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Contents

Getting Started

  • Calling Geometrica | Mathematica notations | Tour of Geometrica | More on quadrics

Points

  • Cartesian point | Parametric point | Bound point
  • Barycenter | Brianchon point | Center | Center of gravity | Center of perspective | Circumcenter | Excenters | Fermat point | Foci | Incenter | Orthocenter | Pole | Vertices

Lines and Planes

  • Cartesian line and plane | Pencil | Euclidean line and plane | Conjugate line | Conjugate plane | Parallel | Normal plane
  • Altitudes | Asymptotes | Axes | Axis of perspective | Bisector | Diagonals | Directrices | Generators | Medians | Normals | Pascal line | Polar | Symmetry planes

Conics

  • Cartesian conic in 2D and 3D | Pencil | Euclidean conic | Conic elements
  • Circle | Circumcircle | Euler circle | Excircles | Incircle | Ellipse | Parabola | Hyperbola

Quadrics

  • Cartesian quadric | Euclidean quadric | Quadric elements
  • Sphere | Circumsphere | Edge sphere | Exspheres | Insphere | Ellipsoid | Quadric cone | Elliptic cylinder | Hyperbolic cylinder | Parabolic cylinder | Elliptic paraboloid | Hyperbolic paraboloid | Hyperboloid of one sheet | Hyperboloid of two sheets

Curves and Surfaces

  • General definitions | Parameter range | Cubic or Bezier fits | Parallel | Ribbon | Offset or paraxial Arc | Helix | Petal | Racetrack | Solid angle
  • General cone and cylinder | Helicoid | Helicoidal staircase | Solid angle 3D | Solid of revolution

Polygons and Polyhedra

  • General definitions | Segment | Elements of polygons and polyhedra | Orthogonal and parallel polygons
  • Triangle | Isosceles | Right angled | Diamond | Parallelogram | Rectangle | Regular polygon | Reference systems
  • Cube | Parallelepiped | Platonic solid | Prismoid | Pyramid | Tetrahedron

Drawing

  • Basic functions in 2D and 3D | Origin of lines and planes | Range of variations of the parameters | Arrow | Legend | Painting a figure | Canonical view point | Control of display
  • Movies and story boards in two and three dimensions

Transformations

  • Affine | Symmetries and inversion | Projection and inverse projection | Rotation | Translation | Zooming | Shear | Splitting a segment | Embedding a 2D object into 3D

Intersections

  • 2D intersections: Line, conic or polygonal line | Line and general curve
  • 3D intersections: Coplanar lines or conics | Plane with line, plane, polygonal line, polyhedron, quadric, cone, cylinder, general curve or surface

Affine Notions

  • Barycentric coordinates | Cartesian coordinates | Cross ratio | Ratio | Sorting points on lines and conics

Metric Notions

  • Angle of vectors, lines and planes | Areas in 2D and of surfaces | Distance between points, parallel lines and parallel planes | Length of curves | Radius of circles and spheres | Eccentricity of conics

Tests and Theorems

  • Collinear points, segments, and lines | Coplanar points, segments, and lines
  • Congruent triangles | Similar triangles | Regular polygon | Platonic solid
  • Point on line, conic, polygonal line, plane, or quadric | Line on plane or quadric | Segment on line, plane, or quadric | Conic on plane or quadric
  • Surface of type cone, cylinder or sphere
  • Line, polygonal line, or conic tangent to a conic | Plane or line tangent to a quadric
  • Identical objects
  • Theorems: Brianchon | Desargues | Euler | Pascal