Haptic perception of curvature depends largely on the kind of touch. An active and dynamic touch is considered to be the most natural way of exploring. In this study, we have designed and evaluated a kinematic platform for curvature perception through active and dynamic touch. This platform can independently orient, elevate, and translate a flat plate; by exploring forward and backward along the flat plate with a finger, users can achieve curvature feeling of extruded objects. The mechanism of platform and the way of touch have maximally respected the cues for curvature perception, especially the slip cue. Psychophysical evaluation demonstrated that the discrimination threshold of curvature for virtual shapes is close to that for real shapes, and the virtual shape is felt equally curved as the real one. The curvature perception of mono-convex surfaces was then expanded to perception of more complex surfaces: large textures, which have a sinusoidal profile. The evaluation has accessed the correspondence between the virtual and real large textures.