W4: Implicit and Parametric Representations

CSCI 3090: Implicit and Parametric Representations

Instructor: Shima Rezasoltani

Faculty of Science, Ontario Tech

Goals of the Class

• Understand the need for curved modeling primitives.

• Describe the differences between implicit and parametric representations of curves.

• Understand piecewise representations and continuity of curves.

Implicit Representations

Definition

• Implicit representations involve providing an equation for the object.

• In 2D, the equation is of the form:

  • f(x,y) = 0

  • This function represents curves where all points satisfying the equation are on the curve.

Challenges with Polygons

• Vertex Specification Woes:

  • Polygons are low-level representations necessitating many shapes for complex objects.

  • Curved objects are inadequately represented by polygons due to their flat nature.

  • Need for higher-level representations that handle curves effectively.

Characteristics of Implicit Representations

• Line Representation:

  • Direction of the line represented by vector.

  • Normal form expression of a line involves using the perpendicular vector to check point placement on the line.

• Circle Representation:

  • Implicit representation for circles as (x - xc)² + (y - yc)² - r² = 0.

  • Expressed in vector notation as ||p - c||² = r², where (xc, yc) is the center.

Displaying Implicit Representations

• Implicit representations are compact, capturing curvature but difficult to visually display.

  • For visualization, the gradient of the function helps in determining normal vectors, which can then be used to trace points on the surface effectively.

• 3D representations are similar; for a sphere, the implicit form is (x - xc)² + (y - yc)² + (z - zc)² - r² = 0.

Parametric Representations

Definition

• Parametric representations offer a more straightforward way to represent curves by using parameters.

  • Typically, in the form:

    • (x,y) = f(t)

• Enables easier drawing compared to implicit forms, while maintaining compactness.

2D Parametric Representation

• Basics:

  • A parameter t varies along the curve, generating points through functions of t.

• Circle Parameterization:

  • Example: [ x(t) = r imes cos(t), , y(t) = r imes sin(t) ] for 0 ≤ t < 2π.

  • Note non-uniqueness of parameterization; alternates can generate the same curve.

Common Types of Parameterizations

• Arc Length Parameterization:

  • Defines points along the curve by arc length s.

• Unit Parameterization:

  • Produces points from 0 to 1, commonly used for computational ease.

  • Circle example: [ x(u) = r imes cos(2πu), , y(u) = r imes sin(2πu) ] for u ranging from 0 to 1.

3D Parametric Representations

• Generalizes to 3D as (x, y, z) = f(t).

• For surfaces, utilize two parameters to create points on 3D surfaces.

Piecewise Curves

• Definition:

  • Represents curves by dividing them into smaller pieces, each defined by simpler functions.

• Advantages:

  • Allows for flexibility and ease of use in creating complex shapes.

  • Facilitates representation of any curve through sufficient pieces, though requires multiple functions.

Continuity in Piecewise Representations

Importance

• Ensures that curve pieces fit together smoothly, addressing the transition points.

Definitions of Continuity

• C0 Continuity: No gaps between pieces; requires f1(1) = f2(0).

• C1 Continuity: Ensures first derivatives align at boundaries, providing smoother transitions.

• C2 Continuity: Equates second derivatives for even smoother connections.

Geometric Continuity

• Describes proportional nth derivatives across two curves, relevant for applications like automotive aerodynamics where G2 continuity is essential.

Summary of Key Learnings

• Differentiated between implicit and parametric representations, highlighting strengths and applications of each in modeling curves and surfaces.