Mumbai university > Comp > SEM 4 > Computer Graphics

Marks: 10M

Year: Dec 2014,Dec 2015

1. Bezier curve is another approach to construct a curve.

2. A Bezier curve is determinate by a defining a polygon.

3. Bezier curve has a number of properties which makes them highly useful and convenient for curve and surface design.

4. Bezier curves are used in the time domain, particularly in animation and user interface design.

5. Generalizations of Bezier curves to higher dimensionsare called Bezier surfaces, of which the Bezier triangle is a special case.

6. The cubic Bezier curve require four sample points, these points completely specify the curve.

7. The figure 10 shows sample Bezier curve.

   

8. The curve begins at first sample point and ends at fourth sample point.

9. If we need another Bezier curve then we need another four sample points.

10. But if we need two Bezier curves connected to each other, then with six sample points we can achieve it.

11. For this, the third and fourth point of first curve should be made same as first and second point of second curve.

12. The equations for Bezier curve are as follows:

$$X = X_4a^3 + 3X_3a^2(1 - a) + 3X_2a(1 - a)^2 + X_1 (1 - a)^3$$

$$Y = Y_4a^3 + 3Y_3a^2(1 - a) + 3Y_2a(1 - a)^2 + Y_1 (1 - a)^3$$

$$Z = Z_4a^3 + 3Z_3a^2(1 - a) + 3Z_2a(1 - a)^2 + Z_1 (1 - a)^3$$

1. We can also construct a Bezier curve by taking midpoints.

   

Properties:

• The basis functions are real.

• The Bezier curve always passes through the first and last control points i.e. curve has same end points as the guiding polygon.

• 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 3.

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

• The direction of the tangent vector at the endpoints is the same as that of the vector determined by first and last segments.

• The curve lies entirely within the convex hull formed by 4 control points.

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

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

• The curve is invariant under an affine transformation.

Generation Bezier surface from Bezier curve:

The parametric bicubic equation for Bezier surface can be given as

Substituting the equation C in equation A we have,

$P (u, t) = U.M_B. G_B. M_B^T. U^T$

In terms of X, Y and Z separately the above equation can be written as

$X (u, t) = U.M_B. G_{BX}. M_B^T. U^T$

$Y (u, t) = U.M_B. G_{BY}. M_B^T. U^T$

$Z (u, t) = U.M_B. G_{BZ}. M_B^T. U^T$