Differentiable curveStudy of curves from a differential point of viewDifferential geometry of curves is the branch of geometry that deals with smoothcurves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach.
Topological spaceIn mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.Interval (mathematics)In mathematics, an interval is the set of all real numbers lying between two fixed endpoints with no "gaps". For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval. An interval may contain neither endpoint (called an open interval), both endpoints (called a closed interval), or either endpoint (called a semi-open or semi-closed interval).
Image (mathematics)In mathematics, the image of a functionf:X→Y{\displaystyle f:X\to Y} is the set of all f(x){\displaystyle f(x)} such that x{\displaystyle x} belongs to the domain of f{\displaystyle f}. The image by f{\displaystyle f} of an element x{\displaystyle x} of the domain of f{\displaystyle f} is f(x){\displaystyle f(x)}, that is, the output corresponding to the input x{\displaystyle x}.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.
Point (geometry)In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist.
Level setIn mathematics, a level set of a real-valued functionf of nreal variables is a set where the function takes on a given constant value c, that is: Lc(f)={(x1,…,xn)∣f(x1,…,xn)=c}.{\displaystyle L_{c}(f)=\left\{(x_{1},\ldots ,x_{n})\mid f(x_{1},\ldots ,x_{n})=c ight\}~.}When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x1 and x2.
Line (geometry)In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve and an idealisation of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher.Continuous functionIn mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting its argument to sufficiently small changes. A discontinuous function is a function that is not continuous.
Parametric equationIn mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or more variables called parameters. In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point. For this case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve.LinearityIn mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined by f ( x ) = ( a x , b x ) {\displaystyle f(x)=(ax,bx)} that maps the real line to a line in the Euclidean plane R2 that passes through the origin.
Euclid's ElementsThe Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid. The Elements is the oldest extant large-scale deductive treatment of mathematics.
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