Locus (mathematics)
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.^{[1]}^{[2]}
History and philosophy
Until the beginning of 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance of the center.^{[3]}
In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the actual infinite was an important philosophical position of earlier mathematicians.^{[4]}^{[5]}
Once set theory became the universal basis over which the whole mathematics is built,^{[6]} the term of locus became rather old fashioned.^{[7]} Nevertheless, the word is still widely used, mainly for a concise formulation, for example:
- Critical locus, the set of the critical points of a differentiable function
- Singular locus, the set of the singular points of an algebraic variety
- Connectedness locus, the subset of the parameter set of a family of rational functions, for which the Julia set of the function is connected
More recently, techniques such as the theory of schemes, and the use of category theory instead of set theory to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points.^{[5]}
Examples in plane geometry
Examples from plane geometry include:
- The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.^{[8]}
- The set of points equidistant from two lines that cross is the angle bisector.
- All conic sections are loci:^{[9]}
- Parabola: the set of points equidistant from a single point (the focus) and a line (the directrix).
- Circle: the set of points for which the distance from a single point is constant (the radius). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a Circle of Apollonius.
- Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
- Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant. The circle is the special case in which the two foci coincide with each other.
Other examples of loci appear in various areas of mathematics. For example, in complex dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps.
Proof of a locus
To prove a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages:^{[10]}
- Proof that all the points that satisfy the conditions are on the given shape.
- Proof that all the points on the given shape satisfy the conditions.
Examples
First example
We find the locus of the points P that have a given ratio of distances k = d_{1}/d_{2} to two given points.
In this example we choose k= 3, A(-1,0) and B(0,2) as the fixed points.
- P(x,y) is a point of the locus
This equation represents a circle with center (1/8,9/4) and radius . It is the circle of Apollonius defined by these values of k, A, and B.
Second example
A triangle ABC has a fixed side [AB] with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal.
We choose an orthonormal coordinate system such that A(-c/2,0), B(c/2,0). C(x,y) is the variable third vertex. The center of [BC] is M( (2x+c)/4, y/2 ). The median from C has a slope y/x. The median AM has slope 2y/(2x+3c).
- C(x,y) is a point of the locus
- The medians from A and C are orthogonal
The locus of the vertex C is a circle with center (-3c/4,0) and radius 3c/4.
Third example
A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus.
In the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is perpendicular to k. The angle between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.
Fourth example
A locus of points need not be one-dimensional (as a circle, line, etc.). For example,^{[1]} the locus of the inequality 2x+3y–6<0 is the portion of the plane that is below the line 2x+3y–6=0.
See also
References
- 1 2 James, Robert Clarke; James, Glenn (1992), Mathematics Dictionary, Springer, p. 255, ISBN 978-0-412-99041-0
- ↑ Whitehead, Alfred North (1911), An Introduction to Mathematics, H. Holt, p. 121, ISBN 978-1-103-19784-2
- ↑ Cooke, Roger L. (2012), "38.3 Topology", The History of Mathematics: A Brief Course (3rd ed.), John Wiley & Sons, ISBN 9781118460290,
The word locus is one that we still use today to denote the path followed by a point moving subject to stated constraints, although, since the introduction of set theory, a locus is more often thought of statically as the set of points satisfying a given collection.
- ↑ Bourbaki, N. (2013), Elements of the History of Mathematics, translated by J. Meldrum, Springer, p. 26, ISBN 9783642616938,
the classical mathematicians carefully avoided introducing into their reasoning the 'actual infinity'
. - 1 2 Borovik, Alexandre (2010), "6.2.4 Can one live without actual infinity?", Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice, American Mathematical Society, p. 124, ISBN 9780821847619.
- ↑ Mayberry, John P. (2000), The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, 82, Cambridge University Press, p. 7, ISBN 9780521770347,
set theory provides the foundations for all mathematics
. - ↑ Ledermann, Walter; Vajda, S. (1985), Combinatorics and Geometry, Part 1, Handbook of Applicable Mathematics, 5, Wiley, p. 32, ISBN 9780471900238,
We begin by explaining a slightly old-fashioned term
. - ↑ George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975
- ↑ Hamilton, Henry Parr (834), An Analytical System of Conic Sections: Designed for the Use of Students, Springer
- ↑ G.P. West, The new geometry: form 1