Locus (mathematics)

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Each curve in this example is a locus defined as the conchoid of a circle centered at point P and the line l. In this example, P is 7cm from l.

In geometry, a locus (plural: loci) is a set of points whose location satisfies or is determined by one or more specified conditions.[1][2]

Commonly studied loci[edit]

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.[3]
  • The set of points equidistant from two lines which cross is the angle bisector.
  • All conic sections are loci:[4]
    • 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.

Proof of a locus[edit]

In order to prove that a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages:[5]

  • 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[edit]

(distance PA) = 3.(distance PB)

First example[edit]

We find the locus of the points P that have a given ratio of distances k = d1/d2 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
\Leftrightarrow |PA| = 3 |PB|
 \Leftrightarrow |PA|^2 = 9 |PB|^2
\Leftrightarrow (x+1)^2+(y-0)^2=9(x-0)^2+9(y-2)^2
 \Leftrightarrow 8(x^2+y^2)-2x-36y+35 =0
\Leftrightarrow \left(x-\frac18\right)^2+\left(y-\frac94\right)^2=\frac{45}{64}

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

Second example[edit]

Locus of point C

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).

The locus is a circle
C(x,y) is a point of the locus
\Leftrightarrow The medians from A and C are orthogonal
\Leftrightarrow  \frac{y}{x} \cdot \frac{2y}{2x+3c} = -1
\Leftrightarrow       2 y^2 + 2x^2 + 3c x = 0
\Leftrightarrow       x^2 + y^2 + (3c/2) x = 0
\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.

Third example[edit]

The intersection point of the associated lines k and l describes the circle

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 \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.

Fourth example[edit]

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[edit]

References[edit]

  1. ^ a b James, Robert Clarke; James, Glenn (1992), Mathematics Dictionary, Springer, p. 255, ISBN 978-0-412-99041-0 
  2. ^ Whitehead, Alfred North (1911), An Introduction to Mathematics, H. Holt, p. 121, ISBN 978-1-103-19784-2 
  3. ^ George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975
  4. ^ Hamilton, Henry Parr (834), An Analytical System of Conic Sections: Designed for the Use of Students, Springer 
  5. ^ G.P. WestThe new geometry: form 1