Relativistic velocity, core circles, and Paul Miller's protractor (II) | Rational Geometry MF143

We introduce an important variant to the unit circle---what we call the core circle, which has diameter the unit interval [0,1]. For understanding the projective line, this core circle is a very useful object, and forms a bridge from projective geometry to Euclidean planar geometry.

Here we establish some basic facts and formulas for this object, and show how quadrances between points on the core circle exactly agree with projective quadrances between the corresponding projective 1-points. We also find a pleasant formula for the quadrea of a triangle on the core circle.

Video Content:
2:58 Review of the unit circle
9:35 The core circle
13:36 Advantage of the core circle
17:53 Parametrization of the core circle
20:38 Expression of parametrization of the core circle
23:58 Three points on the core circle
29:57 Proving the Core circle theorem
34:03 Core circle quadrance theorem

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[jesusthroughmary]

The symmetry between the projective and affine quadrances has, I think, sold me on this whole thing. Simply beautiful.

[adjoaadjavon]

Video Content:
00:00 Introduction
2:58 Rational Parametrization
9:36 The Core Circle
13:36 Advantage
17:54 Alternative Approach
20:38 The Unit Circle Quadrance Theorem
23:59 Examples
29:58 Unit Circle Signed Area Theorem
34:04 Core Circle Triangle Quadrance Theorem

[jacobolus]

This is great! Maybe it’s covered in another video, but have you pointed out somewhere that the line x = 1 and the “core circle” x^2 + y^2 = x are inverses of each-other across the unit circle? And more generally, (affine) points inside the “core disk” invert to points in the plane with x > 1, whereas points outside the “core disk” invert to points with x < 1.

So you can kind of think of "projective quadrance" between two points [1:v1] and [1:v2] on the projective line differing from the affine (v2-v1)^2 kind of definition from your non-relativistic velocity example by being what you get if you first invert the points across the unit circle, then compute the quadrances.

[mnada]

Can I assume that only rational coordinates lay on the core circle ?