Difference between revisions of "Tangent"
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| − | In [[geometry]], the '''tangent''' line (or simply the tangent) to a [[curve]] at a given point is the straight line that "just [[touch]]es" the curve at that [[point]] (in the sense explained more precisely below). As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point. The same definition applies to [ | + | In [[geometry]], the '''tangent''' line (or simply the tangent) to a [[curve]] at a given point is the straight line that "just [[touch]]es" the curve at that [[point]] (in the sense explained more precisely below). As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point. The same definition applies to [https://en.wikipedia.org/wiki/Space_curves space curves] and curves in n-dimensional [https://en.wikipedia.org/wiki/Euclidean_space Euclidean space]. |
| − | 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 [ | + | 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 [https://en.wikipedia.org/wiki/Differential_geometry differential geometry] and has been extensively generalized; |
The word "tangent" comes from the Latin tangere, meaning "to [[touch]]". | The word "tangent" comes from the Latin tangere, meaning "to [[touch]]". | ||
==Tangent line to a curve== | ==Tangent line to a curve== | ||
| − | The [[intuitive]] [[idea|notion]] that a tangent line "just touches" a curve can be made more explicit by considering the [[sequence]] of straight lines ([ | + | The [[intuitive]] [[idea|notion]] that a tangent line "just touches" a curve can be made more explicit by considering the [[sequence]] of straight lines ([https://en.wikipedia.org/wiki/Secant_line secant lines]) passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A. The [[existence]] and [[Unique|uniqueness]] of the tangent line depends on a certain [[type]] of [[mathematical]] smoothness, known as "differentiability". For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the [[vertex]] because the limit of the [[progress]]ion of secant lines depends on the direction in which "point B" approaches the vertex. |
| − | In most cases, the tangent to a curve does not cross the curve at the point of tangency (though it may, when continued, cross the curve at other places away from the point of tangent) This is true, for example, of all tangents to a circle or a [ | + | In most cases, the tangent to a curve does not cross the curve at the point of tangency (though it may, when continued, cross the curve at other places away from the point of tangent) This is true, for example, of all tangents to a circle or a [https://en.wikipedia.org/wiki/Parabola parabola]. However, at exceptional points called [https://en.wikipedia.org/wiki/Parabola inflection points], the tangent line does cross the curve at the point of tangency. An example is the point (0,0) on the graph of the cubic parabola y = x3. |
| − | Conversely, it may happen that the curve lies entirely on one side of a straight line 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 the triangle—where the tangent line does not exist for the reasons explained above. In [ | + | Conversely, it may happen that the curve lies entirely on one side of a straight line 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 the triangle—where the tangent line does not exist for the reasons explained above. In [https://en.wikipedia.org/wiki/Convex_geometry convex geometry], such lines are called supporting lines. |
==Notes== | ==Notes== | ||
# Descartes, René (1954). The geometry of René Descartes. Courier Dover. pp. 95. ISBN 0486600688. | # Descartes, René (1954). The geometry of René Descartes. Courier Dover. pp. 95. ISBN 0486600688. | ||
# R. E. Langer (October 1937). "Rene Descartes". The American Mathematical Monthly (Mathematical Association of America) 44 (8): 495–512. doi:10.2307/2301226. | # R. E. Langer (October 1937). "Rene Descartes". The American Mathematical Monthly (Mathematical Association of America) 44 (8): 495–512. doi:10.2307/2301226. | ||
== External links == | == External links == | ||
| − | * Weisstein, Eric W., "[ | + | * Weisstein, Eric W., "[https://mathworld.wolfram.com/TangentLine.html Tangent Line]" from [https://en.wikipedia.org/wiki/MathWorld MathWorld]. |
| − | * [ | + | * [https://www.mathopenref.com/tangent.html Tangent to a circle] With interactive animation |
| − | * [ | + | * [https://www.vias.org/simulations/simusoft_difftangent.html Tangent and first derivative] - An interactive simulation |
| − | * [ | + | * [https://math.fullerton.edu/mathews/n2003/TangentParabolaMod.html The Tangent Parabola by John H. Mathews] |
[[Category: Mathematics]] | [[Category: Mathematics]] | ||
Latest revision as of 02:02, 13 December 2020
In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point (in the sense explained more precisely below). As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point. The same definition applies to space curves and curves in n-dimensional Euclidean space.
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;
The word "tangent" comes from the Latin tangere, meaning "to touch".
Tangent line to a curve
The intuitive notion that a tangent line "just touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability". For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point B" approaches the vertex.
In most cases, the tangent to a curve does not cross the curve at the point of tangency (though it may, when continued, cross the curve at other places away from the point of tangent) This is true, for example, of all tangents to a circle or a parabola. However, at exceptional points called inflection points, the tangent line does cross the curve at the point of tangency. An example is the point (0,0) on the graph of the cubic parabola y = x3.
Conversely, it may happen that the curve lies entirely on one side of a straight line 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 the triangle—where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines.
Notes
- Descartes, René (1954). The geometry of René Descartes. Courier Dover. pp. 95. ISBN 0486600688.
- R. E. Langer (October 1937). "Rene Descartes". The American Mathematical Monthly (Mathematical Association of America) 44 (8): 495–512. doi:10.2307/2301226.
External links
- Weisstein, Eric W., "Tangent Line" from MathWorld.
- Tangent to a circle With interactive animation
- Tangent and first derivative - An interactive simulation
- The Tangent Parabola by John H. Mathews
