Introduction
Curves
Parametric Equation
Represent the curve as the image of some function \(\vec p:\mathbb R\rightarrow \mathbb R^2\). For example \(\vec p(t) = (\cos(t), \sin(t))\) is the curve of a circle of radius 1
Implicit Equation
Represent the curve as the level set of some function, i.e. for some \(C\in\mathbb R. g:\mathbb R^2\rightarrow\mathbb R^1\), the curve is \(\{(x, y) \mid g(x, y) = C\}\). For example \(\{(x, y)\mid x^2 + y^2 - 1= 0\}\) be the curve of a circle of radius 1.
Surfaces
Functional Surface
Consider a function \(f:\mathbb R^2\rightarrow \mathbb R\), the surface is the image of \(f\). of a height field. However, in this case, we cannot represent a perpendicular surfaces parallel to the z-axis.
Parametric Equation
Similar to a parametric curve, we can represent a surface as the image of \(S:\mathbb R^2 \rightarrow \mathbb R^3\). If we can draw a map from parametric domain to 3D on the small neighborhood of each point (other than the boundary), we call it a manifold (not very topologically definition of the manifold).
Implicit Equation
Define \(g:\mathbb R^3\rightarrow \mathbb R\), then the curve is the level set of \(g\), i.e. \(\{(x, y, z) \mid g(x, y, z) = C\}\)
Tangent and normal
For Curves
For a parametric curve, the tangent \(\frac{\partial \vec p}{\partial t} = \vec t\) and unit tangent \(\hat t = \frac{\vec t}{ \|\vec t\|}\) represents the direction to which stay as on a curve.
For an implicit curve, the normal \(\nabla g\) and the unit normal \(\hat n = \frac{\nabla g}{\|\nabla g\|}\) represents the direction to get off the curve.
Note that \(\hat n \cdot \hat t = 0\), i.e. the tangent and normal are perpendicular.
For Surfaces
The normal is still \(\nabla g\) and the unit normal \(\hat n = \frac{\nabla g}{\|\nabla g\|}\).
However, the tangent is now a space or a plane in 2D, \(\{\hat t \mid \hat t\cdot \hat n = 0\}\), For a parametric curve, the two easy tangent line is \(\hat t_1 = \frac{\partial S}{\partial u}\) and \(\hat t_2 = \frac{\partial S}{\partial v}\), and the tangent plane is then spanned by \(\hat t_1, \hat t_2\), if we have \(\hat t_1\cdot \hat t_2 = \pm 1\), i.e. \(\partial_uS = c\:\partial_vS\), then it is a degenerate point, i.e. at that point, it is a curve instead of a surface.
Geometry + Topology
A surface is then defined by geometry and topology.
Some properties that defines geometry and topology
Geometry | Topology |
---|---|
position | closed-ness |
tangent / normals | orientable |
dimension | number of boundaries (ex. calendar has 2 boundaries) |
"curvature" | number of (topological) holes (ex. a donut has one hole) |
self-intersection |
Discrete Topology
Note that in computer science world, there is no actual "continuous" space, so we need some discrete methods.