This study examines the degree of two-dimensional curvature of solar wind directional discontinuity (DD) surfaces at 1 AU using magnetic field, density, and velocity data from the WIND and IMP-8 spacecraft for a large number (N = 134) of carefully selected events having large discontinuity angles of 90¿ or greater. The discontinuity angle (ω) is measured in the DD's current sheet, the normal (n) to which is estimated by field variance analysis. The fundamental analysis depends on estimates of these DD surface normals at the two spacecraft and the DD's center-times and positions. On average, the transit time from one DD sighting to the other was 36 minutes, and the associated distance along the normal direction was 137 RE. The transition-interval lengths across the DDs are translated into thicknesses and examined for the amount of change between the two spacecraft observing points. The average thickness is relatively large, 14 RE.; the most probable thickness is ≈6 RE. All relevant quantities are examined statistically to establish their distributions, average, and degree of change. A weighted average of the radius of curvature is estimated to be 380 RE, but its most probable value is 290 RE. The average ω is 140¿ with a relatively large spread (σ = 28¿). The average direction of propagation is: longitude (φn) = 194¿ and latitude (θn) = 7¿ (but ⟨∣θn∣⟩ = 27¿), where φn = 0¿ is sunward and θn = 0¿ is the ecliptic plane. Various parameters are studied with respect to DD type, i.e., rotational or tangential discontinuity (RD or TD), defined in terms of the ratio (in percent) of speed of propagation to net speed of the DD surface, where the net speed is the sum of the convection velocity (along n) plus the propagation speed. The RD %-ratio is moderately small, but the TD ratio is very small or zero. The results by this definition of type are favorably compared to those from the more conventional method, which depends on the absolute strength of the normal component of the magnetic field. There is little difference in any average parameter value according to type. However, the average ⟨ω⟩ appears to depend slightly on type with the ⟨ω⟩ for the RDs being smaller. The discontinuity type was shown to change commonly in either direction (TD to RD or RD to TD) between the two observing positions, i.e., ≈40% of the time. It is not clear if these type-changes are spatial or temporal. Shortcomings of the analysis are (1) the need to impose an upper limit on the angular difference of the DD normals between the two observing positions (which eliminated most surfaces of very small radii of curvature), and (2) the inability to distinguish real curvature from shorter-scale surface variations, from only two spacecraft data sets. The results of the study should help to caution us as to the simplistic use of the planar DD surface assumption in projecting, to the distance of Earth's magnetosphere, a distantly observed DD surface (e.g., one near L1), especially for studies that depend on accurately predicting the timing and characteristics of magnetospheric events.