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Intermediate Topic 27 of 49 Graphics, Vision & Media

Bézier Curves and Splines

Smooth curves defined by control points.

How it works

A Bézier curve is a smooth parametric curve defined by control points, with the curve starting at the first point, ending at the last and being pulled toward the others. A cubic Bézier uses four control points and is evaluated by de Casteljau's algorithm or Bernstein polynomials. Splines join several curve segments smoothly: B-splines and NURBS provide local control and exact conics, and Catmull-Rom splines pass through their control points. Continuity is described as C0, C1 or C2, meaning matching position, tangent and curvature.

Where it is used

Vector graphics paths in SVG and fonts, animation easing curves, camera paths, CAD surfaces and road or track design use Béziers and splines.

Common pitfall

Moving one control point changes the whole high-degree Bézier curve, so low-degree segments are joined instead. Joining segments without matching tangents leaves visible corners. Parametrization is not uniform in arc length, so equal steps in t give uneven spacing, requiring arc-length reparametrization for constant speed. Flattening curves for rendering needs careful subdivision tolerances.

Where it fits in the course

Bézier Curves and Splines is topic 27 of 49 in Graphics, Vision & Media, rated Intermediate. It follows Global Illumination. The next topic is Mesh Simplification.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Graphics, Vision & Media quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Bézier Curves and Splines?
Smooth curves defined by control points. It belongs to Graphics, Vision & Media and is an intermediate-level topic.
Where do people use Bézier Curves and Splines?
Vector graphics paths in SVG and fonts, animation easing curves, camera paths, CAD surfaces and road or track design use Béziers and splines.
What is the most common mistake with Bézier Curves and Splines?
Moving one control point changes the whole high-degree Bézier curve, so low-degree segments are joined instead.
What should I learn before this topic?
Start with Global Illumination, the topic just before it in Graphics, Vision & Media.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Graphics, Vision & Media quiz.
Is Bézier Curves and Splines hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Bézier Curves and Splines?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Bézier Curves and Splines?
Global Illumination, Mesh Simplification, Ray Tracing are closely connected. The full list is in the related topics section of this page.
What should I study after Bézier Curves and Splines?
Continue with Mesh Simplification, the next topic in Graphics, Vision & Media.

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