1. Syllabus for Analytic Geometry of Space It is an introductory course, which includes the subjects usually treated in rectangular coordinates. They presuppose as much knowledge of algebra, geometry, and trigonometry as is contained in the major requirement of the entrance examination, and plane analytic geometry

2. Features: –Development of linear system of planes –Plane coordinates –The concept of infinity –The treatment of imaginaries –The distinction between centers and vertices of quadric surfaces.

3. Study the linear algebra and geometry required for mathematical and computer study in space. Learn about curves, their derivatives and applications in 3D space Linear transformations of points and of coordinates The classifications of quadric curves and surfaces

4. Teaching methods –With the helps of drawings and hints –Illustration of graphics and images to help the students to enhance their ability of space imagination –Questions –Experiments with software, e.g., ProE, AutoCAD –Exercises –Intuitive proof of theorems Problem: limited time of 18 study hours.

5. Fig. 1 A torus with textures

6. Fig. 2 A dumbbell of free-form surfaces

7. It mainly includes seven chapters Chapter 1 Coordinates 1. Rectangular Coordinates 2. Orthogonal projection 3. Direction cosines of a line 4. Distance between two points 5. Angle between two directed lines 6. Point dividing a segment in a given ratio 7. Polar coordinates

8. 8. Cylindrical coordinates 9. Spherical coordinates Chapter 2 Planes and Lines 1. Equation of a plane 2. Plane through three points 3. Intercept form of the equation of a plane 4. Normal form of the equation of a plane 5. Reduction of a linear equation to the normal form

9. 6. Angle between two planes 7. Distance to a point from a plane 8. Equation of a line 9. Direction cosines of the line of intersection of two planes 10. Forms of the equations of a line 11. Angle which a line makes with a plane 12. Distance from a point to a line 13. Distance between two non-intersecting lines

10. 14. System of planes through a line 15. Application to descriptive geometry 16. Bundles of planes 17. Plane coordinates* 18. Equation of a point* 19. Homogeneous coordinate of the point and of the plane* 20. Equation of the plane and of the point in homogenous coordinates* 21. Equation of the origin* Where “*” denotes that these sections are selectional

11. Chapter 3 Transformation of Coordinates 1. Translation 2. Rotation 3. Rotation and reflection of axes 4. Euler’s formulas for rotation of axes 5. Degree unchanged by transformation of coordinates

12. Chapter 4 Types of Surfaces 1. Imaginary points, lines, and planes 2. Loci of equations 3. Cylindrical surfaces 4. Projecting cylinders 5. Plane sections of surfaces 6. Cones 7. Surfaces of revolution

13. Chapter 5 Forms of Quadric Surfaces 1. Definition of a quadric 2. The sphere 3. The ellipsoid 4. The hyperboloid of one sheet 5. The hyperboloid of two sheets 6. The imaginary ellipsoid 7. The elliptic paraboloid 8. The hyperbolic paraboloid 9. The quadric cones 10. The quadric cylinders 11. Summary

14. Chapter 6 Classification of Quadric Surfaces 1. The Intersection of a quadric and a line 2. Diametral planes, center 3. Equation of a quadric referred to its center 4. Principal planes 5. Reality of the roots of the discrimination cubic 6. Simplification of the equation of a quadric 7. Classification of quadric surfaces 8. Invariants under motion 9. Proof that I, J, and D are invariant 10. Proof that is invariant 11. Discussion of numerical equations

15. Chapter 7 Some Properties of Quadric Surfaces 1. Tangent lines and planes 2. Normal forms of the equation of a tangent plane 3. Normal to a quadric 4. Rectilinear generators 5. Asymptotic cone 6. Plane sections of quadrics 7. Confocal quadrics through a point* 8. Confocal quadrics tangent to a line* 9. Confocal quadrics in plane coordinates*