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 (note 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

Discrete Topology

Note that in computer science world, there is no actual "continuous" space, so we need some discrete methods.