Specialist Mathematic
Bezier Curve Investigation

Introduction
A Bézier curve is a parametric curve used in computer graphics. Bézier curves are used to model smooth curves in, for example, animation.
Bézier curves were widely publicised in 1962 by the French engineer Pierre Bézier who used them to design car bodies. Paul de Casteljau first developed the curves in 1959.
In the following investigation you will, individually, develop the algebra of Bézier curves and use them to design an object using Bezier Curves.
Task 1
Aim
Method * Linear Bézier curves
Given points P0 and P1, a linear Bézier curve is a straight line between those two points.
The curve is given by B(t) where B(t) represents the moving point between two points P0 and P1, and .
Fig 1
Fig 1

Application

* Quadratic Bézier curves
A quadratic Bézier curve is the path traced by the function B(t), given points P0, P1, and P2.
It departs from P0 in the direction of P1, then bends to arrive at P2 in the direction from P1.
Fig 2
Fig 2
Consider Fig 2.

From Fig 2 above use the points Q0 and Q1 as shown and the linear model from Fig 1 to show that the Bezier curve is defined as - and is therefore a quadratic Bezier curve. (2 degree equation)
Task 1 continued * Cubic Bézier curves * Four points P0, P1, P2 and P3 in the plane or in three-dimensional space define a cubic Bézier curve. * The curve starts at P0 going toward P1 and arrives at P3 coming from the direction of P2. Usually, it will not pass through P1 or P2; these points are only there to provide directional information. * The distance between P0 and P1 determines "how long" the curve moves into direction P2 before turning towards P3. Using the Fig 3 show that the Bezier curve is defined as: and therefore is cubic Bezier curve. (3 degree equation)

Fig 3
Fig 3

* Make a conjecture
From Fig 3 make a conjecture of the mathematical structure of the Bezier curve, B(t),
Such that the Bezier equation shows 5 points P0 ,…, P4 .
Prove your conjecture and supply a diagram to assist your formulation of the equation for B(t).

Task 2
Aim
Method

Using appropriate software develop a shape, logo or design using Bezier curves. Be imaginative. Design a detailed logo or consider an animation character as selected.
Your resulting task must have: * At least 4 Bezier curves, Ranging from degree 2 to degree 6 Bezier equations. * The mathematical development of at least 4 parametric equations describing these curves or some of the curves. This must include the 6 degree Bezier equation. * Discussion of any difficulties you had with your design, and therefore limitations of the model developed.
Application
Conclusion