Did you know how the graphic software,  CAD software make a curve?

Firstly, i though how to make curve as we draw a curve in a plain paper, but it isn’t so simple like that, they don’t have human brain to memorize any function of different shape. Engine on the software must be coded to draw the curve, but the curves it self have various type,  from curve with specific diameter and round, ellipse, etc

There must be a common created code, because we already found it on CAD/Graphic software…

One of it is Bezier Curve, found by Bezier, employee of French auto company Renault. Bezier describe curve can be made by defined the vertices of polygon that construct it.

Bezier selected the blending functionn that have properties (Principles CAD/CAM/CAE Systems: Kun Woo Lee):

1. The curve passes through the first and last vertex of the polygon
2. The tangent vector at the starting point of the curve has same direction as the first segment of the polygon. Similarly, the last segment of the polygon gives the slope of the tangent vector at the ending point.
3. The nth derivative of the curve at the starting point or ending point is determined by the first or last (n+1) vertices of the polygon. In fact, the second property is a special case of this property. This property is used very effectively when two Bezier curves need to be combined so that they satisfy the continuity of the higher order derivatives at the connection point.
4. The same curve is generated when the order of the vertices of the polygon is reversed.

With this properties Bezier use basic polinomial that call, Bernstein Polinomial as blending function:

With above polynomial function applied to vertices of polygon, we get Bezier Formula:

Pi are the vertices cordinate on (x,y) or other cordinates form…

With that Polinomial Function, we can generate bezier curve, the vertices cordinate we call it “control point”…

^Linear Bezier Curve

.

^Quadratic Bezier Curve – 2nd Orde

.

^Cubic Bezier Curve

.

^ Quartic Bezier Curve

.

Credits:

1. Principles of CAD/CAM/CAE Systems – Kun Woo Lee
2. Wikipedia of Bezier Curve