Bezier Curves Investigation Tool

This investigation tool demonstrates the Bezier curves calculations and properties. Almost every property and formula shown in the class can be visualized here. 

 

In order to modify your curve you can :

Press Click to add a control point,

Shift and Click to delete a control point,

drag to modify a control point

The curve can be viewed in different ways

Viewing modes :

Show Lines                           Shows all the lines of De-Casteljo algorith

Show Point                           Show the point on the curve that was generated by the algorithm

Show Curve Only                 Shows only the curve and the control polygon

Show Mid Pnts Meaning       Shows the two Bezier curves with degree N-1 that the two last De-Casteljo are on.

Derivative Viewing modes:

Don't Show Dreiv.                Do not show the bezier derivative vector. 

Show De-Casteljo deriv.      Show the derivative vector generated from the De-Casteljo algorithm.

Show Bezier deriv.               Show the derivative vector and the N-1 Bezier diffrences curve.

Controls:

Clear button                          Resets and clears all drawing area.

Deg Elevation button             Elevates degree of the curve.

Deg Reduction button           Reduce the curve degree.

Scrollbar                              Allows you to see a specific point on the curve. 

One can demonstrate some properties of the Bezier curve

Affine tranformation

              You can move (by dragging with Shift key pressed), zoom (by dragging with Ctrl key pressed), and rotate (by dragging the mouse) the curve and see it is Affine Invariant.

Convex Hall

              You can see the Bezier curve will always stay in its control point's convex hall.

Variation diminishing

              You can define a line (by clicking or dragging with Ctrl key pressed), and see that the number of intersections with the control points polygon (green points) is not less than the number of intersections with the Bezier curve (magenta points).