Geometry 3: Curves

1Describing curved spaces

Various ways to describe curved surfaces.
How do we describe a curved surface?
- By functions (remember that functions are mappings).
- By how it differs from a reference Euclidean space: by how it is bent.
https://en.wikipedia.org/wiki/Hypersurface

By deformation functions.
- Curved surfaces can be described by a homeomorphism from an Euclidean subspace.
- A function whose domain is an Euclidean subspace and whose codomain is a curved subspace of that domain.
  - The domain's ambient space is implied from context.
https://math.stackexchange.com/questions/493075/is-it-correct-to-think-about-homeomorphisms-as-deformations
Understand how to algebraically describe a curved space.
- Example:
  - Consider a circle with origin \(O = (0,0)\) and radius \(r\).
  - Its analytic-algebraic description is \( \SetBuilder{(x,y)}{x^2+y^2=r^2} \).
  - Its synthetic-algebraic description is \( \SetBuilder{x}{d(O,x) = r} \).
Example ways of describing a circle with origin \(O = (0,0)\) and radius \(r\):
- synthetic description: the set of every point \(x\) where \( d(O,x) = r \).
- parametric description: the set of every coordinate tuple \((x(t),y(t))\) for all \(t \in [0,2\pi]\) where:
  \[\begin{align*} x(t) &= r \cos t \\ y(t) &= r \sin t \end{align*} \]
  - See also these Wikipedia articles:
    - parametric equations
    - parametric surface
- algebraic description: the set of every coordinate tuple \((x,y) \in \Real^2\) where
  \[\begin{align*} x^2 + y^2 &= r^2 \end{align*} \]