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

• 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.

• 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

• 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 theoremsProblem: limited time of 18 study hours.

Fig. 1 A torus with textures

Fig. 2 A dumbbell of free-form surfaces

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.       Cylindrical coordinates9.       Spherical coordinates • Chapter 2 Planes and Lines1.       Equation of a plane2.       Plane through three points3.       Intercept form of the equation of a plane4.       Normal form of the equation of a plane5.   Reduction of a linear equation to the

normal form

6.     Angle between two planes7.     Distance to a point from a plane8.     Equation of a line9.    Direction cosines of the line of intersection of

two planes10.   Forms of the equations of a line11.   Angle which a line makes with a plane12.   Distance from a point to a line13.   Distance between two non-intersecting lines

14.   System of planes through a line15.   Application to descriptive geometry16.   Bundles of planes17.   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

• 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

•  Chapter 4 Types of Surfaces  1.       Imaginary points, lines, and planes2.       Loci of equations3.       Cylindrical surfaces4.       Projecting cylinders5.       Plane sections of surfaces6.       Cones7.       Surfaces of revolution 

• Chapter 5 Forms of Quadric Surfaces 1.       Definition of a quadric 2.       The sphere3.       The ellipsoid4.       The hyperboloid of one sheet5.       The hyperboloid of two sheets6.       The imaginary ellipsoid7.       The elliptic paraboloid8.       The hyperbolic paraboloid9.       The quadric cones10.   The quadric cylinders11.   Summary

 • Chapter 6 Classification of Quadric Surfaces 1.       The Intersection of a quadric and a line2.       Diametral planes, center3.       Equation of a quadric referred to its center4.       Principal planes5.       Reality of the roots of the discrimination cubic6.       Simplification of the equation of a quadric7.       Classification of quadric surfaces8.       Invariants under motion9.       Proof that I, J, and D are invariant10.   Proof that is invariant11.   Discussion of numerical equations

 • 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*