arc length, \kappa = \left\| \frac{d\mathbf{T}}{ds} \right\|, representing the instantaneous rate at which the direction of the curve changes.^[1] This concept extends to higher dimensions, where the radius of curvature, $1/\kappa, describes the radius of the osculating circle that best approximates the curve locally.^[2] For surfaces embedded in three-dimensional Euclidean space, curvature is characterized by multiple quantities derived from the second fundamental form, including the principal curvatures \kappa_1 and \kappa_2, which are the maximum and minimum normal curvatures along orthogonal directions at a point. The Gaussian curvature K = \kappa_1 \kappa_2 measures the intrinsic geometry of the surface, determining whether it is elliptic (K > 0), parabolic (K = 0), or hyperbolic (K < 0), and remains unchanged under isometric deformations.^[3] In contrast, the mean curvature H = (\kappa_1 + \kappa_2)/2 captures extrinsic aspects, such as how the surface bends relative to its embedding space, and is crucial for studying minimal surfaces where H = 0.^[4] The foundational theory of surface curvature was established by Carl Friedrich Gauss in his 1827 paper Disquisitiones generales circa superficies curvas, where he introduced Gaussian curvature and proved the Theorema Egregium, demonstrating that K can be computed solely from the first fundamental form using intrinsic measurements like distances and angles on the surface itself, independent of its embedding.^[5] Later developments by Bernhard Riemann generalized curvature to higher-dimensional manifolds via the Riemann curvature tensor, enabling its application in general relativity to describe spacetime geometry.^[6] These concepts distinguish between extrinsic curvature, which depends on the ambient space, and intrinsic curvature, which is observable from within the manifold, profoundly influencing fields from computer graphics to theoretical physics.^[7]

Historical Development

Early Concepts in Geometry

The concept of curvature in early geometry emerged through qualitative distinctions between straight and curved lines, primarily in the works of ancient Greek mathematicians. Around 300 BCE, Euclid in his Elements defined a straight line as one that "lies evenly with the points on itself," implying a uniform alignment that curved lines lack, as they deviate from such evenness. This intuitive contrast highlighted straight lines as the shortest path between points, while curved lines were seen as longer, irregular paths, setting the foundation for understanding bending in geometric figures without quantitative measures. Euclid's approach focused on plane geometry, where circles and other basic curves were treated as distinct from rectilinear forms, emphasizing their properties in constructions like those involving tangents and intersections.^[8] Apollonius of Perga (c. 200 BCE) further developed these ideas through his systematic study of conic sections (parabolas, ellipses, hyperbolas), analyzing their asymptotic behaviors and varying degrees of "bending" relative to axes, which influenced later qualitative understandings of curves.^[9] During the medieval Islamic period, mathematicians advanced these ideas by exploring conic sections, which exhibited varying degrees of bending. Al-Kindi, in the 9th century, discussed the perspective projection of circular wheels appearing as ellipses, recognizing conic curves as bent forms arising from optical distortions of straight-lined circles. This qualitative insight into how curves "bend" under projection contributed to early applications in optics and architecture. Later, in 1070 CE, Omar Khayyam's Treatise on the Demonstration of Problems of Algebra utilized conic sections—parabolas, hyperbolas, and ellipses—to geometrically solve cubic equations, noting their inherent curvature properties that allowed intersections to yield solutions unattainable with straightedge and compass alone. Khayyam classified these curves based on their bending behaviors relative to axes, bridging algebraic problems with geometric intuition.^[10]^[11] In the Renaissance, figures like Johannes Kepler and René Descartes introduced early parametric ideas to describe curved paths more systematically. Kepler, in his 1609 Astronomia Nova, parametrized planetary orbits as ellipses using angular measures from the sun, capturing the curve's bending through time-dependent positions rather than static equations. Descartes, in his 1637 La Géométrie, advanced this by representing curves algebraically and proposing parametric forms, such as for the folium curve in 1638, which expressed points on the curve via a parameter to explore its looped bending. These developments marked a shift toward more dynamic geometric descriptions, laying groundwork for later calculus-based analyses.

Advancements in Differential Geometry

The invention of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century enabled precise analysis of curve properties, including tangents and the osculating circle, which captures the local curvature through second-order approximation at a point.^[12] Newton, in his Principia Mathematica (1687), used fluxions to determine the radius of curvature for planetary orbits, while Leibniz, in correspondence and publications around 1684–1692, introduced the term "osculating" for circles that "kiss" curves with higher-order contact, laying groundwork for differential geometric concepts. Christiaan Huygens, in his 1673 Horologium Oscillatorium, derived an early formula for the radius of curvature using the evolute, providing a pre-calculus quantitative measure.^[12]^[13] These tools shifted curvature from qualitative descriptions to quantitative measures, facilitating subsequent advancements in local curve and surface analysis. Leonhard Euler, in his 1748 Introductio in analysin infinitorum, classified points on curves into those of continuous curvature, inflection points where the tangent crosses the curve and curvature vanishes or changes sign, and cuspidal points of abrupt sharpening, providing a systematic framework for curve singularities.^[14] Building on this, these contributions emphasized curvature's role in approximating curves via conic sections, influencing the development of analytic geometry. Carl Friedrich Gauss's seminal 1827 work Disquisitiones generales circa superficies curvas introduced Gaussian curvature as the product of principal curvatures on a surface, a invariant under rigid motions that quantifies intrinsic bending.^[15] Central to this is the Theorema Egregium, proving that Gaussian curvature can be computed solely from the first fundamental form (the metric tensor), independent of the surface's embedding in Euclidean space, thus distinguishing it from extrinsic measures.^[15] This intrinsic perspective revolutionized geometry, enabling studies of surfaces without reference to ambient space. In the mid-19th century, Jean Gaston Darboux extended curvature theory to higher-dimensional hypersurfaces in his works on n-dimensional geometry (e.g., 1877–1883), developing Darboux derivatives and integrals for curvature invariants in n-dimensions.^[16] Concurrently, Eugenio Beltrami's 1868 essay "Saggio di interpretazione della geometria non euclidea" modeled hyperbolic geometry on surfaces of constant negative curvature, confirming its consistency via pseudospherical embeddings and paving the way for multi-dimensional generalizations.^[17] Bernhard Riemann's 1854 habilitation lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" culminated these efforts by defining curvature on abstract manifolds through a metric tensor, introducing the Riemann curvature tensor to measure deviation from flatness in arbitrary dimensions, foundational for modern differential geometry.^[18]

Curvature of Plane Curves

Definition via Osculating Circle

The osculating circle of a plane curve at a given point is the circle that best approximates the local geometry of the curve near that point by sharing the same position, tangent direction, and second-order contact, meaning it matches the curve up to the second derivative. This circle provides the limiting position as the circle passing through the point and two infinitesimally nearby points on the curve. The radius R of this osculating circle quantifies the local bending of the curve, with smaller radii corresponding to sharper turns.^[19] The curvature \kappa at the point is defined as the reciprocal of this radius, \kappa = \frac{1}{R}. This definition captures the intrinsic tendency of the curve to deviate from a straight line, independent of the specific parametrization, and serves as a foundational measure in differential geometry for plane curves. For a straight line, the osculating circle has infinite radius, yielding zero curvature, while for a circle of radius R, the curvature is constant at \frac{1}{R}./01%3A_Curves/1.03%3A_Curvature) Geometrically, the osculating circle aligns perfectly with the curve at the point in terms of position and first derivative (tangent vector), and its curvature matches the second derivative of the curve, ensuring the highest-order approximation possible with a quadratic form like a circle. This second-order matching distinguishes it from mere tangent lines (first-order) or secant approximations, providing an intuitive visualization of how the curve "kisses" the circle at that point.^[19] Intuitively, curvature measures the rate at which the direction of the tangent vector changes as one traverses the curve along its arc length s. This rate of turning is expressed as \kappa = \frac{d\theta}{ds}, where \theta is the angle that the tangent makes with a fixed reference direction; higher values of \kappa indicate faster rotation of the tangent, corresponding to tighter bending. For example, on a curve like a parabola, the curvature increases as the point moves away from the vertex, reflecting accelerating turning.^[2] To visualize this turning, a curvature comb can be constructed along the curve by drawing short line segments perpendicular to the tangent (in the principal normal direction) at regular intervals, with each segment's length scaled proportionally to the local curvature \kappa. Regions of high curvature appear as denser or longer "teeth" in the comb, highlighting variations in bending, such as smooth arcs versus sharp inflections, which aids in qualitative analysis of curve shape.^[20]

Formulas in Arc-Length Parametrization

In arc-length parametrization, a plane curve is represented by a smooth position vector function \mathbf{r}(s) = (x(s), y(s)) where s denotes the arc length from some initial point, satisfying the condition \|\mathbf{r}'(s)\| = \sqrt{(x'(s))^2 + (y'(s))^2} = 1.^[21] The unit tangent vector to the curve is defined as \mathbf{T}(s) = \mathbf{r}'(s), which has constant magnitude 1 and points in the direction of increasing arc length.^[21] Differentiating the identity \mathbf{T}(s) \cdot \mathbf{T}(s) = 1 with respect to s gives $2 \mathbf{T}(s) \cdot \mathbf{T}'(s) = 0, implying that \mathbf{T}'(s) is orthogonal to \mathbf{T}(s).^[21] The curvature \kappa(s) at a point on the curve measures the instantaneous rate at which the tangent vector rotates and is given by

\kappa(s) = \|\mathbf{T}'(s)\| = \|\mathbf{r}''(s)\|.

This expression follows directly from the geometric definition based on the osculating circle, the circle that best approximates the curve at s by matching position, tangent, and curvature. The radius \rho(s) = 1/\kappa(s) of this circle is the reciprocal of the curvature, and \mathbf{r}''(s) represents the centripetal acceleration toward the circle's center, with magnitude $1/\rho. Specifically, the Taylor expansion \mathbf{r}(s + h) = \mathbf{r}(s) + h \mathbf{T}(s) + \frac{h^2}{2} \mathbf{r}''(s) + o(h^2) aligns with the circle's expansion centered at \mathbf{r}(s) + \rho \mathbf{N}(s), where \mathbf{N}(s) is the inward unit normal, confirming \|\mathbf{r}''(s)\| = 1/\rho.^[21]^[22] Within the Frenet-Serret apparatus adapted to plane curves, the evolution of the tangent vector is described by

\frac{d\mathbf{T}}{ds} = \kappa(s) \mathbf{N}(s),

where \mathbf{N}(s) is the unit principal normal vector, defined as \mathbf{N}(s) = \frac{\mathbf{T}'(s)}{\kappa(s)} (assuming \kappa(s) \neq 0), pointing toward the concave side of the curve. The complementary equation is \frac{d\mathbf{N}}{ds} = -\kappa(s) \mathbf{T}(s), forming a closed system that governs the curve's local geometry.^[23] For oriented plane curves, where the parametrization respects a consistent direction, the signed curvature \kappa(s) distinguishes between left and right turns relative to the orientation and is computed as the determinant

\kappa(s) = \det\begin{pmatrix} x'(s) & x''(s) \\ y'(s) & y''(s) \end{pmatrix} = x'(s) y''(s) - y'(s) x''(s).

This signed version satisfies |\kappa(s)| = \|\mathbf{r}''(s)\| and aligns with the Frenet-Serret normal direction, positive for counterclockwise turns and negative otherwise.^[22]

Formulas in General Parametrization

For a plane curve parametrized by a vector-valued function \mathbf{r}(t) = (x(t), y(t)) in \mathbb{R}^2, where t is an arbitrary parameter, the parametrization is general if the speed v(t) = \|\mathbf{r}'(t)\| \neq 0. The curvature \kappa(t) at a point on the curve is then expressed as

\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3},

where the cross product in the plane is the scalar x'(t) y''(t) - y'(t) x''(t), taken in absolute value for the magnitude.

\] This formula arises from the definition of curvature as the rate of change of the unit tangent vector with respect to arc length, adjusted for the general parameter via the chain rule.\[

To derive this, recall that in arc-length parametrization \mathbf{r}(s), the curvature is \kappa = \left\| \frac{d\mathbf{T}}{ds} \right\|, where \mathbf{T}(s) = \mathbf{r}'(s) is the unit tangent vector with \|\mathbf{T}\| = 1. For general t, the arc length s(t) = \int_{t_0}^t v(u) \, du, so \frac{ds}{dt} = v(t). The unit tangent becomes \mathbf{T}(t) = \mathbf{r}'(t) / v(t), and

\frac{d\mathbf{T}}{dt} = \frac{d}{dt} \left( \frac{\mathbf{r}'(t)}{v(t)} \right) = \frac{\mathbf{r}''(t) v(t) - \mathbf{r}'(t) v'(t)}{v(t)^2}.

The curvature satisfies \kappa = \left\| \frac{d\mathbf{T}}{ds} \right\| = \left\| \frac{d\mathbf{T}}{dt} \right\| / v(t). The magnitude \left\| \frac{d\mathbf{T}}{dt} \right\| equals \|\mathbf{r}'(t) \times \mathbf{r}''(t)\| / v(t)^2, since the component parallel to \mathbf{r}'(t) vanishes in the cross product computation, leaving the perpendicular contribution.$$] Thus, [ \kappa(t) = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)| / v(t)^2}{v(t)} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{v(t)^3} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}.[$$ This expression is invariant under reparametrization, meaning that if \tilde{t} = \phi(t) is a smooth, strictly increasing change of parameter with \phi'(t) > 0, the curvature \tilde{\kappa}(\tilde{t}) computed with \tilde{\mathbf{r}}(\tilde{t}) = \mathbf{r}(t(\tilde{t})) equals \kappa(t), as the transformation scales the derivatives in a way that cancels in the formula, preserving the geometric measure of bending.$$]

Expressions in Coordinate Systems

In specific coordinate systems, the curvature of a plane curve can be expressed by specializing the general formula for parametrized curves, κ = ‖r'(t) × r''(t)‖ / ‖r'(t)‖³, where r(t) = (x(t), y(t)) is a parametrization by a parameter t.^[24] These coordinate-specific forms facilitate direct computation without explicit parametrization. For a plane curve given as the graph of a function y = f(x), where f is twice differentiable, the curvature κ at a point (x, f(x)) is [ \kappa = \frac{|f''(x)|}{(1 + [f'(x)]^2)^{3/2}}.

This formula arises by parametrizing the curve as r(x) = (x, f(x)), computing the first and second derivatives r'(x) = (1, f'(x)) and r''(x) = (0, f''(x)), and substituting into the general curvature expression, which simplifies to the magnitude of the [cross product](/page/Cross_product) |f''(x)| divided by the cube of the speed (1 + [f'(x)]²)^{1/2}.[24] In polar coordinates, for a curve r = r(θ) where r is twice differentiable with respect to the polar angle θ, the curvature κ is

\kappa = \frac{|r(\theta)^2 + 2[r'(\theta)]^2 - r(\theta) r''(\theta)|}{[r(\theta)^2 + [r'(\theta)]^2]^{3/2}}.

To derive this, convert to Cartesian parametrization via x(θ) = r(θ) cos θ and y(θ) = r(θ) sin θ, then apply the general formula; the cross product component yields the numerator, while the speed squared is r² + (r')².[24] For an implicit curve defined by F(x, y) = 0, assuming \nablaF \neq 0 at the point of interest, the curvature κ is

\kappa = \frac{|F_{xx} F_y^2 - 2 F_{xy} F_x F_y + F_{yy} F_x^2|}{(F_x^2 + F_y^2)^{3/2}},

where subscripts denote partial derivatives. This expression is obtained by implicitly differentiating F(x, y) = 0 twice to find relations for the second derivatives, parametrizing locally along the curve using the gradient direction for normalization, and inserting into the parametric curvature formula; the denominator reflects the squared norm of the gradient, ensuring consistency with arc-length scaling.[](https://mathworld.wolfram.com/Curvature.html) ### Specific Examples One classic example is the circle of radius $ R $, parametrized in the plane as $ \mathbf{r}(t) = (R \cos t, R \sin t) $. Its curvature is constant and given by $ \kappa = \frac{1}{R} $, independent of the point on the curve.[](https://mathworld.wolfram.com/Curvature.html) This reflects the uniform bending of the circle, where the osculating circle at any point coincides exactly with the curve itself, emphasizing the circle's role as the curve of constant curvature.[](https://mathworld.wolfram.com/Curvature.html) Consider the parabola $ y = \frac{x^2}{4p} $, a standard conic section with focus at $ (0, p) $. The curvature is $ \kappa(x) = \frac{1/(2p)}{\left(1 + \left( \frac{x}{2p} \right)^2 \right)^{3/2}} $, which achieves its maximum value of $ \kappa = \frac{1}{2p} $ at the vertex $ (0, 0) $.[](https://math.gmu.edu/~rsachs/math215/textbook/Math215Ch2Sec3.pdf) As $ |x| $ increases, $ \kappa $ decreases, approaching zero, indicating that the parabola flattens out away from the vertex while the sharpest bend occurs there, consistent with the osculating circle's radius of $ 2p $ at the vertex.[](https://math.gmu.edu/~rsachs/math215/textbook/Math215Ch2Sec3.pdf) For the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ with $ a > b > 0 $, parametrized as $ \mathbf{r}(t) = (a \cos t, b \sin t) $, the curvature varies along the curve and is given by $ \kappa(t) = \frac{ab}{(a^2 \sin^2 t + b^2 \cos^2 t)^{3/2}} $. It reaches a maximum at the vertices $ (\pm a, 0) $ where $ \kappa = \frac{a}{b^2} $, and a minimum at $ (0, \pm b) $ where $ \kappa = \frac{b}{a^2} $, illustrating how the ellipse bends more sharply at the ends of the major axis than the minor axis.[](https://math.stackexchange.com/questions/527538/how-to-calculate-the-curvature-of-an-ellipse) The [cycloid](/page/Cycloid), generated by a point on a [circle](/page/Circle) of [radius](/page/Radius) $ r $ rolling along the x-axis and parametrized as $ \mathbf{r}(t) = (r(t - \sin t), r(1 - \cos t)) $, exhibits varying [curvature](/page/Curvature) $ \kappa(t) = \frac{1}{2r |\sin(t/2)|} $ for $ t \notin 2\pi \mathbb{Z} $.[](https://math.umd.edu/~jmr/241/curves1.htm) At the cusps, where $ t = 2\pi k $ for [integer](/page/Integer) $ k $, $ \kappa $ approaches [infinity](/page/Infinity), signifying an abrupt directional change and infinite bending [radius](/page/Radius) inverse, which geometrically corresponds to the point's instantaneous [rest](/page/REST) and sharp tip formation during the roll.[](https://math.umd.edu/~jmr/241/curves1.htm) ## Curvature of Space Curves ### General Formulas for Space Curves A space curve is a [smooth](/page/Smooth) [mapping](/page/Mapping) $\mathbf{r}: I \to \mathbb{R}^3$, where $I$ is an interval in $\mathbb{R}$, representing a parametrized path in three-dimensional [Euclidean space](/page/Euclidean_space).[](https://www.whitman.edu/mathematics/calculus_online/section13.03.html) The curvature $\kappa$ at a point on the curve quantifies the instantaneous rate at which the curve deviates from being a straight line, extending the notion from plane curves to three dimensions.[](https://tutorial.math.lamar.edu/classes/calciii/curvature.aspx) When the curve is parametrized by [arc length](/page/Arc_length) $s$, so that $|\mathbf{r}'(s)| = 1$, the curvature simplifies to $\kappa(s) = |\mathbf{r}''(s)|$.[](https://math.hawaii.edu/~lee/calculus/curvature.pdf) Equivalently, if $\mathbf{T}(s) = \mathbf{r}'(s)$ is the unit [tangent vector](/page/Tangent_vector), then $\kappa(s) = \left| \frac{d\mathbf{T}}{ds} \right|$, measuring the magnitude of the rate of change of the [tangent](/page/Tangent) direction with respect to [arc length](/page/Arc_length).[](https://math.gmu.edu/~rsachs/math215/textbook/Math215Ch2Sec3.pdf) For a general parametrization $\mathbf{r}(t)$ where $t$ is not necessarily [arc length](/page/Arc_length), the curvature is given by

\kappa(t) = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3},

provided $\mathbf{r}'(t) \neq \mathbf{0}$.[](https://tutorial.math.lamar.edu/classes/calciii/curvature.aspx) This formula arises from the chain rule relating the arc-length derivative to the parameter $t$, and it remains [invariant](/page/Invariant) under reparametrization.[](https://www.whitman.edu/mathematics/calculus_online/section13.03.html) Geometrically, the curvature $\kappa$ is the reciprocal of the radius of the osculating circle, which is the circle in the osculating plane that best approximates the curve at the given point, matching the curve's position, tangent, and curvature up to second order.[](https://math.hawaii.edu/~lee/calculus/curvature.pdf) In the plane curve case, this reduces to the two-dimensional osculating circle, but the three-dimensional version lies within the plane spanned by the tangent and principal normal vectors.[](https://www.math.uci.edu/~ndonalds/math162a/curves.pdf) ### Curvature via Arc and Chord Lengths One approach to approximating the curvature of a space [curve](/page/Curve) involves selecting three points A, B, and C along the curve and computing a discrete measure based on the [geometry](/page/Geometry) of the [triangle](/page/Triangle) they form. The standard [Menger curvature](/page/Menger_curvature), introduced by [Karl Menger](/page/Karl_Menger) in the 1930s as a coordinate-free way to quantify curve curvature using only the positions of points on the curve, is given by

\kappa(A, B, C) = \frac{4 \cdot \area(\triangle ABC)}{|A-B| \cdot |B-C| \cdot |C-A|},

where $\area(\triangle ABC)$ denotes the area of the triangle formed by A, B, and C, and $|A-B|$, $|B-C|$, $|C-A|$ are the Euclidean chord lengths between the respective points.[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) This quantity equals the reciprocal of the circumradius $R$ of $\triangle ABC$ and serves as an approximation to the curve's curvature at B; as A, B, and C converge to a common point P along the curve, $\kappa(A, B, C)$ approaches the Frenet curvature $\kappa(P)$ at P.[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) A related approximation uses arc lengths in place of chord lengths:

\kappa(A, B, C) = \frac{4 \cdot \area(\triangle ABC)}{\arc(AB) \cdot \arc(BC) \cdot \arc(CA)},

which converges to the Frenet curvature in the limit for closely spaced points, since arc lengths approximate chord lengths locally. However, this variant requires measuring along the curve, unlike the position-based Menger definition.[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) An alternative discrete approximation leverages chord lengths and the turning angle at the middle point. For points A, B, and C close together, let $\Delta\theta$ be the angle at B between chords BA and BC, and let $\Delta s$ be the arc length from A to C. Then, the curvature at B is approximated by

\kappa(B) \approx \frac{2 \Delta\theta}{\Delta s}.

This formula arises because $\Delta\theta$ corresponds to the inscribed angle subtending the arc AC, while the central angle (related to curvature via $\kappa = 1/R$) is twice $\Delta\theta$; for small segments, $\Delta s \approx 2R \Delta\theta$, yielding the factor of 2.[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) In the limit as A and C approach B, this converges to the continuous curvature definition based on the derivative of the tangent vector.[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) These arc- and chord-based methods are particularly valuable in numerical computations, such as [curve reconstruction](/page/Reconstruction) from point clouds or simulations where derivative information is unavailable or noisy, as they rely solely on positional data without requiring parametrization or [differentiation](/page/Differentiation).[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) They offer a robust finite-difference alternative for estimating curvature in [three-dimensional space](/page/Three-dimensional_space), applicable in fields like [computer graphics](/page/Computer_graphics) and [geometric analysis](/page/Geometric_analysis).[](https://www.sciencedirect.com/science/article/pii/S000187081200103X) ### Frenet-Serret Framework The Frenet-Serret framework describes the local [geometry](/page/Geometry) of a smooth space curve in three-dimensional [Euclidean space](/page/Euclidean_space) by attaching an orthonormal frame, known as the Frenet frame, to each point along the curve. This frame consists of three unit vectors: the [tangent vector](/page/Tangent_vector) $\mathbf{T}$, which points in the direction of the curve's velocity; the principal normal $\mathbf{N}$, which indicates the direction of bending; and the binormal $\mathbf{B}$, defined as the [cross product](/page/Cross_product) $\mathbf{B} = \mathbf{T} \times \mathbf{N}$, which is perpendicular to the [osculating plane](/page/Osculating_plane). The evolution of this frame with respect to [arc length](/page/Arc_length) $s$ is governed by the Frenet-Serret formulas, which incorporate the curve's curvature $\kappa$ and torsion $\tau$. These formulas were first introduced by Jean Frédéric Frenet in his 1847 doctoral thesis and independently developed by Joseph Alfred Serret in 1851.[](https://www.numdam.org/item/JMPA_1852_1_17__437_0.pdf) The Frenet-Serret formulas are expressed as a [system of differential equations](/page/System_of_differential_equations):

\frac{d\mathbf{T}}{ds} = \kappa \mathbf{N},

\frac{d\mathbf{N}}{ds} = -\kappa \mathbf{T} + \tau \mathbf{B},

\frac{d\mathbf{B}}{ds} = -\tau \mathbf{N}.

Here, curvature $\kappa$ quantifies the instantaneous rate at which the [tangent vector](/page/Tangent_vector) $\mathbf{T}$ rotates as the curve bends within the [osculating plane](/page/Osculating_plane) spanned by $\mathbf{T}$ and $\mathbf{N}$, analogous to the curvature of plane curves but now embedded in space. Torsion $\tau$, in contrast, measures the rate of twisting of the [osculating plane](/page/Osculating_plane) out of itself, capturing the three-dimensional deviation from planarity. For plane curves, $\tau = 0$, simplifying the formulas to the two-dimensional case where $\mathbf{B}$ remains constant. This framework provides a complete kinematic description of the curve's motion, enabling the reconstruction of the curve from its curvature and torsion functions alone, up to rigid motions.[](https://mathworld.wolfram.com/FrenetFormulas.html) To compute the frame vectors, start with the unit [tangent](/page/Tangent) $\mathbf{T} = \frac{d\mathbf{r}/ds}{|d\mathbf{r}/ds|}$, where $\mathbf{r}(s)$ is the position [vector](/page/Vector) parametrized by [arc length](/page/Arc_length). The principal [normal](/page/Normal) is then $\mathbf{N} = \frac{1}{\kappa} \frac{d\mathbf{T}}{ds}$, assuming $\kappa > 0$, and the binormal follows from the [cross product](/page/Cross_product). Torsion is calculated as $\tau = -\mathbf{B} \cdot \frac{d\mathbf{N}}{ds}$, providing a direct measure of out-of-plane twisting. These computations rely on the curve being [regular](/page/Regular) and non-zero curvature, ensuring the [frame](/page/Frame) is well-defined and smooth.[](https://mathworld.wolfram.com/FrenetFormulas.html) ## Curvature of Surfaces ### Normal Curvature and Curves on Surfaces When considering curves embedded on a surface in three-dimensional [Euclidean space](/page/Euclidean_space), the curvature of such a curve must account for the [geometry](/page/Geometry) of the surface itself. A surface $S$ can be locally parametrized by a [smooth](/page/Smooth) map $\mathbf{r}(u,v): U \subset \mathbb{R}^2 \to \mathbb{R}^3$, where $U$ is an [open set](/page/Open_set), and the curve $\boldsymbol{\gamma}(t)$ lies on $S$ via $\boldsymbol{\gamma}(t) = \mathbf{r}(u(t), v(t))$ for some [smooth](/page/Smooth) functions $u(t)$ and $v(t)$. The unit [normal](/page/Normal) vector $\mathbf{N}$ to the surface at a point $\mathbf{p} = \boldsymbol{\gamma}(t_0)$ is obtained from the [cross product](/page/Cross_product) of the partial derivatives $\mathbf{r}_u \times \mathbf{r}_v$, normalized appropriately.[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf)[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The normal curvature $\kappa_n$ of the curve $\boldsymbol{\gamma}$ at $\mathbf{p}$ measures how the curve bends in the direction perpendicular to the surface. For a curve parametrized by [arc length](/page/Arc_length) $s$, it is defined as $\kappa_n = \langle \boldsymbol{\gamma}''(s), \mathbf{N}(\mathbf{p}) \rangle$, where $\langle \cdot, \cdot \rangle$ denotes the [dot product](/page/Dot_product) and $\boldsymbol{\gamma}''(s)$ is the second [derivative](/page/Derivative). This scalar quantifies the [projection](/page/Projection) of the curve's acceleration onto the surface normal, capturing the extrinsic bending relative to the ambient space.[](https://uregina.ca/~mareal/cs5.pdf)[](https://www.math.utsc.utoronto.ca/c63/notes/kat.c63.lect5.cc.pdf) A key relation connects the normal curvature to the total curvature $\kappa$ of $\boldsymbol{\gamma}$ as a space curve. Specifically, $\kappa_n = \kappa \cos \phi$, where $\phi$ is the angle between the principal normal $\mathbf{n}$ of the curve (from its Frenet frame) and the surface normal $\mathbf{N}$ at $\mathbf{p}$. This decomposition separates the curve's bending into components normal and tangential to the surface, with the tangential part known as geodesic curvature. Since $\cos \phi$ depends on the embedding of the surface in $\mathbb{R}^3$, $\kappa_n$ is an extrinsic quantity, varying with different realizations of the same intrinsic surface metric.[](https://uregina.ca/~mareal/cs5.pdf)[](https://csaws.cs.technion.ac.il/~ehudr/publications/pdf/SurazhskyMSER03i.pdf)[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf) Meusnier's theorem provides a fundamental insight into the consistency of normal curvature across curves on the surface. It states that all curves passing through $\mathbf{p}$ and sharing the same tangent direction at $\mathbf{p}$ possess the same normal curvature $\kappa_n$ at that point, regardless of their subsequent paths on the surface. Geometrically, this implies that the radius of curvature of the osculating circle to any such curve, when projected onto the plane spanned by the [tangent](/page/Tangent) and [normal](/page/Normal) to the surface (the normal plane), equals the radius of the osculating circle of the normal section in that direction. This theorem, originally established in [1776](/page/1776), underscores that $\kappa_n$ depends solely on the tangent direction at $\mathbf{p}$, enabling the study of surface curvature through planar intersections.[](https://www.math.brown.edu/tbanchof/balt/ma106/dtext86.html)[](https://uregina.ca/~mareal/cs5.pdf)[](https://www.maths.dur.ac.uk/users/pavel.tumarkin/past/fall16/DG/outline_term2_11.pdf) ### Principal Curvatures In [differential geometry](/page/Differential_geometry), the principal curvatures of a surface at a point are defined as the maximum and minimum values of the normal curvature over all possible directions in the tangent plane at that point. These curvatures, denoted $\kappa_1$ and $\kappa_2$ (with $\kappa_1 \geq \kappa_2$), arise as the eigenvalues of the shape operator, which is the [derivative](/page/Derivative) of the Gauss [map](/page/Map) and measures how the surface bends away from the tangent plane.[](https://fsw01.bcc.cuny.edu/luis.fernandez01/web/texts/dgcs.pdf) The corresponding eigenvectors of the shape operator define the principal directions at the point. In these directions, the principal curves—curves on [the surface](/page/The_Surface) whose tangent vectors align with the principal directions—have the property that their osculating planes coincide with the planes spanned by [the surface](/page/The_Surface) [normal](/page/Normal) and the principal direction, making the curve's [normal](/page/Normal) parallel to the surface's [normal](/page/Normal). This alignment highlights the extremal [bending](/page/Bending) behavior in those orientations.[](https://fsw01.bcc.cuny.edu/luis.fernandez01/web/texts/dgcs.pdf) Euler's theorem relates the normal curvature $\kappa_n$ in an arbitrary direction making an angle $\theta$ with one of the principal directions to the principal curvatures via the formula

\kappa_n = \kappa_1 \cos^2 \theta + \kappa_2 \sin^2 \theta.

This quadratic form, derived from the second fundamental form restricted to the tangent plane, shows that the normal curvature varies continuously between $\kappa_1$ and $\kappa_2$ depending on the direction, confirming the principal curvatures as the extrema.[](https://scholarlycommons.pacific.edu/euler-works/333/) Geometrically, [the principal](/page/The_Principal) curvatures quantify the rates of [maximum and minimum](/page/Maximum_and_minimum) bending of the surface at [the point](/page/The_Point!), providing [insight](/page/Insight) into its local shape: positive values indicate bending toward the same side of the tangent plane, while opposite signs suggest a saddle-like [configuration](/page/Configuration). Umbilical points occur where $\kappa_1 = \kappa_2$, rendering all directions principal and the surface locally spherical in curvature behavior, though not necessarily in shape.[](https://fsw01.bcc.cuny.edu/luis.fernandez01/web/texts/dgcs.pdf) For a sphere of radius $R$, the principal curvatures are equal and constant, with $\kappa_1 = \kappa_2 = 1/R$ at every point, reflecting isotropic bending. In contrast, for an infinite [cylinder](/page/Cylinder) of radius $R$, one principal curvature is $\kappa_1 = 1/R$ along the circumferential direction, while the other is $\kappa_2 = 0$ along the generator lines, illustrating anisotropic curvature where the surface bends only in one direction.[](https://fsw01.bcc.cuny.edu/luis.fernandez01/web/texts/dgcs.pdf) ### Gaussian and Mean Curvatures Gaussian curvature, denoted $K$, is defined as the product of the principal curvatures $\kappa_1$ and $\kappa_2$ at a point on a surface, $K = \kappa_1 \kappa_2$. This scalar invariant, introduced by Carl Friedrich Gauss in his 1827 work *Disquisitiones generales circa superficies curvas*, captures the intrinsic geometry of the surface. Unlike extrinsic measures that depend on embedding in ambient space, Gaussian curvature is an intrinsic property, as established by Gauss's *Theorema Egregium*, which proves that $K$ can be computed solely from the first fundamental form describing distances and angles on the surface itself. This theorem implies that properties like local isometry to the plane are preserved under bending without stretching or tearing. The sign of Gaussian curvature classifies surface points into three categories: elliptic (where $K > 0$, both principal curvatures have the same sign, resembling a sphere locally), parabolic (where $K = 0$, one principal curvature vanishes, like a cylinder), and hyperbolic (where $K < 0$, principal curvatures have opposite signs, forming a saddle shape). Elliptic points curve in a consistently convex or concave manner, while hyperbolic points exhibit opposing curvatures that allow for negative total curvature. Parabolic points mark transitions where the surface flattens in one direction. This classification aids in understanding global surface topology and behavior under geometric flows. Surfaces with zero Gaussian curvature everywhere are developable, meaning they can be isometrically mapped onto a plane without distortion, such as cones, cylinders, or tangent developables. The vanishing of $K$ ensures the surface is ruled—generated by straight lines—and locally flat in the intrinsic metric, a direct consequence of the *Theorema Egregium*. Mean curvature, denoted $H$, is the average of the principal curvatures, $H = \frac{\kappa_1 + \kappa_2}{2}$, providing an extrinsic measure that depends on the surface's embedding in three-dimensional space. It quantifies the tendency of the surface to minimize or maximize enclosed volume for a given area; for instance, constant mean curvature surfaces are critical points for the area functional under fixed volume constraints, as seen in soap films or bubbles. Unlike Gaussian curvature, $H$ is not preserved under isometric deformations. Representative examples illustrate these concepts. A plane has $K = 0$ and $H = 0$ everywhere, embodying flatness with no intrinsic or extrinsic bending. A [sphere](/page/Sphere) of radius $R$ exhibits constant positive [Gaussian curvature](/page/Gaussian_curvature) $K = 1/R^2$ and [mean curvature](/page/Mean_curvature) $H = 1/R$, reflecting uniform [elliptic geometry](/page/Elliptic_geometry) that encloses maximal [volume](/page/Volume) for its surface area. A [hyperbolic](/page/Hyperbolic) paraboloid ([saddle](/page/Saddle) surface) has negative [Gaussian curvature](/page/Gaussian_curvature) $K < 0$ at its vertex, with $H$ varying but often near zero, highlighting [hyperbolic](/page/Hyperbolic) twisting without net [volume](/page/Volume) enclosure preference. ### Second Fundamental Form and Shape Operator The [first fundamental form](/page/First_fundamental_form) provides the intrinsic metric structure on a surface parametrized by $\mathbf{r}(u,v)$ in $\mathbb{R}^3$, given by $ds^2 = E\, du^2 + 2F\, du\, dv + G\, dv^2$, where $E = \mathbf{r}_u \cdot \mathbf{r}_u$, $F = \mathbf{r}_u \cdot \mathbf{r}_v$, and $G = \mathbf{r}_v \cdot \mathbf{r}_v$.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) This [quadratic form](/page/Quadratic_form) induces the Riemannian metric on the [tangent space](/page/Tangent_space) at each point.[](https://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf) The second fundamental form captures the extrinsic geometry by measuring how the surface bends away from the tangent plane, defined as $\mathrm{II} = e\, du^2 + 2f\, du\, dv + g\, dv^2 = -\mathrm{d}\mathbf{N} \cdot \mathrm{d}\mathbf{r}$, where $\mathbf{N}$ is the unit [normal](/page/Normal) vector field (Gauss map).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The coefficients are computed from the parametrization as $e = \mathbf{N} \cdot \mathbf{r}_{uu}$, $f = \mathbf{N} \cdot \mathbf{r}_{uv}$, and $g = \mathbf{N} \cdot \mathbf{r}_{vv}$, reflecting the normal components of the second partial derivatives.[](https://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf) These terms arise from the decomposition $\mathbf{r}_{uu} = \Gamma^u_{uu} \mathbf{r}_u + \Gamma^v_{uu} \mathbf{r}_v + e \mathbf{N}$ and analogous equations for the mixed and $v$-derivatives, where $\Gamma$ are [Christoffel symbols](/page/Christoffel_symbols).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The shape operator (or Weingarten map) $S: T_p M \to T_p M$ is the linear transformation $S(\mathbf{v}) = -\nabla_{\mathbf{v}} \mathbf{N}$, where $\nabla$ is the directional derivative in $\mathbb{R}^3$ projected onto the tangent plane.[](https://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf) It relates to the second fundamental form via the bilinear form $\mathrm{II}(\mathbf{X}, \mathbf{Y}) = \langle S(\mathbf{X}), \mathbf{Y} \rangle = -\langle \mathrm{d}\mathbf{N}(\mathbf{X}), \mathbf{Y} \rangle$, making $S$ self-adjoint with respect to the first fundamental form.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) In the basis $\{\mathbf{r}_u, \mathbf{r}_v\}$, the matrix of $S$ is given by the inverse of the first fundamental form matrix times the second fundamental form matrix, i.e., $[S] = \begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1} \begin{pmatrix} e & f \\ f & g \end{pmatrix}$.[](https://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf) The eigenvalues of $S$ are the principal curvatures $\kappa_1$ and $\kappa_2$.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) From these, the Gaussian curvature $K$ and mean curvature $H$ follow as $K = \det S = \frac{eg - f^2}{EG - F^2}$ and $H = \frac{1}{2} \trace S = \frac{eG - 2fF + gE}{2(EG - F^2)}$, providing scalar measures derived from the tensorial framework.[](https://www.math.miami.edu/~galloway/dgnotes/chpt5.pdf) ## Curvature in Higher Dimensions ### Intrinsic Curvature in Riemannian Manifolds In Riemannian geometry, a smooth manifold $M$ is equipped with a Riemannian metric $g$, which assigns to each tangent space $T_pM$ at a point $p \in M$ a positive definite inner product $g_p$ that varies smoothly over $M$.[](https://www.math.toronto.edu/mkoster/notes/Riemannian-Geometry.pdf) This metric enables the measurement of lengths, angles, and volumes intrinsically on the manifold, without requiring an embedding into a higher-dimensional Euclidean space.[](https://www.cis.upenn.edu/~cis6100/cis610-15-sl11.pdf) The pair $(M, g)$ forms a Riemannian manifold, providing the foundational structure for studying curved spaces in higher dimensions. The [Levi-Civita connection](/page/Levi-Civita_connection) $\nabla$ on a [Riemannian manifold](/page/Riemannian_manifold) is the unique torsion-free [affine connection](/page/Affine_connection) that is compatible with the metric, meaning it preserves the inner product under [parallel transport](/page/Parallel_transport).[](https://www.cis.upenn.edu/~cis6100/cis61008Riem-conn.pdf) This connection allows for the covariant differentiation of vector fields and tensors in a manner consistent with the [geometry](/page/Geometry) defined by $g$. The [Riemann curvature tensor](/page/Riemann_curvature_tensor) $R$, which quantifies the intrinsic curvature, arises naturally from $\nabla$ and is defined by

R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z

for vector fields $X, Y, Z$ on $M$.[](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/appendc.pdf) This tensor captures how the connection fails to commute, reflecting the manifold's deviation from flatness. The intrinsic nature of the Riemann curvature tensor is evident in its detectability through geodesic behavior alone, without reference to an ambient space. Geodesics are curves that locally minimize length, analogous to straight lines in Euclidean space, and the tensor governs their relative acceleration via the geodesic deviation equation, which describes how nearby geodesics converge or diverge based on the local geometry.[](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) For instance, positive curvature causes geodesics to focus, while negative curvature leads to spreading, providing a purely internal measure of the manifold's shape.[](https://digitalcommons.latech.edu/cgi/viewcontent.cgi?article=1008&context=mathematics-senior-capstone-papers) This generalizes the Gaussian curvature from two-dimensional surfaces to arbitrary dimensions. A key consequence in the two-dimensional case is the Gauss-Bonnet theorem, which states that for a compact oriented Riemannian surface without boundary, the integral of the [Gaussian curvature](/page/Gaussian_curvature) $K$ over the surface equals $2\pi$ times the [Euler characteristic](/page/Euler_characteristic).[](https://math.berkeley.edu/~alanw/240papers00/zhu.pdf) These ideas originated in Bernhard Riemann's 1854 habilitation lecture, "On the Hypotheses Which Lie at the Bases of Geometry," where he first conceptualized manifolds with variable curvature as intrinsic geometric objects.[](https://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf) ### Sectional, Ricci, and Scalar Curvatures In Riemannian manifolds of dimension greater than two, the full [Riemann curvature tensor](/page/Riemann_curvature_tensor) encodes the intrinsic geometry, but scalar measures derived from its contractions provide more accessible invariants for analysis and classification.[](https://link.springer.com/book/9780817634902) The [sectional curvature](/page/Sectional_curvature) generalizes the [Gaussian curvature](/page/Gaussian_curvature) of surfaces to higher dimensions by quantifying the curvature of two-dimensional subspaces in the [tangent space](/page/Tangent_space). For tangent vectors $X$ and $Y$ at a point $p$ that are linearly independent, the [sectional curvature](/page/Sectional_curvature) $K(X, Y)$ of the [plane](/page/Plane) they span is defined as

K(X, Y) = \frac{\langle R(X, Y)Y, X \rangle}{|X|^2 |Y|^2 - \langle X, Y \rangle^2},

where $R$ is the [Riemann curvature tensor](/page/Riemann_curvature_tensor) and $\langle \cdot, \cdot \rangle$ denotes the inner product induced by the metric.[](https://link.springer.com/book/10.1007/978-3-319-91755-9) This expression measures how much the manifold deviates from being flat in that specific [plane](/page/Plane), with positive values indicating elliptic behavior (like spheres), negative values [hyperbolic](/page/Hyperbolic) (like saddles), and zero values flat (like [Euclidean space](/page/Euclidean_space)). When restricted to surfaces, $K(X, Y)$ reduces to the [Gaussian curvature](/page/Gaussian_curvature).[](https://link.springer.com/book/9780817634902) The [Ricci curvature](/page/Ricci_curvature) arises as a further [contraction](/page/Contraction) of the Riemann tensor, averaging the sectional curvatures over all planes containing a given [direction](/page/Direction). For a [unit vector](/page/Unit_vector) $X$, the [Ricci curvature](/page/Ricci_curvature) in that [direction](/page/Direction) is

\text{Ric}(X, X) = \sum_{i=1}^{n-1} K(X, e_i),

where $\{e_i\}$ is an [orthonormal basis](/page/Orthonormal_basis) for the subspace orthogonal to $X$ in the [tangent space](/page/Tangent_space), and $n$ is the [dimension](/page/Dimension) of the manifold.[](https://link.springer.com/book/10.1007/978-3-319-91755-9) The Ricci tensor $\text{Ric}$ is symmetric and bilinear, serving as the trace of the curvature operator with respect to the [metric](/page/Metric). It provides information on the average curvature experienced by geodesics in the [direction](/page/Direction) of $X$, influencing volume growth and convergence properties of the manifold.[](https://link.springer.com/book/9780817634902) The [scalar curvature](/page/Scalar_curvature) $\text{Scal}$ is the complete trace of the Ricci tensor over an [orthonormal basis](/page/Orthonormal_basis) $\{e_i\}_{i=1}^n$ of the [tangent space](/page/Tangent_space):

\text{Scal} = \sum_{i=1}^n \text{Ric}(e_i, e_i) = 2 \sum_{i < j} K(e_i, e_j).

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[PDF] The Gauss-Bonnet Theorem and its Applications - UC Berkeley math
The Gauss-Bonnet theorem is an important theorem in differential geometry. It is intrinsically beautiful because it relates the curvature of a manifold—a ...
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[PDF] On the Hypotheses which lie at the Bases of Geometry. Bernhard ...
It is easy to see that surface whose curvature is positive may a lways be ro lled on a sphere whose radius is unity divided by the s q uare root of the ...
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Riemannian Geometry | SpringerLink
Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students.
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Introduction to Riemannian Manifolds - SpringerLink
This book focuses on ensuring that student develops an intimate acquaintance with geometric meaning of curvature in Riemannian geometry graduate course.
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[math/0303109] Ricci flow with surgery on three-manifolds - arXiv
Mar 10, 2003 · This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions.
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Developable Surface -- from Wolfram MathWorld
A developable surface, also called a flat surface (Gray et al. 2006, p. 437), is a ruled surface having Gaussian curvature K=0 everywhere.
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Generalized Cylinder -- from Wolfram MathWorld
A generalized cylinder is a developable surface and is sometimes called a "cylindrical surface" (Kern and Bland 1948, p. 32) or "cylinder surface" (Harris and ...
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Tangent Developable -- from Wolfram MathWorld
A ruled surface M is a tangent developable of a curve y if M can be parameterized by x(u,v)=y(u)+vy^'(u). A tangent developable is a developable surface.
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Cone -- from Wolfram MathWorld
### Summary of Curvature of a Cone Surface (Gaussian and Mean)
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Minimal Surface -- from Wolfram MathWorld
Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as x=(u,v,h(u,v)) therefore satisfies Lagrange's equation.
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[PDF] Deep learning via Hessian-free optimization - Computer Science
Hessian-free optimization is a 2nd-order method for deep learning, addressing issues with gradient descent and pathological curvature, and is practical for ...