A "mixed cut" gemstone is typically defined by a step-cut pavilion and brilliant crown, and you'll often see these in many traditionally cut gemstones. "Radiant" cuts are the opposite - usually a brilliant style pavilion and step cuts on the crown. It's a fairly new cut and works well to deepen the saturation in light colored stones. Once this characteristic became known to diamond cutters, many light yellow diamonds in the GIA W-X diamond color range were recut with a radiant design because it increased their apparent color saturation, thus giving them a more desirable Fancy Yellow color. In the sapphire example you can see how the radiant cut concentrates and deepens the pink color in otherwise light pink material.