Commonly studied loci
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.
- The set of points equidistant from two lines which cross is the angle bisector.
- All conic sections are loci:
- 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
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:
- 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.
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
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.
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.
A locus of points need not be one-dimensional (as a circle, line, etc.). For example, the locus of the inequality 2x+3y–6<0 is the portion of the plane that is below the line 2x+3y–6=0.
- 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
- 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. WestThe new geometry: form 1