My Bug Has A Program!

The book "Geometric Algebra for Computer Science" (2)

Conformal model of 3D space

figure 1.1 of book (Fig.1.1 of the book)

The conformal model is an ample model of the 3D space. Points, lines, planes, circles, spheres and many known geometrical constructions can be represented with algebraic ways in that model.

There are 2 additional bases of this representation space. One is o which represents "origin" and the other one is ∞ which means "infinity". For readability, write "no" for o and "ni" for ∞. The 3D point (x, y, z) can be represented as the 5D point no + x e1 + y e2 + z e3 + 0.5 α ni where α = x x + y y + z z and e1-3 are "euclidean" bases.

It looks very strange and somewhat artificial but turns out to be a surprisingly fine embedding. It's also a good extension of homogeneous model. In this world, we have the unit "psudoscalar" I = no^e1^e2^e3^ni which behaves like a "complete" element where ^ is the outer(wedge) product. With it, a bit unpopular construction "dual" of A can be defined to be A/I and denoded by A*.

When p,q,r,s are 4 3D points with a general position represented in this model space, S = p^q^r^s represents a sphere on which those 4 points are. It's known that S* = c - 0.5 rho rho ni where c is the center of S and rho is the radius of S. Looks things go well, don't they?

Rotation, translation and another "conformal" transformation can be described with the uniform manner -- "versor" in this model. They give not only theoritical but also computational tools in the real applications. I hear that the Unreal Engin uses the conformal model and the geometric algebra in their lighting computations.

Plunge

There are intresting constructions which are a bit unpopular. "Plunge" is one of them. An example of plunge is here.

plunge

The circle labeled p ^ K* is the plunge of p, S and H. It crosses perpendicularly to the point p, the sphere S and the plane H.

A circle of latitude can be considered as another example of plunge.

plunge

If you live in point p on this sphere surface not poles, then p ^ L* is the circle of latitude passes p where L is the earth's axis. Since L can be a meet of two planes H and H', p ^ L* = p ^ H* ^ H'* which is the plunge of p, H and H'.

plunge

This is Fig.15.5 of the book which shows many circles p ^ L* for various points p. Now L* looks like the field of possible circles, doesn't it?

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The book "Geometric Algebra for Computer Science" (1)

Published: 火 01 5月 2018
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Geometric Algebra

This is an article for the book

"Geometric Algebra for Computer Science", Leo Dorst, Daniel Fontijne, Stephen Mann (ISBN 9780123749420)

which is one of the greatest book I've read. This is NOT a review or somesuch. I'm happy if I can report a tiny bit of flavor of …
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ESP32 0.5ms tick

Published: 日 22 10月 2017
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ESP32 PHY board

Published: 水 18 10月 2017
By Kaz Kojima

In English.

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I'm making a simple esp32 board with ethernet. ESP32 has MAC and esp-idf supports PHYs like LAN87x0 and TLK110 already. Here is the schematic with KiCAD:

I'll create a git repository for that hardware if it works.

Update 2017-10-21

Now it works!

Revised KiCAD files for this board can be …
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Geometric Algebra

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