Geometry of rational motions on pythagorean hodograph curves and pythagorean normal surfaces

This thesis consists of seven chapters in which we investigate the relationships between the Pythagorean hodograph and Pythagorean normal properties of curves, ruled surfaces, and envelope surfaces resulting from the act of a rational motion on a point, a line, and a plane, respectively. The first three sections of the thesis successively cover the introduction, literature review and basic concepts to be used in the thesis are included. In the fourth section, the case where the trajectory curve resulting from the act of rational motion on a point becomes a Pythagorean hodograph curve is explained. In the fifth section, a relationship is found between the property of the trajectory ruled surface becoming a developable surface under the act of rational motion on a line and the Pythagorean hodograph property. In the sixth section, it is discovered that the curve that accepts the family of planes formed by the act of a one-parameter rational motion on a plane as osculating, normal, or rectifying planes, is a Pythagorean hodograph curve. Then, the Pythagorean normal surface is created with the envelope surface of the family of planes formed by the act of a two-parameter rational motion on a plane. Finally, in the seventh section, a discussion is presented for the studies carried out in the thesis and open problems are pointed out while summarizing the results.