Secant line

For information on the secant trigonometric function, see Secant (trigonometry). Common lines and line segments on a circle, including the secant line

In geometry, a secant line of a curve is a line that (locally) intersects two points on the curve. A chord is an interval of a secant line, the portion of the line that lies within the curve. The word secant comes from the Latin word secare, meaning to cut.

Secants can be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, (assuming that the first derivative of the curve is continuous at point P so that there is only one tangent). As a consequence, one could say that the limit, as Q approaches P, of the secant's slope, or direction, is that of the tangent. In calculus, this idea is the basis of the geometric definition of the derivative.

• Elliptic curve, a curve for which every secant has a third point of intersection, from which a group law may be defined
• Quadrisecant, a line that intersects four points of a curve
• Secant plane, the three-dimensional equivalent of a secant line

References

1. Protter, Murray H.; Protter, Philip E. (1988), Calculus with Analytic Geometry, Jones & Bartlett Learning, p. 62, ISBN 9780867200935.
2. Gullberg, Jan (1997), Mathematics: From the Birth of Numbers, W. W. Norton & Company, p. 387, ISBN 9780393040029.
3. Redgrove, Herbert Stanley (1913), Experimental Mensuration: An Elementary Test-book of Inductive Geometry, Van Nostrand, p. 167.