What are Bezier Curves?

• Developed in the 1960s for Renault car body design
• Now standard in numerous office and drawing packages.
• Wide variety of smooth curve shapes available.
• Complex shapes can be made by joining Bezier curves together.

Typical Bezier Curve

• The curve starts at x0, y0 and finishes at x3, y3 (although the maths is symmetrical so x3, y3 can also be regarded as the starting point and x0, y0 as the ending point)
• There are two control points: x1, y1 and x2, y2
• Moving the control point x1, y1 changes the starting direction of the curve, shown by the red line
• Moving the control point x2, y2 changes the finishing direction of the curve, shown by the red line
• The curve moves smoothly from the start to the end point
• The further the distance of the control point from the start/end point, the more the curve sticks to the tangent line before veering off

Application to Synthesis

A single Bezier curve can be used as a single waveform shape provided:

• The start and end points are both at y=0
• The start and end points define the wavelength; x1=0, x3=lambda
• One control point is above the x axis
• One control point is below the x axis

Simple (Static) Synthesis

• Divide the x axis into a number (N) of equally spaced sampling points.
• Sweep from left to right generating a new output at each sample point.
• Ensure that the time between successive samples is constant 1/fs seconds.

The output frequency is given by fs/N

Simple Bezier Synthesis Animation

Start Animation

Dynamic Synthesis

With dynamic synthesis, the control points move during synthesis…

For each output sample (i.e. 48000 times per second!):

• Find the position of the control points
• Calculate the new curve shape
• Find the corresponding sample point on the curve.

Control points can be moved separately and independently.

Control Point Modulation

• A control point can be moved by applying two fixed frequencies to its position (modulations)
• The X and Y position can be individually modulated
• Modulation is specified by 4 parameters:
• Central X or Y position of the point
• Modulation frequency
• Modulation amplitude
• Phase angle

Four separate modulating frequencies can be applied to a single curve! Compare this with other modulation techniques (eg FM) where only one modulating frequency can be applied.

Dynamic Bezier Synthesis - Y Axis Modulation Animation

Start Animation

Effect of Y Axis Modulation

Y Axis modulation at frequency fm adds peaks at:

• fm   f-fm   f+fm

(Height and phase of peaks not yet known)

Dynamic Bezier Synthesis - X Axis Modulation Animation

Start Animation

Effect of X Axis Modulation

X Axis modulation at frequency fm adds peaks at:

• fm
• f±fm   f±2fm   f±3fm...
• 2f±fm 2f±2fm 2f±3fm...
• 3f±fm 3f±2fm 3f±3fm...
• ...

(Height and phase of peaks not yet known)

Dynamic Bezier Synthesis - X and Y Axis Modulation Animation

Start Animation

Polar Modulation

The control points can be specified using an alternative co-ordinate system:

• Polar co-ordinates
• X and Y replaced by Angle and Radius (relative to start and end point of the curve)

Modulation can now be applied to:

• Angular position (degrees)
• Distance (radius)

Again, this allows four separate modulation frequencies

Dynamic Bezier Synthesis - Polar Modulation Animation

Start Animation

Future Work

• Mathematical analysis of outputs
• Height, phase and position of each peak
• Influence of the unmodulated wave
• Finding new modulation modes
• Finding lots of cool sounds

Bob Lang
May 2004