Curves

Curves - Implicit Representation

• Most curvest can be represented implicitly:
  • \[f(x, y) = 0\]
  • Examples:
    • \[ax + by + c = 0\]
    • \[x^2 + y^2 - r^2 = 0\]
  • In 3D, surfaces are represented as:
    • \[f(x, y, z) = 0\]

Curves - Parametric Representation

• The value of each component (x, y, z) depends on an independent variable, u (the parameter).
• \[p(u) = \begin{bmatrix} x(u) \\ y(u) \\ z(u) \end{bmatrix}\]
• \[\frac{dp(u)}{du} = \begin{bmatrix} \frac{dx(u)}{du} \\ \frac{dy(u)}{y(u)} \\ \frac{dz(u)}{z(u)} \end{bmatrix}\]
  • Derivative : Direction = tangent, Magnitude = speed that curve changes with u

Criteria for choosing a representation

• Parametric forms are not unique. Many different representations are possible for a single curve.
• Criteria for a good representation:
  • Local control of shape
  • Smoothness and continuity
  • Ability to evaluate derivatives
  • Stability
  • Ease of Rendering
• Parametric curves that are polynomials in u satisfy many of these criteria:
  • \[x(y) = c_0 + c_1 u + c_2 u^2 + \cdots + c_n u^n = \sum_{k=0}^n c_ku^k\]

Matrix Equation for parametric curve

• We can write the 3 euqations for x(u), y(u), and z(u) in one matrix equation:
  • \[p(u) = \begin{bmatrix} x(u) \\ y(u) \\ z(u) \end{bmatrix} = \begin{bmatrix} \sum_{k=0}^n u^k c_{xk} \\ \sum_{k=0}^n u^k c_{yk} \\ \sum_{k=0}^n u^k c_{zk} \end{bmatrix = \sum_{k=0}^n u^k \begin{bmatrix} c_{xk} \\ c_{yk} \\ c_{zk} \end{bmatrix} = \sum_{k=0}^n u^k c_k\]

Degree of the Polynomial

• Tradeoff:
  • High degree can have rapid changes and lots of turns, but it requires more computation and curve may not be as smooth.
  • Low degree means a smoother curve, but it may not fit the data as well.
• Compromise: Use low degree polynomials with short curve segments. Cubic polynomial work well.

Multiple curve segments

• We usually want to define a curve segment between two endpoints. We define the curve between u=0 and u=1, so that:
  • $p(0) = p_0$ and $p(1) = p_1$
• Longer curves are composed of multiple segments. We would like the connection between curves to be as smooth as possible.

Interpolation

• If one point provides 3 equations and 12 unknowns, then 4 points provide 12 equations with 12 unknowns.
• We can choose any value of u to correspond to the 4 points, so we will choose to divide the interval 0 to 1 evenly(0, 1/3, 2/3, 1).