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.

Therefore, four loose families of more-efficient light transport modelling techniques have emerged: rasterization, including scanline rendering, geometrically projects objects in the scene to an image plane, without advanced optical effects ; ray casting considers the scene as observed from a specific point-of-view, calculating the observed image based only on geometry and very basic optical laws of reflection intensity, and perhaps using Monte Carlo techniques to reduce artifacts ; and ray tracing is similar to ray casting, but employs more advanced optical simulation, and usually uses Monte Carlo techniques to obtain more realistic results at a speed that is often orders of magnitude slower.

Other applications include performing geometrical operations with shapes ( constructive solid geometry ) in CAD, collision detection in robotics and 3-D video games, ray tracing and other computer applications that involve handling of complex spatial scenes.

It is also a useful reference for those wanting to learn how ray tracing and related geometry and graphics algorithms work.

In the differential geometry of curves, an involute ( also known as evolvent ) is a curve obtained from another given curve by attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve ; or in reverse, unwound.

It includes an interactive geometry editor, ray tracing support for graphics rendering and geometric analysis, computer network distributed framebuffer support, scripting, image-processing and signal-processing tools.

Storing objects in a space-partitioning data structure makes it easy and fast to perform certain kinds of geometry queries — for example, determining whether two objects are close to each other in collision detection, or determining whether a ray intersects an object in ray tracing.

This technique often produces results that are superficially similar to those generated by raytracing, but is less computationally expensive since the radiance value of the reflection comes from calculating the angles of incidence and reflection, followed by a texture lookup, rather than followed by tracing a ray against the scene geometry and computing the radiance of the ray, simplifying the GPU workload.

As predicted by the VSEPR model of electron pair repulsion, the molecular geometry of alkenes includes bond angles about each carbon in a double bond of about 120 °.

The quadrivium includes geometry, arithmetic, astronomy, music.

The noncommutative ring theory, besides the rich structure theory, includes the study of rings such as Banach algebras and operator algebras in functional analysis and the representation ring and cohomology rings in geometry.

While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.

In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

In addition to the previously available vertex and pixel shader stages, the API includes a geometry shader stage that breaks the old model of one vertex in / one vertex out, to allow geometry to actually be generated from within a shader, allowing for complex geometry to be generated entirely on the graphics hardware.

* Euclid's Elements, a 13-book mathematical treatise written by Euclid, that includes both geometry and number theory

In convex geometry, a face of a polytope P is the intersection of any supporting hyperplane of P and P. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.

The book also includes proofs in Euclidean geometry.

This includes obtaining police reports of crashes, observing road user behavior, and collecting information on traffic signs, road surface markings, traffic lights and road geometry.

Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidean geometry.

More generally, the Ricci tensor can be defined in broader class of metric geometries ( by means of the direct geometric interpretation, below ) that includes Finsler geometry.

However, inversive geometry is the larger study since it includes the raw inversion in a circle ( not yet made, with conjugation, into reciprocation ).

Inversive geometry also includes the conjugation mapping.

Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry.

It includes areas currently fashionable ( the Chern-Simons theory arising from a 1974 paper written jointly with Jim Simons ), perennial ( the Chern-Weil theory linking curvature invariants to characteristic classes from 1944, after the Allendoerfer-Weil paper of 1943 on the Gauss-Bonnet theorem ), the foundational ( Chern classes ), and some areas such as projective differential geometry and webs that have a lower profile.

Such results show that transformation geometry includes non-commutative processes.

A chemical structure includes molecular geometry, electronic structure and crystal structure of molecules.

The Latin version also includes Florimond de Beaune's Notes brièves, the first important introduction to Descartes ' cartesian geometry.

A larger field sometimes called " arithmetic of algebraic varieties " now includes diophantine geometry with class field theory, complex multiplication, local zeta-functions and L-functions.