"Normal (geometry) In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In three dimensions, a surface normal, or simply normal, to a surface at a point ""P"" is a vector that is perpendicular to the tangent plane to that surface at ""P"". The word ""normal"" is also used as an adjective: a line normal to a plane, the normal component"
"in Lie sphere geometry. Tangent lines to circles In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line ""t"" to a circle ""C"" intersects the circle at a single point"
"Tangent lines to circles In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line ""t"" to a circle ""C"" intersects the circle at a single point T. For comparison, secant"
"Secant line In geometry, a secant of a curve is a line that intersects the curve in at least two (distinct) points. The word ""secant"" comes from the Latin word ""secare"", meaning ""to cut"". In the case of a circle, a secant will intersect the circle in exactly two points and a chord is the line segment determined by these two points, that is the interval on a secant whose endpoints are these points. A straight line can intersect a circle in two, one or zero points. A line intersecting in two points is called a ""secant line"", in one"
"those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was ""a right line which touches a curve, but which when produced, does not cut it"". This old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The intuitive notion that a tangent line ""touches"" a curve can be made more explicit by considering the sequence"
"Circumference In geometry, the circumference (from Latin ""circumferentia"", meaning ""carrying around"") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, ""circumference"" may also refer to the edge itself rather than to the"
"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, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter. Every diameter is a chord, but not every chord is a diameter. The word ""chord"" is from the Latin ""chorda"" meaning ""bowstring"". Among properties of chords of a circle are the following: The midpoints of a"
"is not being treated formally. When geometry was first formalised by Euclid in the ""Elements"", he defined a general line (straight or curved) to be ""breadthless length"" with a straight line being a line ""which lies evenly with the points on itself"". These definitions serve little purpose since they use terms which are not, themselves, defined. In fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms,"
"3-secants of a finite set of points. Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points. Secant line In geometry, a secant of a curve is a line that intersects the curve in at least two (distinct) points. The word ""secant"" comes from the Latin word ""secare"", meaning ""to cut"". In the case of a circle, a secant will intersect the circle in exactly two points and a chord is the line segment determined by these two points, that is the"
"Trochoid A trochoid (from the Greek word for wheel, ""trochos"") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word ""trochoid"" was coined by Gilles de Roberval. As a circle of radius ""a"" rolls without slipping along a line ""L,"" the center ""C"" moves parallel to ""L,"" and every other point ""P"" in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let ""CP = b"". Parametric equations of the trochoid for which ""L"" is"
"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"
"lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed. The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of"
"Cross section (geometry) In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross sections. The boundary of a cross section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is a sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional"
"exactly one line. This concept can be generalized to systems of points and circles, with each circle being determined by three non-collinear points. However, three collinear points determine a line, not a circle. This drawback can be removed by adding a point at infinity to every line. If we call both circles and such completed lines ""cycles"", we get an incidence structure in which every three points determine exactly one cycle. In an affine plane the parallel relation between lines is essential. In the geometry of cycles, this relation is generalized to the ""touching"" relation. Two cycles ""touch"" each other"
"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"
"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"
"shape analysis, the topological skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries. In Euclidean geometry, parallel lines (lines that never intersect) are equidistant in the sense that the distance of any point on one line from the nearest point on the other line is the same for all points. In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form an hypercycle (which is a curve not a line). Equidistant A point is said to be equidistant from a"
"Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of ""the"" diameter rather than ""a"" diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius"
"including Euclid's definition as 'that which has no part' and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points. Euclid described a line as ""breadthless length"" which ""lies equally with respect to the points on itself"". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line"
"Circular segment In geometry, a circular segment (symbol: ⌓) is a region of a circle which is ""cut off"" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc. Let ""R"" be the radius of the circle, ""θ"" the central angle in radians, α is the central angle in degrees, ""c"" the chord length, ""s"" the arc length, ""h"" the sagitta (height) of the"
"Line (geometry) The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the ""[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its"
"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"
"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"""
"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"
"Cyclogon In mathematics, in geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. The polygon need not even be convex: it could even be a star-shaped polygon. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside"
"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"
"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"
"but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i.e., the Euclidean plane, two lines which do not intersect"
"circle and tangent line has a reflection symmetry about the axis of the radius. No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, ""two"" tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of the circle. Thus the lengths of the segments from P to the two tangent points are equal. By the secant-tangent"
"a stationary circle of radius formula_17, have the following properties: where formula_19 is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with ""cusps"" replaced by either ""radial maxima"" or ""radial minima."" Trochoid A trochoid (from the Greek word for wheel, ""trochos"") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word ""trochoid"" was coined by Gilles de Roberval. As a circle of radius ""a"" rolls"
"Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This"
"passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines. The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem,"
"Quadrisecant In geometry, a quadrisecant or quadrisecant line of a curve is a line that passes through four points of the curve. In three-dimensional Euclidean space, every non-trivial tame knot or link has a quadrisecant. Originally established in the case of knotted polygons and smooth knots by Erika Pannwitz, this result was extended to knots in suitably general position and links with nonzero linking number, and later to all nontrivial tame knots and links. Pannwitz proved more strongly that the number of distinct quadrisecants is lower bounded by a function of the minimum number of boundary singularities in a locally-flat"