<aside> 💡 TLDR; Projective (3D) > Affine (considers 2D only) > Euclidean (considers distance and stuff)

Affine planes are projective planes

In Projective space, no parallelism or perpendicularism

Plane notation: {x = } Line notation: {x = , y = } Point notation: (x,y,z)

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Projective Planes

<aside> 💡 I like to think of projection planes as some object in the distance of your camera, its the viewing plane. Based on projective geometry, everything intersects the origin, and we can see the viewing plane from origin. The problem is if we are the plane that is “parallel” to the viewing plane, we can’t see it!

Affine geometry is if we have the plane as above, but we live on it! You live in a 2D world. only two coordinates, x and y. Still no concept of distance in this world.

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What is the projective space?

Elliptic curves on based on 2D planes

We can reformulate it to exist on a projective plane, with 3 dimensions.

That means points can be defined as tuples with 3 coordinates: (X,Y,Z)

This allows the elliptic curve to be described using homogeneous equation which can then be used to calculate the points on the curve.

What do the terms affine and projective mean in math?

What's the difference between an Euclidean plane, an affine plane, and a projective plane?

What is affine plane? two-dimensional geometric structure where two points determine a unique line and where two lines intersect in a single point. It has no measure of angles, distances, or areas and is different from Euclidean geometry. Affine planes are an important concept in linear algebra and mathematics generally.

Axioms:

• Any two distinct points lie on a unique line.
• Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (Playfair's axiom)
• There exist three non-collinear points (points not on a single line).

Projective planes are affine planes BUT dont consider parallelism

Projective planes project 3 affine dimensions, but since we are dealing with elliptic curves, we can compress to 2 dimensions. A projective point in 2D projective plane is written as (a : b :c)