Basic Concepts

Functions

A one-to-one mapping from the domain to the range:

$$f:\mathbb{R}\to \mathbb{R}$$

Note

$\mathbb{R}$ represents the one-dimensional real number domain.

Height Maps

A height map can be described as:

$$f:{\mathbb{R}}^{\mathbb{2}}\to \mathbb{R}$$

A height map is a 2D image where each pixel records a height value. It is commonly used to represent terrain.

2D Parametric Curve

$$f:\mathbb{R}\to {\mathbb{R}}^{\mathbb{2}}$$

In other words: $f(t)=p$ where $p$ is a two-dimensional coordinate point (x, y).

3D Parametric Curve

$$f:\mathbb{R}\to {\mathbb{R}}^{\mathbb{3}}$$

In other words: $f(t)=p$ where $p$ is a three-dimensional coordinate point (x, y, z).

Texture Mapping

The relationship mapping a texture to an object's surface can be described as:

$$f:\mathrm{\Omega }\to {\mathbb{R}}^{\mathbb{3}}$$

Here, $\mathrm{\Omega }$ represents the texture domain (u, v), and ${\mathbb{R}}^{\mathbb{3}}$ represents the spatial domain (x, y, z).

Manifolds

• 1-Manifold: A Curve
• 2-Manifold: A Surface
• 3-Manifold: A Volume

Arc-length Integral

Used to calculate the length of a curve:

$${\int }_a^b\sqrt{(1+f^{\prime }(x{)}^2)}dx$$

Continuity

When the left limit and right limit exist and are equal at $x=a$, then $f(x)$ is continuous at $x=a$, also known as $C^0$ continuity.

$$\underset{x\to a^{-}}{lim}f(x)=f(a)=\underset{x\to a^{+}}{lim}f(x)$$

Or more formally:

For a given $\delta >0$, there exists $ϵ>0$ such that if $x\in (a-ϵ,a+ϵ)$, then $f(x)\in (f(a)-ϵ,f(a)+ϵ)$.

For the 2D case, we use neighborhoods. Neighborhoods can be described using triangles, rectangles, or circles.

Neighborhoods

Calculus of Multiple Variables

Partial Derivatives

$$\frac{\partial }{\partial x}(x^{2}y+y^{3}z-xyz)=2xy-yz$$$$\frac{\partial }{\partial y}(x^{2}y+y^{3}z-xyz)=x^2+3y^{2}z-xz$$$$\frac{\partial }{\partial z}(x^{2}y+y^{3}z-xyz)=y^3-xy$$

Gradient Vector

$$\overrightarrow{\mathrm{\nabla }}f(x)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})$$

Numerical Calculus

Common numerical methods:

• Taylor Series and Expansion
• Numerical Integration
• Lookup Table

This section belongs to the content of Scientific Computation (COMP5930M).