Parametric Representations Lecture Notes

Parametric Representations: Motivation

• Transformations and rendering processes can be simplified using parametric representations.
• Parametric representations can handle complex objects more easily.

Lecture Overview

• Topics:
  • Parametric representation
  • Curves
  • Surfaces
  • Drawing curves

Curves as Functions

• Curves can be represented using functions where changing $x$ affects $y$.
• Examples: $f(x) = x^2$, $f(x) = cos(x)$
• Characteristics: For every $x$, there is only one corresponding $y$. Thus, there are no overlaps.
• Problem: A single $x$ value cannot represent multiple $y$ values.

Parametric Representation of Curves

• Solution: Use a parameter $t$ to represent both $x$ and $y$.
  • $x = x(t)$
  • $y = y(t)$
• $t$ represents the distance along the curve.
  • $t = 0$: start of the curve.
  • $t = 0.5$: halfway point of the curve (assuming total length is 1).
  • $t = 1$: end of the curve.

Parametric Representation of Surfaces

• Curves are 2D (x, y), while surfaces are 3D.
• Use two parameters, $u$ and $v$, to represent a surface.
  • $x = x(u, v)$
  • $y = y(u, v)$
  • $z = z(u, v)$

Parametric Equation of a Line

• Representing a line using two points $(x<em>A, y</em>A)$ and $(x<em>B, y</em>B)$.
• Standard equation: $y = mx + c$
• Parametric representation using $t$:
  • $x = x<em>A + (x</em>B - x_A)t$
  • $y = y<em>A + (y</em>B - y_A)t$
• $x<em>B - x</em>A$ represents the total horizontal displacement.
• $y<em>B - y</em>A$ represents the total vertical displacement.
• $t$ represents the fraction of the total displacement.
• When $t = 0$, $x = x<em>A$ and $y = y</em>A$.
• When $t = 1$, $x = x<em>B$ and $y = y</em>B$.

Example: Finding Parametric Equation

• Given points $(1, 4)$ and $(5, -8)$, find the parametric equations.
  • $x = 1 + (5 - 1)t = 1 + 4t$
  • $y = 4 + (-8 - 4)t = 4 - 12t$
• If $y = 0$, find the value of $x$.
  • $0 = 4 - 12t \,\Rightarrow\, t = \frac{4}{12} = \frac{1}{3}$
  • $x = 1 + 4(\frac{1}{3}) = 1 + \frac{4}{3} = \frac{7}{3}$

Parametric Representation of a Circle

• Equation: $x^2 + y^2 = r^2$
• Parametric representation: Using $t$ to represent the angle, where $t$ represents the percentage of the total 360 degrees.
  • $x = r \cdot cos(\theta)$
  • $y = r \cdot sin(\theta)$

Summary of Parametric Representations

• Single function: $f(x)$
• Parametric curve: Using $t$ to represent $x$, $y$, and $z$.
• Surface: Using $u$ and $v$ to represent $x$, $y$, and $z$.
• $t$ represents distance along the curve.
• $u$ and $v$ represent location on the surface.
• Linear, cubic, or arithmetic functions can be used to represent different shapes.

Bezier Curves

• The shape of the curve is controlled by control vertex points.
• The curve usually lies within the convex hull of the control points.
• The Bezier curve is typically cubic but can be of higher order.
• Function: A Bezier curve is represented as the sum of basis functions.
  • $P = \sum B<em>i(t) P</em>i$
    • Where $P<em>i$ represents control points and $B</em>i(t)$ represents basic functions.
• The basic function is similar to $x(t)$ and gives a weight to each control point.
• The location of points on the curve depends on the weights of the control points.
• The sum of the weights equals one.

Characteristics of Bezier Curves

• Different control points yield different shapes.
• Basic functions are global, meaning every control point impacts the overall shape of the curve.
• Every control point contributes to the overall shape.
• Curves lie inside the convex hull, ensured by the fact that the sum of basic functions equals one, and each basic function is greater than zero.
• Complex curves can be constructed by combining multiple segments.

Smoothly Connecting Bezier Curve Segments

• Multiple segments can be joined using different control points.
• Smoothness is achieved by ensuring collinearity at the joint control point.
• Types of continuity:
  • Zero order: The two segments have a joint control point.
  • First order: The segments not only have a joint control point, but the tangents are equal.
  • Second order: The second derivative is equal at the joint point.
• Geometric continuity requires derivative continuity.

Bezier Surfaces

• Surfaces are 3D, while curves are 2D.
• Surfaces are controlled by parameters $u$ and $v$.
• Every point on the surface is defined by $u$ and $v$.
• Functions involve control points and Bernstein polynomials.
• Surface lies within the convex hull of all control points.
• Transforming a control point significantly impacts the shape of the surface.
• When $u$ and $v$ are constant, it results in Bezier curves.
• Bezier surface can be regarded as several Bezier curves.

Characteristics of Bezier Surfaces

• Increasing control points increases complexity.
• Control is non-local, meaning a single control point can impact the entire surface.
• Surfaces are often constructed from bicubic patches.
• Similar to curves, derivative continuity allows combination of surfaces.

Drawing Curves

• Involves interpolation between control points.
• Given four control points $(P<em>0, P</em>1, P<em>2, P</em>3)$, the first layer of interpolation results in $(Q<em>0, Q</em>1, Q<em>2)$. The second layer results in $(R</em>0, R_1)$, and the final interpolation yields point B on the curve.

Formats for Curve Drawing

• Approximating: Passes through only the starting point but not necessarily through all control points.
• Interpolating: Goes through all control points.
• Cubic splines have continuous first or second derivatives at joint control points.

NURBS (Non-Uniform Rational B-Splines)

• Basic functions are local, meaning changing a single control point only impacts a local part of the curve.
• Offers finer control compared to Bezier surfaces or curves.

NURBS for Surface Definition

• Allows defining the shape of a surface using control points and basic functions.
• Adjusts overall and local shapes.
• Complex curves result in complex 3D meshes.
• Subdivision algorithms have started to replace this generation method.