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Representing curves and spheres


Introduction :-
                         Smooth curves and surfaces must be generated in many computer graphics applications.  Many real world objects are inherently smooth, and much of the computer graphics involves modelling in real world.

The need to represent curves and surfaces :->

The need to represent curves and surfaces arises in 2 cases :

  1. In modelling existing objects example a mountain, a car.
  2. In modelling from scratch where no preexisting physical object is represented.
    In the first case, pertaining to represent existing objects, a mathematical description of the object may be unavailable.  The representation is oftenlly carried by mere approximation of the object with pieces of planes, spheres or other shapes that are easy to describe mathematically and requires that points on our model to be close to the corresponding points on the object.

    In the second case, the user need to create the object in the modeling process, hence the object matches its representation exactly.

    The above two cases exhibit the behavior of surface modelling.  The 3 most common representation for 3D surfaces are :-
  • Polygon mesh surfaces.
  • Parametric surfaces.
  • Quadratic surfaces.
    Polygon and quadric surface provide precise descriptions for simple hidden objects, objects such as polyhedrons and ellipsoids; spline surfaces and construction techniques are useful for designing.  Procedural methods, such as fractal construction and particle systems allow us to give accurate representation of clouds, clumps of grass and other natural, physically based modeling methods using systems of interacting forces can be used to describe the non rigid behavior objects.

Curved lines and Surfaces :-

Displays of 3D curved lines and surfaces can be generated from an input set of mathematical functions, defining the objects of from a set of user specified data points when functions are specified.  A package can project the defining equations of a curve to the display plane and plot pixel positions along the path of the projected function.  For surfaces, a functional description of the object is often tessellated to produce a polygon mesh approximations to the surface.
     When a set of discrete coordinates points is used to specify an object shape, a functional description is obtained that best fits the designated points according to the constraints of the application.

Spline Representation :- 

           These are examples of this class of curves and surfaces.  These methods are commonly used to design new object shapes, to digitize drawings, and to describe animation paths.  Curve and surface equations can be expressed in either parametric and non-parametric form.  However, for computer graphics the parametric forms are more convenient.

Parametric Cubic Curves

       Poly-lines and polygons are first degree, piece wise linear approximations to curves and surfaces respectively unless the curves or surfaces coordinates must be created and stored to achieve reasonable accuracy.  A more compact and more manipulative representation of piece wise smooth curves is developed.  The mathematical development is generalized to surface.

       The general approach is to use functions that are of higher degree than the linear functions.

       The higher degree approximations can be based on any one of the 3 methods :-
  1. Express y and z as explicit functions of x i.e y = F1(x) & z = F2(x).
  2. Model curves as solutions to implicit equations of the form F(x,y,z) = 0.
     The above two mathematical form do permit rapid determination of whether a point lies on the curve or on which side of the curve does the point lie.  However, they are confronted with their own set of problems.  The third method overcomes the problems associated with them.

Geometric Continuity Conditions

  • G0 geometric continuity (Zero order) :-  If two curve segment join together i.e have same co-ordinate positions at the join point.
  • G1 geometric continuity (first order) :-  If the directions but not necessarily the magnitudes of the tangent vectors of the curve are equal at he join point the curve has G1 continuity.
  • G2 geometric continuity ( second order ) :-  If the first order and the second order parametric, parametric derivations of the order 2 curve sections are proportional at their join point.
Note :-  To have G1 and G2 continuities mean that geometric slopes of the segments are equal at join point.  For 2 tangent vectors TV1 & TV2 to have same directions, it is essential to be a scalar multiple of other.
Tv1 = k.Tv2,  k>0
 

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