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The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points. [22] A generalisation for higher powers of distances is obtained if under n {\displaystyle n} points the vertices of the regular polygon P n {\displaystyle P_{n}} are taken. [ 23 ]
An alternative approach that uses the matrix form of the quadratic equation is based on the fact that when the center is the origin of the coordinate system, there are no linear terms in the equation. Any translation to a coordinate origin (x 0, y 0), using x* = x – x 0, y* = y − y 0 gives rise to
The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a : that is, e = c a {\displaystyle e={\frac {c}{a}}} (lacking a center, the linear eccentricity for ...
Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines). Linear interpolation on a set of data points (x 0, y 0), (x 1, y 1), ..., (x n, y n) is defined as piecewise linear, resulting from the concatenation of linear segment interpolants between each pair of data points.
Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton ...
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
With the Cartesian equation it is easier to check whether a point lies on the circle or not. With the parametric version it is easier to obtain points on a plot. In some contexts, parametric equations involving only rational functions (that is fractions of two polynomials) are preferred, if they exist.
The equation of a line: Ax + By = C, with A 2 + B 2 = 1 and C ≥ 0; The equation of a circle: () + = By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a linear equation in point-slope and slope-intercept form.