This page is a record of labs, with links to some lab sheets and other lab materials for Math 487.
Lab 1. January 3, 2001. Conics
Use Sketchpad to construct conics as loci, first from sums and differences of distances to foci, then by focus-directrix definition, finally by locus of centers of circles tangent to line or to a circle.
Lab 2. January 10, 2001. Rotating Coordinates
Part 1 of this lab has 3 directed activities which are intended to give insight into the upcoming formula for rotation in the (x,y) plane. Part 2 was an unstructured activity based on a physical object: given two pink rubber balls and a sheet of overhead film, construct a non-circular ellipse that will fit (at a slant) precisely into the cylinder obtained by rolling the two balls in the overhead transparency film. In other words, the minor axis of the ellipse should equal the diameter of a ball and the major axis can be any length larger than the minor axis.
Lab 3. January 17, 2001. Affine Coordinates
The first part of this lab is to set of the affine coodinates of a point P given 3 points B (the origin), A1, A2. Another way to think of this is to find the point x for which P = Ax + B. The next part of the lab reverses this process and shows how to map the triangle (0,0), (1,0), (0,1) to a general triangle B, A1, A2. This is used to view the image of a circle by this map. Finally, the coordinates for are applied 3 times in a triangle ABC and some rich interrelationships are explored.
Lab 4. January 24, 2001. Bezier curves and area formula using determinants
In the first part of the lab, we see the geometric DeCasteljau algorithm for constructing the parametric curve, the quadratic Bezier curve, (1-t)^2A + 2t(1-t)B + t^2C. This turns out to be a parabola (we have not demonstrated this yet). An analogous construction yields a cubic Bezier curve. In the second part of the lab, we shear a general parallelogram to find a simpler parallelogram with the same area. We deduce from this the formula for the area or the original parallelogram, which is given by a determinant. We also consider what the shear mapping does to a general point P.
For some additional references to this material beside the handout from class, go the the class note web page for Week 4 and follow links to barycentric coordinates and to Bezier curves.
Lab 5. January 31, 2001. From Affine to 3D.
Lab 6. February 7, 2001. Inversion and Orthogonal Circles.
This lab first constructs orthogonal circles using tangent lines and the definition. Then by tracing all the circles through a given point A which are orthogonal to a given circle c, the inversion of point A appears as a second point which is always contained in each orthogonal circle.
Based on this observation, several constructions of inversion are given. Finally, inversion is used to construct circles orthogonal to one or two given circles passing through two or one given points.
Lab 7. February 14, 2001. Images under inversion and Dr. Whatif's Euclidean Geometry.
There will be a short period in the lab in which images under inversion will be explored, using pre-made sketches and scripts.
The main part of the lab will be an series of constructions in Dr. Whatif's Euclidean Geometry (DWEG). In this model, a point O is removed from the plane and a point I is added at infinity. Then DWEG line, a line in the DWEG geometry, is one of two kinds of object (a) a Euclidean circle through O or (b) a line through O (including the point I as a point on the object). Then one can use circle constructions to construct figures in DWEG geometry.
Lab 8. February 21, 2001. The Poincare model for non-Euclidean geometry
This lab parallels the previous lab in that a model is explored to find similarities and differences in this model from the famiar Euclidean plane.
Lab 9. February 28, 2001. Desargues and Pascal theorems in projective geometry.
Lab 10. March, 2001. Poles and Polars. There will also be construction of polyhedral nets.