Curves can look very different when viewed from inside the plane they are drawn upon (using projective geometry), rather than being seen from above (using Cartesian geometry).

To explain, we describe many details of conic curves. In particular we show how these curves can be defined with respect to a focus points, and we show simple mechanical ways to draw conics.

In particular we describe how to use the `string and pins' method (also called the gardeners method) to draw an ellipse, and we show how to produce a very simple contraption for drawing Hyperbola, using household items.

We demonstrate how projective geometry can give powerful insights into the true nature of conics. Our main driving point is that a conic drawn on a plane stretching out ahead of one can look very different to a standard picture (much like rail road tracks appear to meet on the horizon). This could be viewed as a mere optical illusion, it seems to reveal deep truths about the underlying relationships between the conics.

We consider the issue of how conics appear in projective geometry. Using geogebra, we model how it might look to be standing on a grid
that has a giant parabola drawn upon it. The resulting perspective is surprising.

We also show how to construct ellipses manually using the string and pin method

The Wolfram demonstration I used for the conics can be found here

demonstrations.wolfram.com/LocusOfPointsDefinitionOfAnEllipseHyperbolaParabolaAndOvalOf/

It was written by Marc Brodie.

The Geogebra program can be found here:
www.geogebra.org/

I would like to thank Olive Whicher for his illuminating book on projective geometry, and Norman Wildberger for his insightful teachings on the subject.

1. Why Perspective Drawing Works
2. Without Equations, Conics & Spirals
3. Foundations & Tilings in Perspective
4. When Does A Parabola Look Like An Ellipse?
5. Desargues' Theorem Proof
6. Axioms, Duality and Projections
7. Conics Made Easily and Beautifully
8. Harmonic Quadrangles & The 13 Configuration
9. The Line Woven Net
10. Brianchon's Theorem (Pascal's Dual)
11. Five Points Define A Conic
12. Projective Transformations Of Lines
13. Involutions Of The Line
14. Constructing The Dual Of A Quadrangle - The Thirteen Point Configuration
15. Pascal's Hexagrammum Mysticum Theorem
16. Non Euclidean Geometry & Hyperbolic Social Networks

Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.