The so-called “Bezier Curve” is a method of representing a smooth curve in computer graphics. Note that it is a method of curve representation, and not a curve in the usual sense in math contexts.
In computer graphics, there is a need to represent a smooth curve, in particular, finding a representation of a curve of a few given points entered/drawn by the user. One way is to represent a curve by a set of points that lies on the curve. This method obviously has some flaws. For example, it requires a lot collection of points to represent a curve smoothly, and it will be again jagged after magnification.
The Bezier Curve representation is a method to represent a curve between 2 given points, by a polynomial parametric formula, with the additional idea of using a few “control points” that specifies the tangency of the given points.
The Bezier Curve method is named after the engineer Pierre Bézier.
Cubic Bezier Curve
Given 4 points P0, P1, P2, P3, find a parametric formula in polynomial such that it passes P0 with tangent vector[P0,P1], and passes P2 with tangent vector[P3,P2].
Answer: written in complex number notation: z[t]=(1-t)^3*P0 + 3*t(1-t)^2*P1 + 3*t^2(1-t)*P2 + t^3*P3 We verify that z==P0, z'==3*(P1-P0) z=P2, z'==3*(P2-P3)
See: Websites on Plane Curves, Printed References On Plane Curves.
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