Circles and spheres via dot products I | Wild Linear Algebra A 31 | NJ Wildberger

Circles are fundamental geometrical objects that fit naturally into a linear algebra framework using the dot product. For simplicity we discuss circles with center the origin O in this lecture.

Interestingly, there are (at least!) two different ways to introduce the circle from the Euclidean dot product: in terms of a quadratic equation v.v=k or in terms of a bilinear equation: v_1.v_2=k. The second is much less familiar, but has more power to explain tangents and to appreciated Apollonius' theory of pole/polar duality, which is expressed very simply and naturally in the language of linear algebra.

It also allows us to consider not only the example of the usual unit circle, but also the imaginary unit circle, which plays a surprisingly big role in geometry! We finish the lecture with some classical theorems about circles which are fundamental for projective geometry.

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[StijnOomes]

There are a lot of ideas that are coming together here in a simple and elegant way. Thanks Norman, it is really inspiring!

[aleksanderaksenov]

Thank you Sir!!!Please make more profound course on algebra itself especially on algebraic group theory and homotopy theory

[pickeyberry]

Video Content

00:00 Introduction
10:07 A circle is an equation of the form
16:17 Unit circle
22:32 Null circle / zero circle
25:38 Imaginary unit circle
28:59 Fundamental facts about circles
35:08 Theorem(Apollonius)