GEOMETRY

GEOMETRIC SPACES

MODULE AXIOMS

Objects: Applications
  1. History

Theory
  1. Introduction
  2. Axioms of geometric spaces
  3. Axioms of incidence
  4. Axioms of order
  5. Axioms of congruence
  6. Axioms of continuity
  7. Axiom of parallelism
  8. Projective space
  9. Affine space

Assessments
  1. Test1
  2. Answers1
  3. Test2
  4. Answers2
  5. Test3
  6. Answers3
  7. Assessment

MODULE COORDINATE SYSTEMS

Objects: Theory
  1. Cartesian coordinate system
  2. Polar coordinate system
  3. Cylindrical coordinate system
  4. Spherical coordinate system
  5. Homogeneous coordinate system

Examples
  1. Example1-Polar coord from Cartesian
  2. Example2-Cartesian coord from polar
  3. Example3-Cylindrical coord from Cartesian
  4. Example4-Cartesian coord from cylindrical
  5. Example5-Spherical coord from Cartesian
  6. Example6-Cartesian coord from spherical
  7. Example7-Homogeneous coord from Cartesian
  8. Example8-Cartesian coord from homogeneous

Problems
  1. Problem1-Polar coord from Cartesian
  2. Problem2-Cartesian coord from polar
  3. Problem3-Cylindrical coord from Cartesian
  4. Problem4-Cartesian coord from cylindrical
  5. Problem5-Spherical coord from Cartesian
  6. Problem6-Cartesian coord from spherical
  7. Problem7-Homogeneous coord from Cartesian
  8. Problem8-Cartesian coord from homogeneous

Assessments
  1. Assessment

MODULE GEOMETRIC TRANSFORMATIONS

Objects: Applications
  1. Motion
  2. Deformations and fractals

Theory
  1. Geometric transformations
  2. Euclidean transformations
  3. Transformations of the Euclidean space
  4. Composite Euclidean transformations
  5. Affine transformations
  6. Projective transformations

Examples
  1. Example1-Symmetries
  2. Example2-Revolutions
  3. Example3-Translations
  4. Example4-Composed metric transformations
  5. Example5-Scalings
  6. Example6-Helical movements
  7. Example7-Affinities
  8. Example8-Collineations

Problems
  1. Problem1-Symmetries
  2. Problem2-Revolutions
  3. Problem3-Translations
  4. Problem4-Composed metric transformations
  5. Problem5-Scalings
  6. Problem6-Helical movements
  7. Problem7-Affinities
  8. Problem8-Collineations

Assessments
  1. Test1
  2. Answers1
  3. Test2
  4. Answers2
  5. Test3
  6. Answers3
  7. Assessment

MODULE BASIC GEOMETRIC FIGURES

Objects: Theory
  1. Introduction
  2. Superposition of lines
  3. Metric properties in the Eucludean plane
  4. Metric properties in the Eucludean space
  5. Metric space
  6. Line in the Cartesian plane
  7. Problems on lines in the Cartesian plane
  8. Line in the Euclidean space
  9. Plane in the Euclidean space
  10. Superposition of 2 or 3 planes

Examples
  1. Example1-Centre of line segment, distance of 2 points
  2. Example2-Equations of line through 2 points
  3. Example3-Superposition of 2 lines
  4. Example4-Superposition of line and plane
  5. Example5-Distance of parallel lines
  6. Example6-Angle formed by 2 lines
  7. Example7-Equations of line
  8. Example8-Equations of plane
  9. Example9-Pierce point of line and plane
  10. Example10-Superposition of 2 or 3 planes

Problems
  1. Problem1-Centre of line segment, coord of vector
  2. Problem2-Equations of line through 2 points
  3. Problem3-Superposition of 2 lines
  4. Problem4-Superposition of line and plane
  5. Problem5-Distance of parallel lines
  6. Problem6-Angle formed by 2 lines
  7. Problem7-Equations of line
  8. Problem8-Equations of plane
  9. Problem9-Pierce point of line and plane
  10. Problem10-Superposition of 2 or 3 planes

Assessments
  1. Assessment

CURVES

MODULE CIRCLE

Objects: Applications
  1. Earth equator circumference
  2. Circles arround us

Theory
  1. Circle - introduction
  2. Circle in the Cartesian plane
  3. Superposition of line and circle
  4. Tangent to the circle
  5. Line and circle in the Cartesian plane
  6. Power of point to circle
  7. Circle in Affine plane
  8. Problems on circle in the Cartesian plane

Examples
  1. Example1-Equation of circle
  2. Example2-Circle radius and coordinates of centre
  3. Example3-Equation of tangent line
  4. Example4-Coordinates of intersection point of line and circle
  5. Example5-Equation of tangent in circle point
  6. Example6-Equation of tangent in given direction
  7. Example7-Equation of tangent through given point

Problems
  1. Problem1-Coordinates of circle centre
  2. Problem2-Equation of circle
  3. Problem3-Equation of tangent to circle

Assessments
  1. Assessment
  2. Answers

MODULE ELLIPSE

Objects: Applications
  1. Reflecting property of ellipse
  2. Elliptical orbits

Animations
  1. Point moving on ellipse
  2. Shape of ellipse
  3. Focal angles
  4. Tangent to ellipse
  5. Auxhiliary circle
  6. Control circle
  7. Conjugate diameters

Theory
  1. Ellipse-definition
  2. Ellipse in the Euclidean plane
  3. Superposition of line and ellipse
  4. Tangent to the ellipse
  5. Line intersecting the ellipse
  6. Ellipse in the Cartesian plane
  7. Superposition of line and ellipse - calculations
  8. Ellipse in the Affine plane

Examples
  1. Example1-Coordinates of vertices and foci
  2. Example2-Equation of ellipse
  3. Example3-Coordinates of vertices and foci, graph of ellipse
  4. Example4-Coordinates of intersection point of line and ellipse
  5. Example5-Equation of tangent in ellipse point
  6. Example6-Equation of tangent in given point
  7. Example7-Equation of tangent through given point
  8. Example8-Equation of tangent in given direction
  9. Example9-Intersection points of line and ellipse in homog. coord.

Problems
  1. Problem1-Coordinates of ellipse centre, vertices and foci
  2. Problem2-Equation of ellipse
  3. Problem3-Applied problems
  4. Problem4-Equation of tangent to ellipse

Assessments
  1. Assessment
  2. Answers

MODULE PARABOLA

Objects: Applications
  1. Reflecting property of parabola
  2. Osculating circle of parabola

Animations
  1. Parabola-definition
  2. Shape of parabola
  3. Tangent to parabola
  4. Tangent to parabola in vertex
  5. Directrix of parabola
  6. Tangent properties of parabola

Theory
  1. Parabola-definition
  2. Parabola in the Euclidean plane
  3. Superposition of line and parabola
  4. Tangent to the parabola
  5. Parabola in the Cartesian plane
  6. Superposition of parabola and line
  7. Superposition of line and parabola - calculations

Examples
  1. Example1-Coordinates of vertex and focus
  2. Example2-Equation of parabola
  3. Example3-Coordinates of vertex and focus, equation of directrix, graph of parabola
  4. Example4-Coordinates of intersection points of line and parabola
  5. Example5-Equation of tangent in given point
  6. Example6-Equation of tangent through given point
  7. Example7-Equation of tangent in given direction

Problems
  1. Problem1-Coordinates of parabola vertex and focus, equation of directrix
  2. Problem2-Equation of parabola
  3. Problem3-Applied problems
  4. Problem4-Equation of tangent to parabola

Assessments
  1. Assessment
  2. Answers

MODULE HYPERBOLA

Objects: Applications
  1. Hyperbolic trajectories
  2. Reflecting property of hyperbola

Animations
  1. Hyperbola-definition
  2. Shape of hyperebola
  3. Tangent to hyperbola
  4. Auxhiliary circle
  5. Control circle

Theory
  1. Hyperbola-definition
  2. Hyperbola in the Euclidean plane
  3. Superposition of line and hyperbola
  4. Tangent to the hyperbola
  5. Line intersecting hyperbola
  6. Hyperbola in the Cartesian plane
  7. Superposition of hyperbola and line - calculations
  8. Hyperbola in the Affine plane

Examples
  1. Example1-Coordinates of vertices and foci
  2. Example2-Equation of hyperbola
  3. Example3-Coordinates of vertices and foci, graph of hyperbola
  4. Example4-Coordinates of intersection point of line and hyperbola
  5. Example5-Equation of tangent in hyperbola point
  6. Example6-Equation of tangent in given point
  7. Example7-Equation of tangent through given point
  8. Example8-Equation of tangent in given direction

Problems
  1. Problem1-Coordinates of hyperbola vertices and foci, equation of asymptotes
  2. Problem2-Equation of hyperbola
  3. Problem3-Applied problems
  4. Problem4-Equation of tangent to hyperbola

Assessments
  1. Assessment
  2. Answers

MODULE CONIC SECTIONS

Objects: Applications
  1. Historical survey
  2. Conic sections arround us

Animations

  1. Conic sections

Theory
  1. Conic sections in space
  2. Conic sections in the Affine plane
  3. Regular conic sections
  4. Singular conic sections
  5. Conic section and line
  6. Conic sections in the Euclidean plane

Examples
  1. Example1-Type classification of conic section
  2. Example2-Type classification of conic section
  3. Example3-Type classification of conic section, centre
  4. Example4-Type classification of conic section, centre
  5. Example5-Equation of tangent to conic section
  6. Example6-Superposition of conic section and line
  7. Example7-Conic section, vertices, axes
  8. Example8-Conic section, vertices, axes

Problems
  1. Problem1-Type classification of conic section
  2. Problem2-Type classification of conic section
  3. Problem3-Equation of tangent to conic section
  4. Problem4-Superposition of conic section and line
  5. Problem5-Axes, centre and vertices of conic section

Assessments
  1. Test1
  2. Answers1
  3. Test2
  4. Answers2
  5. Test3
  6. Answers3
  7. Assessment