Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, which can then be imagined as a formal system without any intrinsic real-world meaning.

The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

In algebraic geometry, in contrast, there is an intrinsic definition of tangent space at a point P of a variety V, that gives a vector space of dimension at least that of V. The points P at which the dimension is exactly that of V are called the non-singular points ; the others are singular points.

* In differential geometry, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton – Jacobi equation

After the discovery of the intrinsic definition of curvature, which is closely 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 large.

By allowing also isometric ( or near-isometric ) deformations like bending, the intrinsic geometry of the object will stay the same, while sub-parts might be located at very different positions in space.

In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all.

To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point.

In differential geometry, the Ricci flow is an intrinsic geometric flow.

Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data.

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional

Though is possible to define the angles by geometry, normally the intrinsic rotations equivalence or the extrinsic rotations equivalence are used instead.

The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations, because a regular curve in a Euclidean space has no intrinsic geometry.

One can say that the extrinsic meaning of a singular point isn't in question ; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point.

The sample covariance matrix ( SCM ) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R < sup > p × p </ sup >; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator.

This fact is intrinsic to the geometry of an equilateral triangle — a direct result of the basic trigonometric functions.

Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety.

Most importantly, an embedding is merely a shape, while a potential plot has a distinguished " downward " direction ; thus turning a gravity well " upside down " ( by negating the potential ) turns the attractive force into a repulsive force, while turning a Schwarzschild embedding upside down ( by rotating it ) has no effect, since it leaves its intrinsic geometry unchanged.

In differential geometry, the Calabi flow is an intrinsic geometric flow — a process which deforms the metric of a Riemannian manifold — in a manner formally analogous to the way that vibrations are damped and dissipated in a hypothetical curved n-dimensional structural element.

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation.

The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed.

In analytic geometry, the ellipse is defined as the set of points of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation

It was also found that the field equation governing the dilaton ( derived from differential geometry ) was the Schrödinger equation and consequently amenable to quantization.

This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of space-time through the Dirac algebra.

* In differential geometry, see Liouville's equation

In geometry, a hyperplane of an n-dimensional space V is a " flat " subset of dimension n − 1, or equivalently, of codimension 1 in V ; it may therefore be referred to as an ( n − 1 )- flat of V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly ; in all cases however, any hyperplane can be given in coordinates as the solution of a single ( due to the " codimension 1 " constraint ) algebraic equation of degree 1 ( due to the " flat " constraint ).

For example, mechanics and mathematical analysis were commonly combined into one subject during the 18th century, united by the differential equation concept ; while algebra and geometry were considered largely distinct.

Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically.

In geometry, curve sketching ( or curve tracing ) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot.

So if is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation.

So, for example, concepts like singularities ( the most widely known of which in general relativity is the black hole ) which can not be expressed completely in a real world geometry, can correspond to particular states of a mathematical equation.

A Cartan geometry is a generalization of a smooth Klein geometry, in which the structure equation is not assumed, but is instead used to define a notion of curvature.

The inertial trajectories of particles and radiation ( geodesics ) in the resulting geometry are then calculated using the geodesic equation.

Robert Hooke ( 1635 – 1703 ), who mathematically analyzed the universal joint, was the first to note that the geometry and mathematical description of the ( non-secular ) equation of time and the universal joint were identical, and proposed the use of a universal joint in the construction of a " mechanical sundial ".

Adamowicz does not treat the time-space components of the metric, the spin-torsion field equation, spin-orbit coupling and the non-symmetric momentum tensor, the geometry of torsion, or quantum mechanical spin.

In this reformulation, spacetimes are sliced up into spatial hyperslices in a rather arbitrary fashion, and the vacuum Einstein field equation is reformulated as an evolution equation describing how, given the geometry of an initial hyperslice ( the " initial value "), the geometry evolves over " time ".

However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori.

In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves ( or surfaces ) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition.

* Chord ( geometry ), a line segment joining two points on a curve

In physics and geometry, the catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

At each point, the derivative of is the slope of a Line ( geometry ) | line that is tangent to the curve.

Analytically, x can also be raised to an irrational power ( for positive values of x ); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.

η is the pump efficiency, and may be given by the manufacturer's information, such as in the form of a pump curve, and is typically derived from either fluid dynamics simulation ( i. e. solutions to the Navier-stokes for the particular pump geometry ), or by testing.

At each point, the derivative is the slope of a Line ( geometry ) | line that is tangent to the curve.

In geometry, the tangent line ( or simply the tangent ) to a plane curve at a given point is the straight line that " just touches " the curve at that point — that is, coincides with the curve at that point and, near that point, is closer to the curve that any other line passing through that point.

It is based on the Koch curve, which appeared in a 1904 paper titled " On a continuous curve without tangents, constructible from elementary geometry " ( original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire ) by the Swedish mathematician Helge von Koch.

* Fractals derived from standard geometry by using iterative transformations on an initial common figure like a straight line ( the Cantor dust or the von Koch curve ), a triangle ( the Sierpinski triangle ), or a cube ( the Menger sponge ).

* Arc ( geometry ), a segment of a differentiable curve

* In hyperbolic geometry they " curve away " from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular ; these lines are often called ultraparallels.

* In elliptic geometry the lines " curve toward " each other and intersect.

The definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.

From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry ; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.