One major theme we have had is the geometry of circles and associated constructions of tangent circles (see the constructions in GTC for conics) and orthogonal circles (recent labs and assignments). The concepts of inversion, power of a point, and radical axis play key roles.
The central figure of this study has been the figure of a pencil of circles and a second pencil of circles orthogonal to the circles of the first pencil. Two circles (or a circle and a line) determine a pencil. An elliptic pencil of circles through points A and B has a hyperbolic pencil as orthogonal pencil. This hyperbolic pencil is the set of Apollonian circles defined by A and B. The orthogonal pencil of a parabolic pencil is another parabolic pencil.
A second theme has been the study of new geometries using models built from our knowledge of circle geometry. A third theme has been the study of the regular polyhedra and related concepts in 3D geometry. Also the conic sections appeared as a topic in Euclidean geometry and then returned as a part of projective geometry.
Orthogonal Circles
The study of orthogonal circles made connections among circles of Apollonius, inversion and harmonic division, power of a point and radical axis. Basic Constructions of "Three" to construct a circle through p points orthogonal to q circles, where p+q = 3 werre studied extensively. This was the basis of all the geometry of DWEG models, Stereographic projection, and Poincare model.
Conic Sections
These were introduced as plane sections of a cone, with connection made to the metric definition using the Dandelin spheres. The special case of a cylinder was studied by a model in class. The approach was that taken in Ogilvy. Conics were constructed with Sketchpad using the "paper-folding" construction which traces the points on the conic as a locus of centers of tangent circles.
Polyhedra and 3D Geometry
All the regular polyhedra were introduced, as well as prisms and antiprisms. The relationship between cubes and inscribed tetrahedra and between tetrahedra and octahedral "holes" was studied with models. Also, dodecahedra were built from "tents" on cubes. Distances were computed. Dihedral angles were computed. The symmetries of the cube and the regular tetrahedron were studied in some depth.
Inversive geometry
The points are the points of the Euclidean plane with one point at infinity added. Lines and circles are both considered "inversive circles". The "congruence" transformations of this geometry are circle inversions and line reflections and compositions of such transformations. Angles are preserved by these transformations and "inversive circles" are mapped to "inversive circles". (Also, cross-ratio is preserved, but we did not explored that much.)
DWEG geometry
This is a non-standard model of Euclidean geometry. The points are the points of the Euclidean plane, with one point O in the plane removed and with the point I at infinity added. DWEG-lines are "inversive circles" through O. Congruence transformations of the geometry are the compositions of inversions in DWEG-lines. Angles are measured by the Euclidean angle measure between circles. Steps of equal distance can be marked off by reflecting in parallel mirrors. Circles with center A through P can be defined by reflections of P in mirrors through A. These circles turn out to be Apollonian circles.
Spherical Geometry
We have explored the concept of great circles and spherical triangles, including the formula relating angle sum (spherical excess) to area. We saw that stereographic projection of a sphere onto the plane maps circles to circles. Using the image of the equator, it is possibe to construct in the plane image of the spherre the stereographic image of the antipodal point of any point P, and thus it is possible to construct the great circle through 2 (non-antipodal) points, the poles of a great circle, reflections in circles, and also perpendicular bisectors, etc. We also studied the images of inscribed polyhedra, namely the cube and the regular tetrahedron.
The Poincare disk model of non-Euclidean Geometry
The points are points inside a Euclidean circle C. Lines are arcs or segments of supporting circles or lines which are orthogonal to C. Line reflection is given by inversion in the supporting circles or lines. Angles are measured by the Euclidean angle measure between circles. Steps of equal distance can be marked off by reflecting in parallel mirrors. Circles with center A through P can be defined by reflections of P in mirrors through A. These circles turn out to be Apollonian circles.
We learned about ultra-parallel lines and (asymototic or critical) parallel lines.
Abstract non-Euclidean geometry
We did not go too deeply into this subject, but we did learn about the concept of angle defect and the additive properties of the defect.
Projective Geometry
We have learned the definition of harmonic division and its relation to inversion and also the ruler constructions with a quadrilateral, including an example of perspective drawing of telephone poles. We have been introduced to central projections and how figures can be transformed by these projections. We have also learned the theorems of Desargues and Pascal (see Ogilvy) and how to draw a conic through 5 points using the Pascal Theorem. We have also seen how the polar of a line can be constructed with straightedge alone..