At a given moment, if the right ascension and declination of the sun are $A$ and $D$ and the Greenwich Sidereal Time is $G$, the latitude $y$ and longitude $x$ of the points on the terminator (line separating day and night) is given by the following equation derived from the formula for altitude and not accounting for refraction by the atmosphere:
$$\sin y \sin D + \cos y \cos D \cos(G - x - A) = 0$$
Now for the present purposes $A$, $D$ and $G$ are all constant. Taking $K = G - A$, the above equation becomes:
$$\sin y \sin D + \cos y \cos D \cos(K - x) = 0$$
If it is useful to divide by $\cos D$, then it becomes:
$$\tan y \sin D + \cos D \cos(K - x) = 0$$
How can I express this equation (or the previous, if that is better) in terms of a parameter $t$ so that as $t$ varies from $0$ to $1$, I would get the latitude and longitude pairs of all the points?
NOTE: Why I want to do this is, I can easily express $y$ in terms of $x$ as $y = \arctan(-\cos(K - x) / \tan D)$ but due to the nature of the curve, stepping through equidistant values of $x$ does not result in even reasonably uniformly distributed points on the curve. I am thinking that by using a parameter, I can get better distribution.