This is an alphabetical index of articles related to curves used in mathematics.

- Acnode
- Algebraic curve
- Arc
- Asymptote
- Asymptotic curve
- Barbier's theorem
- Bézier curve
- Bézout's theorem
- Birch and Swinnerton-Dyer conjecture
- Bitangent
- Bitangents of a quartic
- Cartesian coordinate system
- Caustic
- Cesàro equation
- Chord (geometry)
- Cissoid
- Circumference
- Closed timelike curve
- concavity
- Conchoid (mathematics)
- Confocal
- Contact (mathematics)
- Contour line
- Crunode
- Cubic Hermite curve
- Curvature
- Curve orientation
- Curve fitting
- Curve of constant width
- Curve of pursuit
- Curves in differential geometry
- Cusp
- Cyclogon
- De Boor algorithm
- Differential geometry of curves
- Eccentricity (mathematics)
- Elliptic curve cryptography
- Envelope (mathematics)
- Fenchel's theorem
- Genus (mathematics)
- Geodesic
- Geometric genus
- Great-circle distance
- Harmonograph
- Hedgehog (curve)
- Hilbert's sixteenth problem
- Hyperelliptic curve cryptography
- Inflection point
- Inscribed square problem
- intercept, y-intercept, x-intercept
- Intersection number
- Intrinsic equation
- Isoperimetric inequality
- Jordan curve
- Knot
- Limit cycle
- Linking coefficient
- List of circle topics
- Loop (knot)
- M-curve
- Mannheim curve
- Meander (mathematics)
- Mordell conjecture
- Natural representation
- Opisometer
- Orbital elements
- Osculating circle
- Osculating plane
- Osgood curve
- Parallel (curve)
- Parallel transport
- Parametric curve
- Perimeter
- Pi
- Plane curve
- Pochhammer contour
- Polar coordinate system
- Prime geodesic
- Projective line
- Ray
- Regular parametric representation
- Reuleaux triangle
- Ribaucour curve
- Riemann–Hurwitz formula
- Riemann–Roch theorem
- Riemann surface
- Road curve
- Sato–Tate conjecture
- secant
- Singular solution
- Sinuosity
- Slope
- Space curve
- Spinode
- Square wheel
- Subtangent
- Tacnode
- Tangent
- Tangent space
- Tangential angle
- Torsion of curves
- Trajectory
- Transcendental curve
- W-curve
- Whewell equation
- World line

In classical mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

In geometry, the **tangent line** to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve *y* = *f*(*x*) at a point *x* = *c* if the line passes through the point (*c*, *f* ) on the curve and has slope *f*'(*c*), where *f*' is the derivative of *f*. A similar definition applies to space curves and curves in *n*-dimensional Euclidean space.

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In mathematics, the **winding number** of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

In geometry, a **geodesic** is commonly a curve representing in some sense the shortest path between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

**Differential geometry of curves** is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In differential geometry of curves, the **osculating circle** of a sufficiently smooth plane curve at a given point *p* on the curve has been traditionally defined as the circle passing through *p* and a pair of additional points on the curve infinitesimally close to *p*. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all **tangent circles** at the given point that approaches the curve most tightly, was named *circulus osculans* by Leibniz.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In mathematics, a **bitangent** to a curve *C* is a line *L* that touches *C* in two distinct points *P* and *Q* and that has the same direction as *C* at these points. That is, *L* is a tangent line at *P* and at *Q*.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

**Geometrical design** (**GD**) is a branch of computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumesand is closely related to geometric modeling. Core problems are curve and surface modelling and representation. GD studies especially the construction and manipulation of curves and surfaces given by a set of points using polynomial, rational, piecewise polynomial, or piecewise rational methods. The most important instruments here are parametric curves and parametric surfaces, such as Bézier curves, spline curves and surfaces. An important non-parametric approach is the level-set method.

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Images, videos and audio are available under their respective licenses.