Page 5: Types of Curves: How to Choose?
Spring 2026 Sample Solution
Over the past few pages, we saw the three main approaches to modeling curves.
Here is a quick summary:
| Representation | Pros | Cons | Best For |
|---|---|---|---|
| Parametric |
• Easy to Draw: Generating points for rendering is trivial (loop u from 0 to 1). • Control: Intuitive for design; control points directly manipulate shape. • Finite: Naturally represents finite segments with endpoints. |
• Inside/Outside Test: Hard to determine if a point is on the curve or inside a loop. • Intersections: Calculating intersections requires finding the roots of complex polynomials. |
• Font design (TrueType) • Vector drawing (SVG, Illustrator) • Animation paths |
| Implicit (e.g., Circles, Level Sets) |
• Inside/Outside Test: Trivial (just check if ). • Intersections: Finding intersections is simpler. • Topology: Handles splitting/merging naturally. |
• Hard to Draw: Requires complex algorithms to find boundary points. • Infinite: Naturally represents infinite shapes; hard to define a “segment." • Hard to Control: Difficult to manipulate shape intuitively. |
• Ray tracing (intersections) • Fluid simulation (blobs) • CSG (Constructive Solid Geometry) |
| Subdivision Iterative refinement |
• Topology Independent: Works on arbitrary meshes (no parameterization needed). • Scalable: Level of detail (LOD) is built-in; stop when pixels are small. • Uniform: Handles “extraordinary points” better. |
• Approximation: The limit surface often has no simple analytic formula. • Evaluation: Point evaluation at arbitrary parameters is complex. • Artifacts: Can suffer from “ripples” near extraordinary vertices. |
• 3D Character modeling (Pixar/Disney) • Terrain generation • Organic shapes |
In this workbook, we’ll focus on parametric curves. These are the main representations used in 2D computer graphics because they are easy to draw and convenient for designing shapes.
Later in the course, we’ll revisit these three representations when we consider surfaces in 3D.
Now might be a good time to make a checkpoint:
Next: Page 6 - Curve Properties