x = h + X cos θ + Y cos (θ + ½π) = h + X cos θ − Y sin θ,
and
y = k + X cos (½π − θ) + Y cos θ = k + X sin θ + Y cos θ.
Be careful to observe that these formulae do not apply to every conceivable change of reference from one set of rectangular axes to another. It might have been required to take O′X, O′Y′ for the positive directions of the new axes, so that the change of directions of the axes could not be effected by rotation. We must then write −Y for Y in the above.
Were the new axes oblique, making angles α, β respectively with the old axis of x, and so inclined at the angle β − α, the same method would give the formulae
x = h + X cos α + Y cos β, y = k + X sin α + Y sin β.
18. The Conic Sections.—The conics, as they are now called, were at first defined as curves of intersection of planes and a cone; but Apollonius substituted a definition free from reference to space of three dimensions. This, in effect, is that a conic is the locus of a point the distance of which from a given point, called the focus, has a given ratio to its distance from a given line, called the directrix (see [Conic Section]). If e : 1 is the ratio, e is called the eccentricity. The distances are considered signless.
Take (h, k) for the focus, and x cos α + y sin α − p = 0 for the directrix. The absolute values of √{(x − h)² + (y − k)²} and p − x cos α − y sin α are to have the ratio e : 1; and this gives
(x − h)² + (y − k)² = e² (p − x cos α − y sin α)²
as the general equation, in rectangular coordinates, of a conic.