Assuming that $X$ has a two-dimensional normal distribution certain conditional distributions of $X$ given that $X$ lies on a hyperbola or a parabola are found. Two of these distributions, related respectively to the parabola and the hyperbola, resemble the von Mises distribution, which can be obtained as a conditional distribution of $X$ given that $X$ lies on a circle. It is, however, proved that the assumptions leading to the conditional distribution in the hyperbolic case are not analogous to those leading to the von Mises distribution.