The uncertainty of a 2D point usually is described by its 2 × 2 covariance matrix, assuming its coordinates are jointly Gaussian distributed with density pxy(x,y). It can be visualized by its standard ellipse, with its two semiaxes a and b (assuming a ≧ b), and the direction α of the major axis. The variances of the two coordinates, i.e. the two diagonal elements, at the same time are the variances of the two 1D marginal distributions px(x) and py(y) perpendicular to the x- and the y-coordinate, also Gaussians. The bounding box around the standard ellipse has sides twice the standard deviations. Given the direction φ to a known point the standard deviation σdφ of the distance to that point results from the marginal distribution perpendicular to this direction, as can be seen when choosing φ=0° or 90°: the standard deviations σx and σx are the pedal points of the origin on the sides of the enclosing bounding box. The function r(φ) = σd(φ) of the standard deviation of a distance to a fixed point as a function of the direction φ is the distance of the origin to the tangent of the ellipse, being perpendicular to the direction φ and called a pedal curve of the ellipse, here w.r.t to the centre.
The animation shows the uncertainty of the point coordinates by the density pxy(x,y) (darker is larger) and its marginals px(x) and py(y) along x and y, by the standard ellipse (blue) with its parameters. Its form can be controlled by the two points A and B being the end of the major and minor semi axis. The principle parameters are shown on the left. The direction φ to some point may be chosen on the circle in the upper left. As a result, the corresponding tangent point T, the pedal point P and the standard deviation σdφ are given.