Bezier curve is an approach for the construction of the
curve. A Bezier curve is determined by a defining polygon. Bezier curves have a
number of properties that make them highly useful and convenient for curve and
surface design. They are also easy to implement. Bezier curves are widely used
in various CAD systems and in general graphics packages.

**APPROACH FOR CONSTRUCTING BEZIER CURVE**

In this approach the Bezier curve can be constructed
simply by taking midpoints. In midpoint approach midpoints of the lines
connecting four control points (A, B, C, D) are determined (AB, BC, CD). Line
segments connect these midpoints and their midpoints ABC and BCD are
determined. Finally these two midpoints are connected and its midpoint ABCD is
determined.

The point ABCD on the Bezier curve divides the original
curve into two sections. This makes the points A, AB, ABC and ABCD are the
control points of the first section and the points ABCD, BCD, CD and D are the
control points for the second section. By considering two sections separately
we can get two more sections for each separate section i.e. the original Bezier
curve gets divided into for different curves. This process can be repeated to
split the curve into smaller sections so short that they can be replaced by
straight lines or even until the sections are not bigger than individual
pixels.

ALGORITHM

1. Get four control points say A (x

_{A}, y_{A}), B (x_{b}, y_{B}), C (x_{c}, yc) and D (x_{d}, y_{D}).
2. Divide the curve represented by points A,
B, C and D in two sections

x

_{AB}= (x_{A}+ x_{b}) / 2
y

_{AB}= (y_{A}+ y_{b})/2
x

_{BC}= (x_{B}+ x_{C}) / 2
y

_{BC}= (y_{B}+ y_{C}) / 2
x

_{CD}=( x_{C}+ x_{D}) / 2
y

_{CD}= (y_{C}+ y_{D}) / 2
x

_{ABC}= (x_{AB}+ x_{BC}) / 2
y

_{ABC}= (y_{AB}+ y_{BC}) / 2
x

_{BCD}= (x_{BC}+ x_{CD}) / 2
y

_{BCD}= (y_{BC}+ y_{CD}) / 2
x

_{ABCD}= (x_{ABC}+ x_{BCD}) / 2
y

_{ABCD}= (y_{ABC}+ y_{BCD}) / 2
3. Repeat the step 2 for section A, AB, ABC
and ABCD and section ABCD, BCD, CD and D.

4. Repeat step 3 until we have sections so
short that straight lines can replace them.

5. Replace small
sections by straight lines.

6. Stop.

**MATHEMATICAL APPROACH FOR BEZIER CURVE**

The equation for the Bezier Curve is given as

P(u) = (1 - u)

^{3}P_{1}+ 3 u (1 - u)^{2}P_{2}+ 3 u^{2}(1 - u) P_{3}+ u^{3}P_{4}for 0 ≤ u ≤ 1**PROPERTIES OF BEZIER CURVE**

1. The basic functions
are real.

2. Beizer
curve always passes through the first and last control points i.e. curve has
the same end points as the guiding polygon.

3. The degree of the polynomial defining the
curve segment is one less than the number of defining polygon point. Therefore,
for 4 control points, the degree of the polynomial is three, i.e. cubic
polynomial.

4. The curve generally follows the shape of
the defining polygon.

5. The direction of the tangent vector at the
end points is the same as that of the vector determined by first and last
segments. Same vector is determined by both and point and first & last
segment.

6. The curve lies entirely within the convex
hull formed by four control points.

7. The convex hull property for a Bezier curve
ensures that the polynomial smoothly follows the control points.

8. The curve exhibits the variation
diminishing property. This means that the curve does not oscillate about any
straight line more often than the defining polygon.

9. The curve is invariant under an affined
transformation.

**EXAMPLE**

Construct
the Bezier curve of order 3 and with polygon vertices A (1, 1), B (2, 3), C (4,
3) and D (6, 4).

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