Locus (mathematics)#History and philosophy

Until the beginning of the 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 from the center.{{citation |title=The History of Mathematics: A Brief Course |first=Roger L. |last=Cooke |edition=3rd |publisher=John Wiley & Sons |year=2012 |isbn=9781118460290 |url=https://books.google.com/books?id=CFDaj0WUvM8C&pg=PT534 |contribution=38.3 Topology |quote=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.}}

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.{{citation |title=Elements of the History of Mathematics |first=N. |last=Bourbaki |author-link=Nicolas Bourbaki |translator=J. Meldrum|translator-link=John D. P. Meldrum |publisher=Springer |year=2013 |isbn=9783642616938 |page=26 |url=https://books.google.com/books?id=4JprCQAAQBAJ&pg=PA26 |quote=the classical mathematicians carefully avoided introducing into their reasoning the 'actual infinity'}}.

Once set theory became the universal basis over which the whole mathematics is built,{{citation |title=The Foundations of Mathematics in the Theory of Sets |volume=82 |series=Encyclopedia of Mathematics and its Applications |first=John P. |last=Mayberry |publisher=Cambridge University Press |year=2000 |isbn=9780521770347 |url=https://books.google.com/books?id=mP1ofko7p6IC&pg=PA7 |page=7 |quote=set theory provides the foundations for all mathematics}}. the term of locus became rather old-fashioned.{{citation |title=Combinatorics and Geometry, Part 1 |volume=5 |series=Handbook of Applicable Mathematics |first1=Walter |last1=Ledermann |first2=S. |last2=Vajda |publisher=Wiley |year=1985 |isbn=9780471900238 |page=32 |quote=We begin by explaining a slightly old-fashioned term}}. 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.
• Zero locus or vanishing locus, the set of points where a function vanishes, in that it takes the value zero.
• 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.{{citation |title=Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice |first=Alexandre |last=Borovik |publisher=American Mathematical Society |year=2010 |isbn=9780821847619 |contribution=6.2.4 Can one live without actual infinity? |url=https://books.google.com/books?id=hEPSAwAAQBAJ&pg=PA124 |page=124}}.

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.George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975.
• The set of points equidistant from two intersecting lines is the union of their two angle bisectors.
• All conic sections are loci:{{citation |first1=Henry Parr |last1=Hamilton |title=An Analytical System of Conic Sections: Designed for the Use of Students |publisher=Springer |year=1834}}.
• Circle: the set of points at constant distance (the radius) from a fixed point (the center).
• Parabola: the set of points equidistant from a fixed point (the focus) and a line (the directrix).
• 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

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.

To prove a geometric shape is the correct locus for a given set of conditions,

one generally divides the proof into two stages: the proof that all the points that satisfy the conditions are on the given shape, and the proof that all the points on the given shape satisfy the conditions.G. P. West, The new geometry: form 1.

File:Locus apollonius.svg

Find the locus of a point P that has a given ratio of distances k = d₁/d₂ to two given points.

In this example k = 3, A(−1, 0) and B(0, 2) are chosen as the fixed points.

: P(x, y) is a point of the locus

: $\backslash Leftrightarrow\; |PA|\; =\; 3\; |PB|$

: $\backslash Leftrightarrow\; |PA|^2\; =\; 9\; |PB|^2$

: $\backslash Leftrightarrow\; (x\; +\; 1)^2\; +\; (y\; -\; 0)^2\; =\; 9(x\; -\; 0)^2\; +\; 9(y\; -\; 2)^2$

: $\backslash Leftrightarrow\; 8(x^2\; +\; y^2)\; -\; 2x\; -\; 36y\; +\; 35\; =\; 0$

: $\backslash Leftrightarrow\; \backslash left(x\; -\; \backslash frac18\backslash right)^2\; +\; \backslash left(y\; -\; \backslash frac94\backslash right)^2\; =\; \backslash frac\{45\}\{64\}.$

This equation represents a circle with center (1/8, 9/4) and radius $\backslash tfrac\{3\}\{8\}\backslash sqrt\{5\}$. It is the circle of Apollonius defined by these values of k, A, and B.

File:Locus3a.svg

A triangle ABC has a fixed side [AB] with length c.

Determine the locus of the third vertex C such that

the medians from A and C are orthogonal.

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).

File:Locus3.svg

:C(x, y) is a point of the locus

:$\backslash Leftrightarrow$ the medians from A and C are orthogonal

:$\backslash Leftrightarrow\; \backslash frac\{y\}\{x\}\; \backslash cdot\; \backslash frac\{2y\}\{2x\; +\; 3c\}\; =\; -1$

:$\backslash Leftrightarrow\; 2\; y^2\; +\; 2x^2\; +\; 3c\; x\; =\; 0$

:$\backslash Leftrightarrow\; x^2\; +\; y^2\; +\; (3c/2)\; x\; =\; 0$

:$\backslash Leftrightarrow\; (x\; +\; 3c/4)^2\; +\; y^2\; =\; 9c^2/16.$

The locus of the vertex C is a circle with center (−3c/4, 0) and radius 3c/4.

File:Geassocieerde rechten.svg

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 $\backslash alpha$ 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.

A locus of points need not be one-dimensional (as a circle, line, etc.). For example, the locus of the inequality {{math|2x + 3y – 6 < 0}} is the portion of the plane that is below the line of equation {{math|1=2x + 3y – 6 = 0}}.

• Algebraic variety
• Curve
• Line (geometry)
• Set-builder notation
• Shape (geometry)

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Category:Elementary geometry