Page 121 - DCAP504_Computer Graphics

Computer Graphics

Explicit or Parametric Curve
It provides a mapping from a single parameter to a set of points on the curve. The parameter provides
an index to curve points.
Generative or Procedural Curve
The procedural curve descriptions are very different from the first two types of curves.

Subdivision schemes and fractals.

Hermit Curve

Hermit curves are described by the start points, end points and tangents at those points. To obtain a
point on the hermit curve, each of the control points are multiplied by some function and added. The
functions are called basis functions. The basis functions when applied to all the components of the
hermit curve act in the similar way as Bezier and B spline curves.
Hermit curve equation
The equation for a hermit curve that interpolates on (x k,x k+1) implements the following equation:
P(x) = h 00(t)p k+h 10(t)(x k+1 – x k)m k+h 01(t)p k+1+h 11(t)(x k+1 – x k)m k+1

Where, t=(x-x k)/ (x k+1-x k) and h is the basis functions.
These curves do not change due to transformations of rotation, translation and rotation. If these
transformations are applied to the endpoints and tangent vectors, they will automatically get applied to
the transformations of the entire curve.
Bezier Curve
In the case of Bezier curves the endpoints are defined by two control points. The tangents at the end
points are controlled in a geometric manner.
Bezier curve equation:
A Bezier curve of degree n can be defined as

n 
Here,    is a binomial coefficient.

 1 
For linear interpolation,

For quadratic interpolation,

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