Page 122 - DCAP504_Computer Graphics

Unit 8: 3-D in Computer Graphics

Properties of Bezier curves
1. The initial and final control points are interpolated.
2. The tangent to the curve at the first control point is along the line joining the first and second
control points.
3. The tangent at the last control point is along the line joining the second last and last control points.
4. The curve lies entirely within the convex hull of its control points.
B-Spline Curve

A B-spline curve P (t) can be defined as

Here,
{Pi: i=0, 1, n} represent the control points.
K represents the order of the polynomial segments of the B-spline curve. The order k means that the
curve is composed of piecewise polynomial segments of degree k-1.
N i,k (t) - this term is the normalized B-spline blending functions. They can be described by order k and
by the sequence of real numbers that do not reduce.
Periodic B-spline

The B-spline that closes on itself is called the periodic B-spline. It requires that the first ‘n’ control points
are identical to the last ‘n’, and the initial ‘n’ parameters in the knot vector are identical to the finishing
intervals.
8.1.4 Add-ons: Polygon Rendering

Polygon rendering is used to create images that are 3-D in nature. In normal practice, objects that arise
when their surface is modeled are triangular polygons and they are next rendered in a wire frame
model. Rendering is the process that used to model the objects by specifying their surfaces using
polygons.
It consists of three types of shading, namely:
1. Constant Shading

2. Gouraud Shading
3. Phong Shading
Now let us discuss each of them.
Constant Shading
This is the simplest shading model. It is also known as ‘faceted shading’ or ‘flat shading’. A polygon
here is shaded by applying an illumination model and determining the intensity value so that it can be
used to shade an entire polygon model. The intensity value is held across polygon to create the
polygon’s shade. This approach is considered valid if the following assumptions are true:
1. Light source is placed at infinity, with the light source vector kept constant across polygon face.
2. Viewer at infinity, with normal vector kept constant across polygon face.
3. The actual surface being modeled is represented by the polygon and not its approximation.
In case, any of the first two assumptions are not correct, some method must be used to determine the
value of L and V.

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