The use of prismatoid bodies is well known. The aim of this project is to study the relationship between the geometric characteristics of straight prisms with an equilateral semi-regular polygon at the base and the corresponding twisted antisprisms obtained by their torsion deformation. In this work, we assume that twisted antiprisms are convex and non-convex prismatoids with equilateral semi-regular polygons (polygons with an even number of vertices, all angles of which are equal, and the sides are equal through one) in bases having triangular faces which formed as a result of rotation of the upper polygon of the base by some angle φ around the center of the inscribed circle. Within the framework of different models the changes of geometrical characteristics the paper investigates relationship between the geometric parameters of the twisted antiprism and the initial straight prism. In this work, the form of a polygon in the equatorial section of a twisted antiprism is investigated. It turned out that for a mathematical model with a constant height, if the cotangent of the angle between the side edge and the base plane of a straight prism is set aside in the polar coordinate system depending on the angle φ, then the Guido Grandi rose equation holds with the determining coefficient ½. For model with a constant length of the prismatoid edges, which were the diagonals of the face of a straight prism Grandi rose is deformed. If the length of the side edge of a right prism is fixed, then the end of the edge moves along the Viviani’s curve. Some theorems are formulated and proved. Key words: prismatoid, twisted antiprism, equilateral semi-regular polygons, rectangular and polar coordinate systems, Viviani’s curve, Guido Grandi rose.