Williams holds a double major in English and Asian Studies with honors from the University of Kansas. He was drawn to the University of Utah by a "Humanistic Computation" summer seminar held by Jef Raskin. He joined the graduate Computer Science program at the University of Utah in 1972.
At this time in the early 1970s, the University of Utah was the hub for much of the pioneering work in computer graphics.
Lance left Utah (without completing his degree) in 1974 to join the New York Institute of Technology (NYIT). While at NYIT, Williams invented the mipmapping technique for texture filtering, which is ubiquitously used today by graphics hardware for PCs and video games.
Lance was later awarded his doctorate from Utah based on a rule allowing someone who published three seminal papers in his field to bind them together as his thesis.
The three papers are Casting Curved Shadows on Curved Surfaces (1978), Pyramidal Parametrics (1983) and View Interpolation for Image Synthesis (1993).