# Unit-4:Representing curves and surfaces

In computer graphics, curves and surfaces are typically represented using mathematical models such as NURBS (Non-Uniform Rational B-Splines), Bézier curves, and Bézier surfaces. These models allow for precise control over the shape of the curve or surface and make it easy to manipulate and animate them. The parameters of these models can be manipulated to create a wide variety of shapes and styles, which makes them a popular choice in computer graphics and animation. To use these models in a CGMA (computer graphics and media applications) pipeline, one typically needs to have a strong understanding of the mathematical principles behind them, as well as the tools and software packages available for working with them

## Polygon meshes parametric

A polygon mesh is a type of surface representation in computer graphics and animation that is made up of a set of polygons. These polygons can be either triangular or quadrilateral, and are defined by a set of vertices. The vertices are the points in space that define the shape of the polygons.

Parametric representation of a polygon mesh refers to the use of mathematical equations to define the position of each vertex in the mesh. This allows for precise control over the shape and form of the mesh, as well as the ability to easily modify and animate the mesh over time.

Example: Let’s consider a torus (donut-shaped object) as an example. A torus can be represented as a parametric polygon mesh by using a mathematical equation to define the position of each vertex in the mesh.

One common way to create a torus is to use a parametric equation that defines the position of each vertex based on its polar and azimuthal angles. The polar angle determines the radius of the torus, while the azimuthal angle determines its position around the circumference of the torus.

To create the polygon mesh, we would divide the surface of the torus into a set of small polygons, each defined by a set of vertices. The position of each vertex is then determined using the parametric equation.

The resulting polygon mesh will have a precise, controlled shape that can be easily modified and animated over time. This makes parametric polygon meshes a powerful tool for creating complex shapes and objects in computer graphics and animation

## cubic curves

Cubic curves are a type of curve representation in computer graphics and animation that are defined by a set of cubic polynomials. Cubic curves are a type of spline curve, which means that they are defined by a set of control points and are smooth and continuous between those control points.

Cubic curves are often used in CGMA for tasks such as modeling and animating smooth curves, such as those found in characters’ limbs or objects’ shapes. Cubic curves are also used in motion graphics to create smooth, animated transitions between keyframes.

Example: Let’s consider a simple animation of a bouncing ball as an example. To create the animation, we could use cubic curves to define the path of the ball as it moves through the air.

We would start by defining a set of keyframes, each representing the position and velocity of the ball at a specific point in time. We would then use cubic curves to smoothly interpolate between these keyframes, defining the path of the ball as it moves through the air.

The resulting animation would have a smooth, continuous path for the ball, with no jagged or abrupt changes in its trajectory. This makes cubic curves a powerful tool for creating smooth, natural-looking animations in CGMA