MAT6026 Geometry of Rigid MotionsBahçeşehir UniversityDegree Programs MATHEMATICS (TURKISH, PHD)General Information For StudentsDiploma SupplementErasmus Policy StatementNational Qualifications
PhD. TR-NQF-HE: Level 8 QF-EHEA: Third Cycle EQF-LLL: Level 8

Course Introduction and Application Information

Course Code Course Name Semester Theoretical Practical Credit ECTS
MAT6026 Geometry of Rigid Motions Spring 3 0 3 8
This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

Basic information

Language of instruction: Turkish
Type of course: Departmental Elective
Course Level:
Mode of Delivery: Face to face
Course Coordinator : Assoc. Prof. ERSİN ÖZUĞURLU
Recommended Optional Program Components: None
Course Objectives: To examine general motions in real space, by carrying plane and sphere motions in real space to dual sphere. Then to make students have a geometric based scope on the studies in physics, engineering and robotics.

Learning Outcomes

The students who have succeeded in this course;
Be able to define sliding vectors
Be able to solve problems related to sliding vectors
Be able to understand motion in space
Be able to calculate velocities in plane motion.
Be able to express motion in space.
Be able to calculate motion in space.

Course Content

•Sliding vectors and examination of tensors •1 and 2 parameter motion of plane and sphere; Pole curves, Pfaffian vectors •Euler – Savary formulas and their correspandance in sphere which are called Darboux vectors and relations between them.

Weekly Detailed Course Contents

Week Subject Related Preparation
1) Defining kinematics, geometric properties of vectors and sliding vectors
2) Vector and analytic expressions of sliding vector, comoment, geometric interpretation and results. Vector coordinates of a line, detailed studies on it and related results. Introduction to tensors.
3) Composition of tensors, moments, equivalence class, moment with respect to an axis, analytic expressions, invariant and axis. Special tensors and examing them. Warignon theorem, related theorems and results.
4) Operations on tensors, which leave them equivalent to themselves, reduction and examining of tensors. Algebraic structure of tensors. Comoment, automoment, invariant of a system, related theorems and results.
5) Line to zero moment, polar plane, conjugate lines, Plucker coordinates related theorems and problems. Vector production tensors; theorem and problems
6) Definition of kinematics. Kinematics in plane. Constructing plane motion of one-parameter, translation and rotation. Derivation equations of motion and interpretations. Relative velocity.
7) Angular velocity, absolute and drift velocity. Composition of velocities and interpretation. Pole of instantaneous rotation orbits of poles. The relation of plane motion with the locus of pole points.
8) Rolling of pole curves without sliding. Inverse motion, examples of motion. Moving coordinate system and norming them. Pffafian forms.
9) Constructing motion in moving planes. Expressions of relative, drift and absolute velocity and composition of velocities. Expressing pole points with Pffafian forms. Chain of plane moving with respect to each other. Pole of rotation, line of pole.
10) Canonic relative system. Differential forms and interpretation of them. Orbit curve and its curvature. Center of curvature. Euler- Savary formula.
11) Point correspondence in plane motion. Examples and applications of Euler-Savary formula. Introduction to sphere kinematics. Motions around a fixed pointy sphere motions. Showing motion on sphere. Introduction to sphere motions of one- parameter.
12) Velocities in sphere motion of one parameter (absolute, relative and drift velocities), Pffafian vector. Composition of motions, related theorems and results. Reducing sphere motion
13) Canonic relative system. Pol curves and rolling of them without sliding. Theorems and results about moving spheres of same center and moving with respect to each other.
14) Correspondence of Euler- Savary formulas of orbit curves in sphere motion. Space kinematics (introduction to kinematics in space)


Course Notes / Textbooks: Kinematik Dersleri, Muller, H., R., Ankara Ün., 1963.
References: .

Evaluation System

Semester Requirements Number of Activities Level of Contribution
Quizzes 2 % 5
Homework Assignments 2 % 5
Presentation 1 % 20
Midterms 1 % 30
Final 1 % 40
Total % 100
Total % 100

ECTS / Workload Table

Activities Number of Activities Duration (Hours) Workload
Course Hours 14 3 42
Study Hours Out of Class 14 2 28
Presentations / Seminar 1 40 40
Homework Assignments 2 10 20
Quizzes 2 10 20
Midterms 1 25 25
Final 1 25 25
Total Workload 200

Contribution of Learning Outcomes to Programme Outcomes

No Effect 1 Lowest 2 Low 3 Average 4 High 5 Highest
Program Outcomes Level of Contribution