W4: Parametric Curves

CSCI 3090

Course title: Parametric CurvesInstructor: Shima RezasoltaniInstitution: Faculty of Science, Ontario Tech

Overview of Parametric Curves

Goal of the Class

• Differentiate between local and global control to understand how adjustments to control points affect the shape and continuity of curves.

• Determine if a curve interpolates a given point, denoting how closely the curve can follow specific data points.

• Understand how blending functions combine to create various types of curves for diverse applications in computer graphics and geometric modeling.

Parametric Representations

Function Requirement

• A function, f(t), is needed to describe points along a curve through a continuous mapping of parameter t.

• Trigonometric functions can become cumbersome for complex shapes, leading to a preference for polynomials due to their mathematical properties and ease of differentiation and integration.

Canonical Form

• Polynomial Representation

  • General polynomial form:[ f(t) = \Sum (a_i \cdot t^i) ]where i=0 to n.

  • In this context, the coefficients a_i are vectors that specify the coordinates (x, y) of points on the curve, enabling more straightforward computations of curve points over varied values of t.

Blended Form

Vector and Polynomial Usage

• In this blended representation, ci are vectors that represent control points, and bi(t) are polynomials known as basis or blending functions.

• The flexibility in choosing these blending functions aids in fitting curves to datasets and enhances control over curve shapes for specific applications.

Transitioning from Canonical Form to Blended Form

• Canonical:[ f(u) = \Sum (a_i \cdot u^2) ]

• Blended:[ f(u) = \Sum (c_i \cdot b_i(u)) ]

  • For canonical, each b_i(u) simplifies to u^i, linking polynomial degree to its geometric implications.

Linear Case

Linear Polynomial Formula

• Basic linear representation:[ f(u) = p0 + (p1 - p0) \cdot u ]

  • Control points p0 and p1 guide the shape and direction of the curve, offering straightforward interpolation between two points.

• At u=0: f(0) = p0 and at u=1: f(1) = p1, confirming endpoints.

Control Points

• The arrangement of control points directly impacts the resulting curve.

• Different configurations of control points yield distinct blending functions while preserving the canonical representation.

• Example Control Point Configurations:

  • Midpoint with one endpoint

  • Endpoint with a vector to the other endpoint

  • Midpoint, slope, and distance to an endpoint, which affects curvature significantly.

Solving for Coefficients

• To determine curve behavior accurately, equate representations to solve for coefficients.

• General relationships involve constructing a matrix form to tackle more intricate situations systematically.

Constraint Matrix

• Develop equations using known values to constrain the curve's shape, referred to as the constraint matrix (C).

Blending Matrix

• By inverting the constraint matrix, one obtains the blending matrix, denoted as B, which dictates how control points influence the curve.

• Example: For a line segment parameterized by the halfway point and endpoint, the impact of choosing specific points and parameters can be analyzed mathematically.

Polynomial Degree

• While linear polynomial functions provide a foundation, cubic polynomials are often preferred in practical applications due to:

  • Their ability to provide sufficient smoothness and contouring as curves transition through points.

  • The trade-offs exist, especially regarding the control and smoothness of more complex curves, which may require higher-degree polynomials.

Continuity Across Curve Segments

• Knots: These are critical value points where curve segments transition (e.g., at parameters 0, 1, 2, ... n).

• Approaches to maintaining continuity across segments include:

  • Shared Point Scheme: Ensures segments meet at defined endpoints.

  • Dependency Scheme: Controls continuity in slope and tangent across segments.

  • Explicit Equation Scheme: Defines curves through continuous function definitions across segments.

Local vs. Global Control

• Local Control: Altering one control point only affects a localized area of the curve, making localized adjustments easier and more intuitive.

• Global Control: A change to a single control point can influence the entire curve, complicating adjustments and necessitating a comprehensive understanding of all points.

Interpolating vs. Approximating Curves

• Interpolating Curves: Designed to pass through all given control points, ensuring exact fit for all points.

• Approximating Curves: These curves do not necessarily pass through every control point but provide a general shape that aligns closely with the data, allowing for smoother connectivity between points.

Properties of Cubics

• Ideal cubic curves should:

  • Be defined piecewise as cubic functions, ensuring each section is smooth.

  • Interpolate a set of control points while maintaining local control and C2 continuity (continuous first and second derivatives).

  • Recognize that no single curve can perfectly satisfy all conditions due to inherent trade-offs in curve design and mathematical constraints.

Summary

Key topics from the lecture include:

• Parametric Representations

• Blending Functions

• Control Points

Next Class

• Focus on advanced curves; review current concepts before the next session to ensure understanding and preparedness for more complex topics.