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Abstract.Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.

This article describes an evolution of training programs to use first-person interaction in virtual reality (VR) situations to teach safety skills to children with autism spectrum disorder (ASD) and fetal alcohol spectrum disorder (FASD). Multiple VR programs for children aged 2 to 9 were built and tested between 1992 and 2007. Based on these results, a learning design evolved that uses practice in virtual space with guidance and correction by an animated character, strategic limitations on allowed actions to force correct patterning, and customization of worlds and responses to simplify user controls. This article describes program evolution by comparing design details and results as variations in behavioral responses between disorders, differences in skill set complexity between different safety skills being taught, and improved technology required changes in the virtual training methodology. A series of research projects are summarized in which the VR programs proved effective for teaching children with ASD and FASD new skills in the virtual space and, where measured, most children generalized the actions to the real world.