Project 5: NURBS Surface Generator

Build a NURBS surface evaluator that generates a mesh from control points and knot grids.

Project Overview

Attribute Value
Difficulty Level 3: Advanced
Time Estimate 2-3 weeks
Main Language C++
Alternative Languages C, Rust, Python
Knowledge Area Parametric surfaces
Tools Mesh viewer, plotting tool
Main Book “The NURBS Book” by Piegl & Tiller

What you’ll build: A surface evaluator that outputs a triangle mesh approximation of a NURBS surface.

Why it teaches computational geometry: Surfaces are how CAD represents smooth solids. This makes them concrete.

Core challenges you’ll face:

  • Extending basis functions to two parameters
  • Managing control point grids and weights
  • Producing stable, uniform sampling

Real World Outcome

You will generate a surface mesh that can be visualized in a mesh viewer. The surface should reflect the control net and weights.

Example Output:

$ ./nurbs_surface --control surface.json --u 40 --v 40 --output surface.obj
Degree (u,v): 3,3
Control points: 6 x 6
Wrote surface mesh to surface.obj

Verification steps:

  • View the surface mesh and compare against the control net
  • Adjust weights and observe shape changes

The Core Question You’re Answering

“How do CAD systems generate smooth surfaces from a grid of control points?”

This project extends curve intuition into full surface modeling.

Concepts You Must Understand First

Stop and research these before coding:

  1. Tensor-product surfaces
    • How do you combine two sets of basis functions?
    • Book Reference: “The NURBS Book” by Piegl & Tiller, Ch. 5
  2. Control nets
    • How do control points influence local surface shape?
    • Book Reference: “Curves and Surfaces for CAGD” by Gerald Farin, Ch. 6
  3. Surface sampling
    • How do you choose u/v resolution for mesh output?
    • Book Reference: “Geometric Modeling” by Michael E. Mortenson, Ch. 9

Questions to Guide Your Design

  1. Sampling strategy
    • Will you sample uniformly in parameter space?
    • How will you detect high-curvature regions?
  2. Output mesh
    • How will you triangulate the sampled grid?
    • How will you generate normals for rendering?

Thinking Exercise

Surface Parameter Grid

Imagine a 4x4 control net. Sketch how many quads appear when you sample a 10x10 grid in parameter space.

Questions while working:

  • How many triangles does that produce?
  • How does sampling density affect performance?

The Interview Questions They’ll Ask

Prepare to answer these:

  1. “How do NURBS surfaces differ from NURBS curves?”
  2. “What is a tensor-product surface?”
  3. “How do control nets affect surface shape?”
  4. “Why do you need tessellation for display?”
  5. “How do you compute normals for a parametric surface?”

Hints in Layers

Hint 1: Starting Point Extend your curve basis evaluation into two dimensions.

Hint 2: Next Level Compute weighted sums across the control net for each (u,v).

Hint 3: Technical Details Triangulate the sampled grid consistently to avoid cracks.

Hint 4: Tools/Debugging Visualize the control net alongside the generated surface.

Books That Will Help

Topic Book Chapter
NURBS surfaces “The NURBS Book” by Piegl & Tiller Ch. 5
Control nets “Curves and Surfaces for CAGD” by Gerald Farin Ch. 6
Tessellation “Geometric Modeling” by Michael E. Mortenson Ch. 9

Implementation Hints

  • Start with a simple surface like a bilinear patch.
  • Keep your evaluator separate from the mesh writer.
  • Provide controls to vary resolution for quality vs speed.

Learning Milestones

  1. First milestone: You can evaluate points on a surface grid.
  2. Second milestone: You can export a mesh that renders correctly.
  3. Final milestone: You can explain how control nets shape surfaces.