3.1. Parametric Representation

Surfaces are often defined as:

$S(u, v) = [x(u, v), y(u, v), z(u, v)]$ where $(u, v)$ are parameters.

Parametric representation is a way to define surfaces using parameters. Instead of describing a surface by a single equation, we use two parameters, $u$ and $v$, to represent points in three-dimensional space. This means that we can express the coordinates of any point on the surface as functions of these parameters.

For example, the equations $x(u, v)$, $y(u, v)$, and $z(u, v)$ give the x, y, and z coordinates of a point depending on the values chosen for $u$ and $v$.

Think of a parametric representation like a GPS navigation system that tells you route options. Instead of a single map that shows one layout at a time, it allows you to enter different parameters (like departure and destination points) to discover various routes (points on the surface) you can take.

Parametric form:

$S(u, v) = P_0 + u(P_1 - P_0) + v(P_2 - P_0)$ (for triangle, $u, v \geq 0, u + v \leq 1$).

A planar surface is the simplest type of surface in parametric representation. Here, we define the surface using three coplanar points, denoted as $P_0$, $P_1$, and $P_2$. The parameters $u$ and $v$ determine how we mix these points to find any point on the triangular surface defined by them.

In this equation, $u$ and $v$ vary within certain constraints to ensure we stay within the triangle formed by these three points. This results in a smooth surface that can cover any area within that triangle.

Imagine an artist creating a triangular canvas. Each corner of the triangle represents a point in the equation. The artist can choose any point on the canvas (within those three corners), adjusting their brush stroke based on how much they want to lean towards one corner or another, similar to how $u$ and $v$ help determine which point on the surface to reach.

Generated by rotating a planar curve about an axis.

Parametric form (for rotating profile $r(z)$ around z-axis):

$x = r(z) \cos \theta, \quad y = r(z) \sin \theta, \quad z = z$ where $\theta \in [0, 2\pi)$.

A surface of revolution is created when you take a two-dimensional curve and rotate it around an axis. This process generates a three-dimensional shape. The parameters $r(z)$ and $heta$ dictate the profile of the surface. Here, $r(z)$ provides the radius at any height $z$, and $heta$ determines the circular coordinate around the axis of rotation.

As $heta$ varies from $0$ to $2 ext{π}$, the curve results in a complete revolution, giving us a solid figure like a cylinder or sphere based on the original curve's shape.

Think about making a clay pot on a potter's wheel. As the wheel spins (just like rotating around the axis), you're shaping the clay along a curve. The height of the pot corresponds to $z$, while how far out the clay extends from the center is given by $r(z)$. You can create various shapes depending on how you manipulate the clay as it revolves, akin to how adjusting the curve changes the resulting surface.

Coons Patch: Used for interpolating four boundary curves. Blends boundary conditions smoothly to fill the patch.

Bicubic Patch: Parametric surface defined by cubic polynomials in both $u$ and $v$. Equation: $S(u, v) = \sum_{i=0}^3 \sum_{j=0}^3 a_{ij} u^i v^j$ Capable of modeling smooth, organic freeform surfaces.

Coons patches and bicubic patches are both techniques for defining surface geometry in CAD. A Coons patch allows you to create a surface by specifying four boundary curves, ensuring the surface smoothly fits these curves. This is particularly useful in modeling situations where you have multiple surfaces that need to connect seamlessly.

A bicubic patch, on the other hand, is defined using cubic polynomials, enabling smooth transitions over both parameter directions $u$ and $v$. This allows for more complex shapes and fine-tuning of surface details.

Imagine you are a quilt maker, and you want your quilt to fit a particular shape. The Coons patch would be like selecting four edges of fabric that you want to connect. You carefully sew them together so that the surface (the top of your quilt) flows nicely across those edges. The bicubic patch is like adding decorative stitches or patterns across the entire quilt to make the surface not just connection points but a beautifully complex design throughout.

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