Page 115 - DCAP504_Computer Graphics

Computer Graphics

The operations that can be performed on splines other than changing its shape are:
1. Adding points along the curve
2. Examining the radius of curvature along the curve
3. Showing the constraints

4. Changing the fit method
5. Closing open splines
Splines are used in computer animation to:
1. Define Spatial Shapes - the shapes of two or three-dimension objects. Here, the points on the
surface of the object are the knots.

2. Define the Path of an Object through Space – Here, the points on the path are the knots.
3. Define Eases – Ease is the velocity of the movement along a path. There are two ways of defining
the ease. Ease in is the slow movement at the beginning of the path and faster at a later stage. Ease
out being the reverse.

In general, Modeling applications use spline. They are also used to pass the curve smoothly through the
parameters. These parameters are known as sparse time.
Spline Representation
As we know spline is a flexible wooden strips tool that can be used to produce a smooth curve through
a specified set of points. These points are represented mathematically with a cubic polynomial function
called spline curves. The points in a spline have more than one coordinate. Splining of points together is
to spline the entire x , y, and z coordinates together. So, it is expected to present a solution for one
coordinate and apply the same on others during the splining process.
The following equation set can be used to describe a cubic polynomial that has to be fitted between
every pair of control points:
x(p)=axp +bxp +cxp+dx
3
2
y(p)=ayp +byp +cyp+dy (1)
2
3
z(p)=azp +bzp +czp+dz where 0<=p<=1
3
2
The values of all the four coefficients are determined for every n curve section within n+ 1 point. The
numerical value for each of the coefficients is obtained by setting boundary conditions for all the joints
between curve sections. Common methods that are implemented to set the boundary conditions are
discussed in section 6.12 for cubic interpolation splines.
8.1.2 Surfaces
Surface in simple words is known as a family of curves. A solid surface is obtained by merging infinite
number of curves without allowing any gap between any two curves.

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