Ellipses

Ellipses: Incircle and Circumcircle

Any curve formed by squashing or stretching a circle in such a way that all of its dimensions in one direction are decreased by a fixed ratio, p, or increased by a fixed ratio, 1/p, is called an ellipse (of proportion p). Every ellipse has a circumscribed circle and an inscribed circle of which it is a squashed or stretched version respectively. The diameters of these circles, joining their points of contact with the ellipse are called the major axis and minor axis of the ellipse (they are axes of symmetry). If the radii of the circles are a and b respectively than p = b/a, which can take any value between 0 and 1.

There is a rectangle that has the same circumscribed and symmetrically inscribed circles as the ellipse and the same axes of symmetry. Its longer sides are tangent at the ends of the minor axis. The ratio of the sides of this rectangle may be termed the shape, s, of the ellipse. Applying the theorem of Pythagoras to the quadrants of the rectangle, we find s = b/c where c is the half-length of the rectangle, and c = √(a² − b²), so s = b/√(a² − b²) = p/√(1 − p²). Thus s².p² = s² − p², and p = s/√(1 + s²).

An ellipse is the locus of points whose distances from two given points R and S add to a constant value, r + s = 2a. The foci R and S are the points where the fitted rectangle crosses the major axis. The ratio of the distance between the foci (2c) to the major axis (2a) is called the eccentricity of the ellipse, so e = c/a = √(a² − b²)/a = √(1 − p²). Thus e² + p² = 1 and p = √(1 − e²). An ellipse can be drawn by means of a loop of thread of length 2a held taut about a pencil at P and pins at R, S. The normal at P bisects the angle SPR.

If P is any point on the ellipse (other than the ends of the major axis, A, A') and P' its diametrically opposite point then the tangents at P and P' are parallel and meet the escribed circle at YZY'Z' forming a rectangle. The other sides YZ' and Y'Z pass through the foci of the ellipse. Conversely, given any point S inside a circle by drawing SY, to any point Y on the circle, and YZ perpendicular to SY, the lines YZ thus generated envelope an ellipse.

Ellipses: Circumrectangles

If we draw any diameter PP' through the centre O of an ellipse and tangents at its ends, then the diameter parallel to these tangents is the conjugate diameter. The tangents form an enclosing parallelogram. Each diameter bisects the chords parallel to its conjugate. This parallelogram is found always to have the same area. This can be understood, since the ellipse is a projection of a circle and the parallelograms are projections of squares circumscribed to the circle. [Source: M. E. J. Gheury de Bray, Elementary Hyperbolics, pp.184-5.]

Tangents at right angles meet on the director circle, whose radius is √(a² + b²). These tangents include one set that form a square, whose diameters are the major and minor axes.

The chord through the focus parallel to the minor axis, the latus rectum, is of length 2.b²/a. It is twice the harmonic mean of the segments of any chord through the focus: SL = 2.SP.SQ/(SP + SQ). The tangents at the end of a focal chord intersect on a straight line called the directrix for that focus. OX = a/e. OS.OX = a². The ellipse is the locus of a point that moves so that its distances from a fixed point S and a fixed line XM are in a fixed ratio, e, less than unity: SP = e.PM.

Ellipses: Trammel property

An ellipse can be drawn by means of a pointer attached to a rod whose ends slide in perpendicular grooves. This device is known as a trammel of Archimedes. By a theorem in kinematics any 'trammel' curve, or curve described by a linkage, is a roulette described by the carried point when the body-locus of the instantaneous centre of the carrier link rolls on its space-locus. In the case of the trammel of Archimedes the motion is that of a circle with the rod as diameter, rolling on the inside of a fixed circle of twice the size.

The trammel line does not coincide with the normal, except along the axes.

Ellipses: Frégier property

Frégier's theorem: If MN is a variable chord of an ellipse such that MPN = 90° then MN passes through a fixed point on the normal at P.

Ellipses: Theorem of Brianchon

If the six sides of a hexagon are tangents to a conic then the lines joining pairs of opposite vertices are concurrent.

Ellipses: Theorem of Pascal

If the six vertices of a hexagon are on a conic then the points of intersection of pairs of opposite sides are collinear. This is the dual of Brianchon's theorem.