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# Curves

In mathematics a Curve is "a line that does not need to be straight". With that definition a large number of different curves exist each of which can be described by a mathematical formula.

Different mathematical curves

In computer graphics we're most interested in curves that can be described by mathematical formulas that are called polynomial functions. In order to be able to draw arbitrarily curved lines often multiple such curve-segments are combined to a Spline. The term "Spline" originates from an old draftsmen tool called a Flat Spline.

Curves that are piecewise defined by polynomial functions are called Splines

There are many different types of splines (hermite, bézier, catmull-rom, b-spline,..). but they all share the following key properties:

• they are continuous at the Knots
• Control Points influence their shape
Splines with their segments, control points and knots

## Bézier Spline

Quadratic Bézier Curve with its Control Point

Cubic Bézier Curve with its two Control Points

The most popular Bézier curves are quadratic and cubic in nature as higher degree curves are expensive to draw, evaluate and not so intuitive to handle.

Characteristics

• they cannot exactly describe circles and ellipses
• they are a special cases of B-Spline
##### Related nodes

BezierEditor (2d)
BezierEditor (3d)

BezierEditor nodes are operating with cubic Bézier splines.

## B-Spline

An open B-Spline does not go through its control points

A clamped B-Spline which at least goes through its first and last control point

B-Spline is short for "basis spline".

Characteristics

• they do not go through their control points
• they have C2 continuity!
• they are a special case of NURBS
• in order to have a sharp corner in a B-Spline it takes three control-points at the same position
• it is hard to insert a control point without changing the curve
• it is hard to make a straight line
• depending on its degree, changing one point changes large parts of the curve
• they cannot exactly describe circles and ellipses.

In order to make it more intuitive to handle a B-Spline can also be constructed in a way it at least goes through its first and last control point.

## NURBS

A NURBS with its weighted control points

NURBS (short for "non uniform rational b-spline") are essentially B-Splines with a weight for each control point. If all its weights equal 1 the NURBS is a B-Spline.

• NURBS generalizes both B-splines and Bézier curves
• they can accurately represent circles and ellipses

Sources

# Shoutbox

~6d ago

joreg: Workshop on 30.11: Strategies for Sound Reactive Graphics: How to control everything through sound Signup here: https://thenodeinstitute.org/courses/ws23-vvvv-02-how-to-control-everything-through-sound/

~15d ago

joreg: The vvvv winter semester course program is out, starting with a free course on November 23rd: https://thenodeinstitute.org/ws23-vvvv-intermediates

~16d ago

LCA: ravazque, this guy is working on this v3 since ever. check: https://nuitrack.com/

~26d ago

~2mth ago

karistouf: done with vvvv beta :) https://vimeo.com/872242439

~2mth ago

joreg: Mapping festical call for projects: https://mappingfestival.com/en/call-for-projects

~2mth ago

joreg: Starting October 16: vvvv beginner class winter 23/24 Sign up here: https://thenodeinstitute.org/courses/ws23-vvvv-beginner-class/

~3mth ago

~3mth ago