Circular surfaces are smooth one-parameter families of circles. This paper includes three main purposes about circular surfaces and roller coaster surfaces defined as circular surfaces whose generating circles are lines of curvature. The first one is to reconstruct equations of spacelike circular surfaces and spacelike roller coaster surfaces by using unit split quaternions and homothetic motions. The second one is to parametrize timelike circular surfaces and give some geometric properties such as striction curves, singularities, Gaussian and mean curvatures. Furthermore, the conditions for timelike roller coaster surfaces to be flat or minimal surfaces are obtained. The last one is to express split quaternionic and matrix representations of timelike circular surfaces and timelike roller coaster surfaces. Moreover, to support the theory studied in the paper some illustrated examples are presented.