Ellipse — Chambers's Encyclopaedia

Ellipse, a geometrical curve and figure, intermediate to the circle and parabola (see CONIC SECTIONS). It is of great importance in Astronomy, being the shape of the orbit described by a planet under the action of gravitation. Mathematically, the ellipse is a closed curve, every point of which has the sum of its distances from two fixed points always the same. These two fixed points are called the foci; and the diameter drawn through them is the major axis; the minor axis bisects the major at right angles. The distance of either focus from the middle of the major axis is the eccentricity. The less the eccentricity, as compared with the axis, the nearer the figure approaches to a circle. When the foci coincide the ellipse becomes a circle, and when they are infinitely apart it becomes a parabola. The tangent at any point of the curve is always equally inclined to the two focal distances; and any diameter bisects all the chords which are parallel to the tangents at its extremities.

The Trammel or Elliptic Compass affords the easiest way of drawing an ellipse. It depends on the principle that when a line of fixed length moves so that its extremities are always on two fixed perpendicular lines, any point in it must describe an ellipse. A simple practical method is by passing a loop of thread over two pins stuck in the foci, Ss in the diagram, the length of the loop being equal to SB. If the point of a pencil be put into the loop P, and moved round so as to keep it stretched, the pencil will trace an ellipse AEBD. There are also various ways of approximating to the figure by the use of circular axes.

The equation to an ellipse (see GEOMETRY, ANALYTICAL), referred to its centre as origin, and to its major and minor axes as rectangular axes, is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where $a$ and $b$ are the semi-major and semi-minor axes respectively. From this equation it may be shown, by the integral calculus, that the area of an ellipse is equal to $\pi ab$ ; or is got by multiplying the product of the semi-major and semi-minor axes by 3.1416. It may also be shown that the length of the circumference of an ellipse is got by multiplying the major axis by the quantity $\pi \left\{ 1 - \left( \frac{1}{2} \right)^2 \frac{e^2}{1} - \left( \frac{1.3}{2.4} \right)^2 \frac{e^4}{3} - \left( \frac{1.3.5}{2.4.6} \right)^2 \frac{e^6}{5} - \dots \right\}$ to which there is an excellent practical approximation, viz., $\frac{\pi}{2} \left( a + b \sqrt{2(a^2 + b^2)} \right)$ . The eccentricity, $e$ , is $\sqrt{1 - \frac{b^2}{a^2}}$ and the ellipticity is the ratio $a - b$ to $a$ .