4 ms·

Yeah, I was looking over the materials and thinking, where’s the Algebraic Geometry? I don’t think that Algebraic Geometry (as opposed to algebraic geometry in

by dhosek 4y ago

Yeah, I was looking over the materials and thinking, where’s the Algebraic Geometry? I don’t think that Algebraic Geometry (as opposed to algebraic geometry in the sense that the author here is using it) has any applications to computer graphics, but definitely can be applied to other aspects of computer science (I can imagine it being potentially useful in cryptographic applications), although my own Algebraic Geometry knowledge is more limited than I would like it to be.

kweingar 4y ago
Algebraic geometry is also used in robotics
kevinventullo 4y ago
Projective space, syzygies, Bezier curves, cubic curves aren’t algebraic geometry?
- kcexn 4y ago
  I'd say they're more algebraic representations of regular geometries. Algebraic Geometry as a mathematical field is interested in solving highly non-trivial geometric problems (think, from differential geometry, functional analysis, etc.), using tools from Abstract Algebra (think Galois Fields, Lie Groups etc.) It saddles a bridge between traditional geometry, and abstract algebra, and allows insights from one field of mathematics to be applied to the other. As such, it allows practitioners skilled in these tools to make many useful inferences about incredibly complicated systems. It's also incredibly dense. In part because many of the tools of algebra are incredibly involved. But also, in part because to define an algebraic object in a way that is equivalent to a geometric object, sometimes requires a fairly complicated definition.
  - edflsafoiewq 4y ago
    Algebraic geometry existed before Grothendieck lol. It's still algebraic geometry even it looks boring next to etale cohomology of infinity stacks or whatever.
    - kcexn 4y ago
      Even before Grothendieck's scheme theory, algebraic geometry was already pretty focused on varieties in rings of polynomials and other such constructions. The heart of the domain is still using abstract algebraic arguments to solve geometric problems. E.g.; Euclid's method for finding the midpoint of a line is to draw two concentric circles centred at the vertices with radius the length of the line. The straight line that passes through the two intersection points of the circles, also passes through the midpoint of the line. This is the same as saying the midpoint of a line is the intersection of an algebraic variety with a root at one vertex, and another algebraic variety with a root at the other vertex. You don't need schemes, or projective curves, or local rings to prove it.

by dhosek 4y ago

kweingar 4y ago

Algebraic geometry is also used in robotics

kevinventullo 4y ago

Projective space, syzygies, Bezier curves, cubic curves aren’t algebraic geometry?

kcexn 4y ago
I'd say they're more algebraic representations of regular geometries. Algebraic Geometry as a mathematical field is interested in solving highly non-trivial geometric problems (think, from differential geometry, functional analysis, etc.), using tools from Abstract Algebra (think Galois Fields, Lie Groups etc.) It saddles a bridge between traditional geometry, and abstract algebra, and allows insights from one field of mathematics to be applied to the other. As such, it allows practitioners skilled in these tools to make many useful inferences about incredibly complicated systems. It's also incredibly dense. In part because many of the tools of algebra are incredibly involved. But also, in part because to define an algebraic object in a way that is equivalent to a geometric object, sometimes requires a fairly complicated definition.
- edflsafoiewq 4y ago
  Algebraic geometry existed before Grothendieck lol. It's still algebraic geometry even it looks boring next to etale cohomology of infinity stacks or whatever.
  - kcexn 4y ago
    Even before Grothendieck's scheme theory, algebraic geometry was already pretty focused on varieties in rings of polynomials and other such constructions. The heart of the domain is still using abstract algebraic arguments to solve geometric problems. E.g.; Euclid's method for finding the midpoint of a line is to draw two concentric circles centred at the vertices with radius the length of the line. The straight line that passes through the two intersection points of the circles, also passes through the midpoint of the line. This is the same as saying the midpoint of a line is the intersection of an algebraic variety with a root at one vertex, and another algebraic variety with a root at the other vertex. You don't need schemes, or projective curves, or local rings to prove it.