Q. Give the difference between interpolation spleen and approximation spine. Also mention the geometric and parametric continuity conditions in these curves.
Ans. A spleen surface can be described with two sets of orthogonal spleen curves. There are several different kinds of spleen specification that are used in graphics applications. Each individual specification simply refers to a particular type of polynomial with certain specified boundary conditions. Spleens are used in graphics application to design curve and surface shapes to digitize drawings for computer storage and to specify animation paths for the object of the camera in a scene. Typical CAD application for spleens includes the design of automobile bodies' aircraft and spacecraft surface and ship hulls. Interpolation and Approximation Spleens We specify a spleen curve by giving a set of coordinate position called control points which indicate the general shape of the curve. These control points are then fitted with piecewise continuous parametric polynomial functions in one of two ways. When polynomial section are fitted so that the curve passes through each control point as in fig 1 the resulting curve is said to interpolate the set of control points. One the other hand when the polynomials are fitted to the general control point path without necessarily passing through any control point the resulting curve is said to approximate the set of control points. Interpolation curves are commonly used to digitize drawings or to specify animation paths. Approximation curves are primarily used as design tools to structure object surfaces. Fig.2 shows an approximation spleen surface created for a design application. Straight lines connect the control point positions above the surface. A spleen curve is defined modified and manipulated with operations in the control points a designer can set up an initial curve. After the polynomial fit is displayed for a given set of control points designer can then reposition some of all of the control points restructure the shape of the curve. In addition the curve can be translated rotated or scaled with transformation applied to the control points. CAD packages can also insert extra control points to aid a designer in adjusting the curve shapes. The convex polygon boundary that encloses a set of control points is called the convex hull. One way to envision the shape of a convex hull is to imagine a rubber band stretched around the positions of the control points so that each control point is either on the perimeter of the hull or inside it. Convex hulls provide a measure for the deviation of a curve or surface from the region bounding the control points. Some spleens are bounded by the convex hull thus ensuring that the polynomial smoothly follow the control points without erratic oscillations. Geometric and Parametric Continuity
Geometric Continuity * Go: Curves are joined
* G1: First derivatives are proportional at the join point. The curve tangents thus have the same direction, but not necessarily the same magnitude. i.e., C1' (1) = (a, b, c) and C2' (0) = (k*a, k*b, k*c).
*G2: First and second derivatives are proportional at join point.
Parametric Continuity
* C0: Curves are joined
* C1: First derivatives equal
* C2: First and second derivatives are equal. If is taken to be time this implies that the acceleration is continuous.
* C n: n^{th} derivatives are equal.
As their names imply geometric continuity requires the geometry to be continuous while parametric continuity requires the geometry to be continuous while parametric continuity requires that the underlying parameterization be continuous as well. Parametric continuity of order n implies geometric continuity of order n but not vice versa.
Spleens
* Splices are cubic curves which maintain C2 continuity.
* Natural Spleen - interpolates all of its control points. - Equivalent to a thin strip of metal forced to pass through control points. - No local control.
* B-spleen -local control. - does not interpolate control points.
The following is an example of a five segment B-spleen curve. The points which indicate the ends of the individual curve segments and thus the join points are known as the knots.