2. Plane Equation Method:
Plane equation method is another method for representing the polygon surface for 3D -object. The information about the spatial orientation of object is described by its individual surface which is obtained by the vertex co-ordinate values and the equation of each plane gives the description of each surface. The equation for a plane surface can be expressed in the form
Where is any point on the plane, and A, B, C, D are constants describing the spatial properties of the plane. The values of A, B, C, D can be obtained by solving a set of three plane equations using co-ordinate values of 3 non collinear points on the plane.
Let and are three such points on the points on the plane. then
(1)
(2)
(3)
The solution of these equations can be obtained in determinant form using Cramer's rule as.
For any point
if then is not on the plane
if the point is inside plane
i.e. invisible side
if the point lies outside the surface.
Wire frame Representation:
In this method, a 3D object is represented as a list of straight lines, each of which is represented by its two end points and This method only shows the skeletal structure of the objects.
It is simple and can see through the object and fast method. But independent line data structure is very inefficient. In this method, the scenes represented are not very realistic.
Blobby objects: Some objects do not maintain a fixed shape but change their surface characteristics in certain motions or when proximity to other objects. e.g. molecular structures, water droplets, other liquid effects, melting objects, muscle shape in human body etc. these objects can be described as exhibiting "blobbiness" and referred as blobby objects.
Several models have been developed for representing blobby objects as distribution functions over a region of space. One way is to use Gaussian density function.
where T=Threshold and a and b are used to adjust amount of blobby ness.
Other methods for generating blobby objects use quadratic density function as
if
if
0 if r>d.
Spline Representation:
Spline: A spline is a flexible strip used to produce smooth curve through a designated set of points. A curve drawn with these set of points is spline curve.
Mathematically, spline curves are described as piece-wise cubic polynomial function. In computer Graphics, a spline surface can be described with two sets of orthogonal spline curves. Splines are used in graphics application to design curve and surface shapes, to digitize drawings for computer storage and to specify animation paths. typical CAD applications for spline include the design of automobile bodies, aircraft and spacecraft surface etc.
Cubic Spline: Is most often used to set up paths for objects motions or to provide a representation for an existing object or drawing. Cubic polynomial offer a reasonable compromise between flexibility and speed of computation. Cubic spline require less calculations compares to higher order polynomials and less memory. Compared to lower polynomials cubic spline are more flexible for modeling arbitrary curve shape.
Given a set of control points, cubic interpolation splines are obtained by fitting the input points with a piecewise cubic polynomial curve that passes through every control points.
Suppose we have n+1 control points specified with co-ordinates
A cubic interpolation fit of these points is
We can describe the parametric cubic polynomial that is to be filled between each pair of control points with the following set of equations.
Bezier Curve and surface
This spline approximation method was developed by the French Engineer Pierre Bezier for use in the design of automobile body. Bezier splines have a no of properties that make them highly useful and convenient for curve and surface design. They are easy to implement. For this resion, Bezier spline are widely available in various CAD systems.
In General Bezier curve can be fitted to any number of control points. The no of control points to be approximated and their relative position determine the degree of Bezier polynomial. The Bezier curve can be specified with boundary conditions, with characterizing matrix or blending functions.
But for general blending function specification is most convenient.
suppose we have n+1 control points positions These co-ordinate points can be blended to produce the following position vector p(u) which describes path of and approximating Bezier polynomial function between and .
(1)
The Bezier blending functions the Bernstein polynomial.
where
The vector equation (1) represents a set of three parametric equations for individual curve condition
· Bezier curve is a polynomial of degree one less than control points.
Properties of BEZIER Curve
· It always passes through initial and final control points.
i.e. Boundary condition.
· Values of the parametric first derivatives of a Bezier curve at the end point can be calculated from control points as.
· The slope at the beginning of the curve is along the line joining the first two points and slope at end of curve is along the line joining last two points.
Parametric second derivatives of a Bezier curve at end points are.
B-Spline Curve:
There are most widely used class of approximating splines. B-spline have a general expression for the calculation of Co-ordinate positions along a curve in a blending function as:
Where is input set of n+1 points.
B-spline have two advantages.
1. Can be any degree, independent of no.of. control points.
2. Allow local control over the shape of surface.
Disadvantage: (Solid-object repn)
Hierarchical tree structures, octrees are used to represent solid objects in some graphics
system, medical imaging and other applications that require displays of objects cross
sections commonly use this method.
An octree encoding scheme divides region of 3D space into octants and stores eight
data elements in each node of the tree. Individual elements are called volume elements or
voxels. When all voxels in an octant are of same type, this type value is stored in
corresponding data elements. an heterogeneous octants are subdivided into octants again.
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