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 (xA, yA) and (xB, yB).
  • Standard equation: y = mx + c
  • Parametric representation using t:
    • x = xA + (xB - x_A)t
    • y = yA + (yB - y_A)t
  • xB - xA represents the total horizontal displacement.
  • yB - yA represents the total vertical displacement.
  • t represents the fraction of the total displacement.
  • When t = 0, x = xA and y = yA.
  • When t = 1, x = xB and y = yB.

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 Bi(t) Pi
      • Where Pi represents control points and Bi(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 (P0, P1, P2, P3), the first layer of interpolation results in (Q0, Q1, Q2). The second layer results in (R0, 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.