1. Foundations of Algebraic Curves
You’ll start by learning what an algebraic curve is and how to classify them by degree. You’ll explore symmetry and simple transformations, giving you the tools to understand more complex curves later.
2. Conic Sections (Circles, Ellipses, Parabolas, Hyperbolas)
These are the classic curves everyone studies. You’ll learn:
How to write their equations
How to find centers, foci, and axes
How they appear in the real world, from satellite orbits to mirrors
3. Cubic Curves (Degree 3)
Things get more exciting here! You’ll explore curves with loops and twists, like the Folium of Descartes and Cissoid of Diocles. You’ll see how a simple cubic equation can make surprising shapes.
4. Quartic Curves (Degree 4)
Even more intricate curves await! You’ll study lemniscates, Cassini ovals, and other quartic curves, learning how equations create loops, nodes, and interesting patterns.
5. Transformations and Geometry of Curves
Here, you’ll learn to move, rotate, and scale curves. You’ll also see how changing parameters in an equation changes the shape, making it easy to explore families of curves dynamically in GeoGebra.
6. Historical and Modern Perspectives
Discover the stories behind famous curves and how they solved classic problems. You’ll also see how algebraic curves connect to modern science, technology, and design.
7. Interactive Exploration with GeoGebra
Finally, you’ll put your knowledge into action:
Build and manipulate curves with sliders
Visualize dynamic behavior and loci
Explore calculus connections like tangents and slopes (if you want a challenge!)
By the end of the book, you’ll have a strong understanding of both the algebra behind curves and the geometry you can see and interact with. More importantly, you’ll have fun discovering the hidden beauty in math!
