Conics (ellipses, hyperbola, parabola) can be understood in terms of this simple construction method from projective geometry,
The idea is to create a projection from one line to another, and then draw (purple) lines joining points on the initial line to the images of such points under projection. This can be used to create ellipses, parabola and hyperbola.
The famous projective geometer Jacob Steiner described how each conic curve can be viewed as the envelope of lines connecting points which are related by a projection. The video also introduces the idea of a projection (as a sequence of perspectives). To make this I used the (free) geogebra software.

www.geogebra.org/

Although one could create such an image manually by drawing many lines joining points of the blue line to the position they get projected to. I also investigate how circles change with the movement of initial point, and obtain something that looks similar to the magnetic field lines of a small magnet.

Thanks to Norman Wildberger for his inspiring lectures on this subject, and Olive Whicher for her fascinating book on projective geometry.

We explain what perspectives are, what projections are, and how this method can be done in geogebra. We also show other interesting constructions which can be made along similar lines.

Also, I have made this video number 5 in my series (the previous video `Projective Geometry: Axioms, Duality and Projections' seemed premature).

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.