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COMPUTER GRAPHIC

Unit-4:Representing curves and surfaces

Polygon meshes

A polygon mesh is a type of computer graphics technique used for creating 3D models.

It is a collection of vertices, edges and faces that define the shape and surface of a 3D object. It is often used in computer games, animation, virtual reality, and computer-aided design (CAD).

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curves

A curve is an infinitely large set of points. Each point has two neighbors except endpoints. Curves can be broadly classified into three categories − explicit, implicit, and parametric curves.

  1. Implicit curves
  2. Explicit curves
  3. Parametric curves
  4. Bezier curves
  5. B-spline curves
Implicit Curves

An implicit curve or surface is the set of zeros of a function of 2 or 3 variables. We use implicit curve functions to define lines and planes.


Parametric curves

Curves have a parametric form called parametric curves. A curve in the plane is said to be parameterized if the set of coordinates on the curves (x,y,z) is represented as a function of a variable t.

The variable t is called a parameter and the relations between x,y,z, and t are called a parametric equation

Bezier curves

A bezier curve is particularly a kind of spline generated from a set of control points by forming a set of polynomial functions.

Discovered by the french engineer Pierre bezier. These functions are computed from the coordinates of the control points.

These curves can be generated under the control of other points. Tangents by using control points are used to generate curves.

Different types of curves are Simple, Quadratic, and Cubic.

  1. Simple curve: Simple bezier curve is a straight line from the point.


2. Quadratic curve: Quadratic bezier curve is determined by three control points.


3. Cubic curve: The cubic bezier curve is determined by four control points.

Properties of Bezier Curve:

  1. Bezier curves are widely available and used in various CAD systems, in general graphics packages such as GL
  2. The slope at beginning of the curve is along the line joining the first two control points and the slope at the end of the curve is along the line joining the last two points
  3. Bezier curve always passes through the first and last points i.e p(o)=po, p(1,=pnlie)
  4. The curves lies entirely within the convex hall formed by the four control points
  5. The slope at the beginning of the curve is along the line joining the first two control points and the slope at the end of the curve is along the line joining the last two points.
  6. The degree of polynomial defining the curve segment is one less than the no of defining the polygon.

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