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शुक्रवार, 24 अप्रैल 2020

BCA BCA 4th Sem Notes Representing Curves & Surfaces

BCA 4th Sem Notes Representing Curves & Surfaces

by on in
  • UNIT-I 
~The Advantages of Interactive Graphics
~Representative Uses of Computer Graphics 
~Classification of Application Development of Hardware and software for computer Graphics
~Overview, Scan:
~Converting Lines
~Scan Converting Circles
~Scan Converting Ellipses









  • UNIT-II 
    • ~Hardcopy Technologies
    ~Display Technologies
    ~Raster-Scan Display System
    ~Video Controller
    ~Random-Scan Display processor
    ~Input Devices for Operator Interaction
    ~Image Scanners
    ~Working exposure on graphics tools like Dream Weaver, 3D Effects etc
    ~Clipping
    ~Southland- Cohen Algorithm
    ~Cyrus-Beck Algorithm
    ~Midpoint Subdivision Algorithm










  • UNIT-III 
    • ~Geometrical Transformation
    ~2D Transformation
    ~Homogeneous Coordinates and Matrix Representation of 2DTransformations 
    ~composition of 2D Transformations
    ~The Window-to-Viewport
    Transformations








  • UNIT-V                           

    • ~Multimedia Definition
    ~CD-ROM and the multimedia highway
    ~Computer Animation
    (Design, types of animation, using different functions)








  • UNIT-VI  

    • ~Uses of Multimedia
    ~Introduction to making multimedia –
    ~The stage of Project
    ~hardware & software requirements to make good multimedia skills
    ~Training opportunities in Multimedia Motivation for Multimedia usage

    Representing Curves & Surfaces

    Different types of objects are drawn on the screen. An object can be drawn by using curves many times

    Types of Curves

    1.  explicit
    2.  implicit
    3.  parametric curves
    Implicit Curves
    Implicit curve representations define the set of points on a curve by employing a procedure that can test to see if a point in on the curve. Usually, an implicit curve is defined by an implicit function of the form −
                              f(x,y)=0
    The multivalued curves can be represented such as for a single x value, many y values. The very common example is a circle, and is implicitly represented as:
                           x2 + y2 - R2 = 0

    Explicit Curves

    The function y = f(x) can be plotted as a curve, which is known as an explicit representation of the curve. For each of the values of x, a single value y is normally computed by the function.

    Parametric Curves


    Parametric curves are the curves that have parametric values. Practically parametric curves are mostly used. The form of a two-dimensional parametric curve is:
                                                             P(t) = f(t), g(t) or P(t) = x(t), y(t)

    Bezier Curves

    The curves are generated under the control of other points. Curve is generated by using approximate tangents. Mathematically the curve is represented as:


    Where n is the polynomial degree, i is the index, and t is the variable.
    The simplest Bézier curve is the straight line from the point P0 to P1. Three control points determine a quadratic Bezier curve and four control points determine the cubic Bezier curve.

    Properties of Bezier Curves

    • takes the shape of the control polygon consisting of segments that join the control points.
    • They pass through the first and last control points.
    • The convex hull of the control points contains these curves.
    • The degree of the polynomial defining the curve segment is one less than the number of defining polygon points. Therefore, for 4 control points, the degree of the polynomial is 3, i.e. cubic polynomial.
    • A Bezier curve generally follows the shape of the defining polygon
    • The direction of the tangent vector at the endpoints is the same as that of the vector determined by the first and last segments.
    • The convex hull property for a Bezier curve ensures that the polynomial smoothly follows the control points.
    • No straight line intersects a Bezier curve more times than it intersects its control polygon.
    • Under an affine transformation, they are very invariant.
    • Bezier curves exhibit global control means moving a control point alters the shape of the whole curve.
    • A Bezier curve can be subdivided at a point t=t0 into two Bezier segments which join together at the point corresponding to the parameter value t=t0

    B-Spline Curves



    The Bezier-curve produced by the Bernstein basis function has limited flexibility

    • The order of the resulting polynomial defines the curve and the order is fixed by the number of specified polygon vertices.
    • The second limiting characteristic is that the value of the blending function is nonzero for all parameter values over the entire curve.
    The B-spline basis contains the Bernstein basis as a special case. The B-spline basis is non-global.

    A B-spline curve is defined as a linear combination of control points Pi and B-spline basis function Ni, k t given by
    C(t)=∑i=0nPiNi,k(t), n≥k−1, 
    Where,
    • {pi: i=0, 1, 2….n} are the control points
    • k is the order of the polynomial segments of the B-spline curve. Order k means that the curve is made up of piecewise polynomial segments of degree k - 1,
    • the Ni,k(t) are the “normalized B-spline blending functions”. They are described by the order k and by a non-decreasing sequence of real numbers normally called the “knot sequence”.

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