PART $1$ Bezier curves programming $[10$ marks$]$

B$$zier curves are a class of parametric curves defined using a set of control points, and a parameter $t$ in

$0,1.$ The analytic parametric equation of a B$$zier curve defined by a control polygon of $n+1$ points

$P_{i}i=0,$cdots,$n$ is given through Bernstein polynomials $B_i^n(t)$ as follows: $C(t)={\displaystyle \underset{i=0}{\overset{n}{}}}{B}_i^n(t)P_i$ Where

$B_i^n(t)=\frac{n!}{i!(n-i)!}(1-t{)}^{n-t}{t}^t$ is the $i^{th}$ Bernstein polynomial of degree $n.$

Given a value of parameter $t,$ the B$$zier curve can be evaluated either using the above parametric

equation or using the Decasteljau subdivision algorithm.

The objective of this homework is to develop a tool for visualization and editing of B$$zier curves of any

degree $n.$ A C$++$ class should be defined to encapsulate the data and all the functions necessary for easy

manipulation of Bezier curves. Special attention should be paid to the design of your application and to

the graphic user interface that allows the activation of the desired functionality.

Recommended functionalities:

Appropriate data structures to store the curve properties $($number of control points, set of control

points, etc.$)$

A Menu to allow navigation through the application functionalities and select the function to launch.

The mouse control function allows the definition of the control polygon, the selection and motion of

one control point into another position to enhance the curve design

A function BezPoint$()$ to compute and return the point on the Bezier curve using the parametric

equation.

A function CasteljauPoint$()$ to compute and return a point on the B$$zier curve using the Decasteljau

algorithm.

A function CasteljauSubdivid$()$ to compute and return the two sub$-$curves of a Bezier curve using the

Decasteljau algorithm.

A function drawBez$()$ to visualize a B$$zier curve with $($or without$)$ its control polygon.

A function to edit the curve: use the mouse to select and move one control point into another position,

then trace the resulting curve.