"built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones: The inverse function exists as well: Chord (geometry) A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just ""secant"", is the infinite line extension of a chord. More generally, a chord is a line segment joining two points on any curve,"
"Tangent In geometry, the tangent line (or simply tangent) 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 at a point on the curve if the line passes through the point on the curve and has slope where ""f"" is the derivative of ""f"". A similar definition applies to space curves and curves in ""n""-dimensional Euclidean space. As it"
"where the line crosses the formula_37 axis. Depending on the context, either formula_42 or the point formula_138 is called the formula_37-intercept. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Informally, it is a 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 at a point on the curve if the line passes through the point on the curve and has slope where ""f"" is"
"More generally, there is a ""k""-dimensional tangent space at each point of a ""k""-dimensional manifold in the ""n""-dimensional Euclidean space. Tangent In geometry, the tangent line (or simply tangent) 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 at a point on the curve if the line passes through the point on the curve and has slope where ""f"""
"a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point. Closely related meanings include the graph of a function (for example, Phillips curve) and a two-dimensional graph. Interest in curves began long before"
"Crunode In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ""ordinary double point"". For a plane curve, defined as the locus of points ""f""(""x"", ""y"") = 0, where ""f""(""x"", ""y"") is a smooth function of variables ""x"" and ""y"" ranging over the real numbers, a crunode of the curve is a singularity of the function ""f"", where both partial derivatives formula_1 and formula_2 vanish. Further the Hessian matrix of second derivatives"
"Intersection (Euclidean geometry) In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the intersection of two distinct lines, which either is one point or does not exist if the lines are parallel. Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved"
"Conchoid (mathematics) A conchoid is a curve derived from a fixed point ""O"", another curve, and a length ""d"". It was invented by the ancient Greek mathematician Nicomedes. For every line through ""O"" that intersects the given curve at ""A"" the two points on the line which are ""d"" from ""A"" are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius ""d"" and center ""O"". They are called conchoids because the shape of their outer branches resembles conch shells. The simplest expression uses polar coordinates with ""O"" at the origin. If"
"of thumb approximations only. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—""amphicyrtic"" (Gr. , on both sides, κυρτός, convex) or ""cissoidal"" (Gr. κισσός, ivy), biconvex; ""xystroidal"" or ""sistroidal"" (Gr. ξυστρίς, a tool for scraping), concavo-convex; ""amphicoelic"" (Gr. κοίλη, a hollow) or ""angulus lunularis"", biconcave. The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only"
"Contour line A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph of the function ""f""(""x"", ""y"") parallel to the ""x"", ""y"" plane. In cartography, a contour line (often just called a ""contour"") joins points of equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys"
"Vertex (geometry) In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. The ""vertex"" of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight ""sides"" meeting at one place. A vertex is a corner point of a"
"will have both positive and negative eigenvalues. Crunode In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ""ordinary double point"". For a plane curve, defined as the locus of points ""f""(""x"", ""y"") = 0, where ""f""(""x"", ""y"") is a smooth function of variables ""x"" and ""y"" ranging over the real numbers, a crunode of the curve is a singularity of the function ""f"", where both partial derivatives formula_1 and formula_2 vanish."
"are often defined in terms of directed smooth curves. These provide a precise definition of a ""piece"" of a smooth curve, of which a contour is made. A smooth curve is a curve with a non-vanishing, continuous derivative such that each point is traversed only once ( is one-to-one), with the possible exception of a curve such that the endpoints match (). In the case where the endpoints match the curve is called closed, and the function is required to be one-to-one everywhere else and the derivative must be continuous at the identified point (). A smooth curve that is"
"the intersection number of the graph of a function with the diagonal may be computed using homology via the Lefschetz fixed point theorem; the self-intersection of the diagonal is the special case of the identity function. Diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ""diagonal"" derives from the ancient Greek διαγώνιος ""diagonios"", ""from angle to angle"" (from διά- ""dia-"", ""through"", ""across"" and γωνία ""gonia"", ""angle"", related to ""gony"" ""knee""); it was used by"
"Curve In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that its curvature need not be zero. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that"
"length"" (Def. 2), while a ""straight"" line is defined as ""a line that lies evenly with the points on itself"" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement ""The extremities of a line are points,"" (Def. 3). Later commentators further classified lines according to various schemes. For example: The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction. These curves include: A fundamental advance in the theory of curves was the advent of analytic geometry"
"given line and fixed point. Glissette A glissette is a curve determined by either the locus of any point, or the envelope of any line or curve, that is attached to a curve that slides against or along two other fixed curves. A basic example is that of a line segment of which the endpoints slide along two perpendicular lines. The glissette of any point on the line forms an ellipse. Similarly, the envelope glissette of the line segment in the example above is an astroid. Any conchoid may be regarded as a glissette, with a line and one of"
"Convex curve In geometry, a convex curve is a curve in the Euclidean plane which lies completely on one side of each and every one of its tangent lines. The boundary of a convex set is always a convex curve. Any straight line ""L"" divides the Euclidean plane into two half-planes whose union is the entire plane and whose intersection is ""L"" . We say that a curve ""C"" ""lies on one side of ""L"""" if it is entirely contained in one of the half-planes. A plane curve is called convex if it lies on one side of each of"
"passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is ""going in the same direction"" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that ""just touches"" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. The word ""tangent"" comes from the Latin ""tangere"", 'to touch'. Euclid makes several"
"Diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ""diagonal"" derives from the ancient Greek διαγώνιος ""diagonios"", ""from angle to angle"" (from διά- ""dia-"", ""through"", ""across"" and γωνία ""gonia"", ""angle"", related to ""gony"" ""knee""); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as ""diagonus"" (""slanting line""). In matrix algebra, a diagonal of a square matrix is"
"surfaces. Curve In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that its curvature need not be zero. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means"
"Orthoptic (geometry) In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: Generalizations: Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation formula_2. The slope at a point of the parabola is formula_3. Replacing formula_4 gives the parametric representation of the parabola with the tangent slope as parameter: formula_5 The tangent has the equation formula_6 with the still unknown formula_7, which can be determined by inserting the coordinates of the parabola point. One gets formula_8"
"curves. Convex curve In geometry, a convex curve is a curve in the Euclidean plane which lies completely on one side of each and every one of its tangent lines. The boundary of a convex set is always a convex curve. Any straight line ""L"" divides the Euclidean plane into two half-planes whose union is the entire plane and whose intersection is ""L"" . We say that a curve ""C"" ""lies on one side of ""L"""" if it is entirely contained in one of the half-planes. A plane curve is called convex if it lies on one side of each"
"tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an ""inflection point"". Circles, parabolas, hyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic function, which has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine. Conversely, it may happen that the curve lies entirely on one side of a straight line"
"convex curve is a convex curve that does not contain any line segments. Equivalently, a strictly convex curve is a curve that intersects any line in at most two points, or a simple curve in convex position, meaning that none of its points is a convex combination of any other subset of its points. Every convex curve that is the boundary of a closed convex set has a well-defined finite length. That is, these curves are a subset of the rectifiable curves. According to the four-vertex theorem, every smooth convex curve that is the boundary of a closed convex set"
"point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk. It will meet it on one side or the other. This proves that the complement of the curve has at most two connected components. On the other hand using the Cauchy integral formula for the winding number, it can be seen that the winding number is constant on connected components of the complement of the curve, is zero near infinity and increases by 1"
"Edge (geometry) In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which"
"Linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. If the two known points are given by the coordinates formula_1 and formula_2, the linear interpolant is the straight line between these points. For a value ""x"" in the interval formula_3, the value ""y"" along the straight line is given from the equation of slopes which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with ""n"" = 1. Solving"
"consisting of the joins of corresponding points of two related ranges (points on a line) on different bases (the lines the points are on). Such an envelope is called a line conic (or dual conic). In the real projective plane, a point conic has the property that every line meets it in two points (which may coincide, or may be complex) and any set of points with this property is a point conic. It follows dually that a line conic has two of its lines through every point and any envelope of lines with this property is a line conic."
"the same points, but at right angles. Hypercycle (geometry) In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the ""axis"", ""center"", or ""base line"" of the hypercycle. The lines perpendicular to the ""axis"", which is also perpendicular to"
"Strophoid In geometry, a strophoid is a curve generated from a given curve ""C"" and points ""A"" (the fixed point) and ""O"" (the pole) as follows: Let ""L"" be a variable line passing through ""O"" and intersecting ""C"" at ""K"". Now let ""P"" and ""P"" be the two points on ""L"" whose distance from ""K"" is the same as the distance from ""A"" to ""K"". The locus of such points ""P"" and ""P"" is then the strophoid of C with respect to the pole ""O"" and fixed point ""A"". Note that ""AP"" and ""AP"" are at right angles in this"
"point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are collinear. The ""shortness"" and ""straightness"" of a line, interpreted as the property that the distance along the line between any two of its points is minimized (see"
"Intersection curve In geometry, an intersection curve is, in the most simple case, the intersection line of two non-parallel planes in Euclidean 3-space. In general, an intersection curve consists of the common points of two ""transversally"" intersecting surfaces, meaning that at any common point the surface normals are not parallel. This restriction excludes cases where the surfaces are touching or have surface parts in common. The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c)"