 Geometric Inversion

presented by Kyle Schultz

This instructional unit was inspired by an instructional unit of the same title created by Pam Turner for EMAT 6690 in the spring of 1995. My intentions are to preserve Pam's work, to update it to reflect a decade's worth of technological advances, to improve it where possible, and to make it available to a wider audience. This unit uses two software packages: Geometer's Sketchpad and Graphing Calculator.

A Note to the Teacher: The instructional unit that can be accessed through the links below contain suggestions for questions and topics intended for student exploration. These items will be underlined. Solutions and proofs for these questions, as well as notes for the teacher, are formatted in italics.

 Introduction to Geometric Inversion Conceptual Foundation Analytic Geometry

 Constructing Inverses Introduction Method 1: Proportions Method 1a: Proportions Method 2: Similar Triangles Method 3: P Outside Gamma Method 4: P Inside Gamma Method 4a: P Inside Gamma

 Inverting Geometric Figures Introduction Inverting Circles Inverting Lines Inverting More Circles Inverting Other Conic Sections

 The Analytic Geometry of Inverting Curves Introduction Curve #1: an ellipse centered at the origin Curve #2: an ellipse not centered at the origin Curve #3: a parabola with vertex at the origin Curve #4: a parabola with vertex not at the origin Curve #5: a hyperbola centered at the origin and tangent to gamma Curve #6: a hyperbola centered at the origin Curve #7: a hyperbola not centered at the origin

 Stereographics Introduction Cross-Section of Construction

 Applications Introduction 1. Influence of Inversion on Segment Length 2a. Ptolemy's Theorem 2b. An Extension 2c. Fermat's Problem 2d. Cubic from an Inscribed Quadrilateral 3. Concentric Circles 4. Extended Diameter 5. Three Chords in a Circle