polygons to quadratics

1. From Polygons to Quadratics:
Understanding 3D Object
Representation in Computer
Graphics

2. OUTLINE
•Introduction
•Polygon Surfaces
•Quadratic Surfaces
•Comparison between Polygon and Quadratic Surfaces

3. INTRODUCTION
A polygon is a closed shape consisting of a sequence of line segments. Each line segment is
joined to the next at its endpoint, and the last line segment connects back to the first.
A frequently used class of objects are the quadric surfaces, which are described with
second-degree equations (quadratics). They include spheres, ellipsoids, tori,
paraboloids, and hyperboloids.
3d representation:
Growing popularity of 3D displays is leading to growing demand to create3D models of
scenes from photographs. From a single image, 3D reconstruction requires interpreting
the pixels of the image as surfaces with depths and orientations.

4. 3d object representation has divided in to 2 types
• Boundary Representations (B-reps) −
• It describes a 3D object as a set of surfaces that separates the object interior from the
environment.
• Space–partitioning representations −
• It is used to describe interior properties, by partitioning the spatial region containing
an object into a set of small, non-overlapping, contiguous solids (usually cubes).
3D object representation.

5. 3D Object Representation :
Polygon Surfaces :
• The most commonly used representation for a 3D graphics object.
• It is a set of surface polygons that enclose the object interior.
• Set of polygons are stored for object description.
• This simplifies and speeds up the surface rendering and display of object since all
surfaces can be described with linear equations

6. 3d Object Representation :
Polygon Surfaces Three ways to represent polygon surfaces
1. Polygon Tables
2. Plane Equations
3. Polygon Meshes

7. Polygon Tables The object is store by using three tables
1. Vertex Table
2. Edge table
3. Polygon-Surface table

8. Polygon tables

9. Plane Equations
• The equation for plane surface can be expressed as Ax + By + Cz + D = 0
• Where (x, y, z) is any point on the plane, and the coefficients A, B, C, and D are
constants describing the spatial properties of the plane.
• We can obtain the values of A, B, C, and D by solving a set of three plane
equations using the coordinate values for three non collinear points in the plane.
Let us assume that three vertices of the plane are (x1 , y1 , z1 ), (x2 , y2 , z2 ) and
(x3 , y3 , z3 ).
• Let us assume that three vertices of the plane are (x1 , y1 , z1 ), (x2 , y2 , z2 ) and
(x3 , y3 , z3 ).
Plane Equations
• Let us solve the following simultaneous equations for ratios A/D, B/D, and C/D.
You get the values of A, B, C, and D. (A/D) x1 + (B/D) y1 + (C/D) z1 = -1 (A/D) x2 +
(B/D) y2 + (C/D) z2 = -1 (A/D) x3 + (B/D) y3 + (C/D) z3 = -1

11. Polygon Meshes
Polygon Meshes 3D surfaces and solids can be approximated by a set of polygonal and line
elements. Such surfaces are called polygonal meshes. In polygon mesh, each edge is shared by at
most two polygons. The set of polygons or faces, together form the “skin” of the object.
It is used in computer games and animations,. Virtual reality.
It is used in medical department , to create 3d objects.
It can be used to create 3d models for games and films.
It is used in medical imaging. It is used to create 3D models of the human body for
medical research and surgery planning.
It is used in computer games and animation. Due to its lightweight nature, it is well-
suited for real-time applications. It is also used in virtual realityIt is used in computer
games and animation. Due to its lightweight nature, it is well-suited for real-time
applications. It is also used in virtual reality is used in computer games and animation.
Due to its lightweight nature, it is well-suited for real-time applications. It is also used in
virtual reality
It is used in computer games and animation. Due to its lightweight nature, it is well-
real-time applications. It is also used in virtual reality

12. Quadric surfaces
A frequently used class of objects are the quadric surfaces, which are described with
second-degree equations (quadratics). They include spheres, ellipsoids, tori,
paraboloids, and hyperboloids.
● Quadric surfaces, particularly spheres and ellips are common elements of
graphics scenes, and they are often available in Quadric Surfaces graphics packages as
primitives horn which more complex objects can be constructed
sphere
A sphere is the graph of an equation of the form x2 + y2 + z2 = p2 for some real number p. The radius of the sphere
is p .

13. ellipsoidal
• An ellipsoidal surface can be described as an extension of a spherical surface, where the radii in
three mutually perpendicular directions can have different values.
• The Cartesian representation for points over the surface of an ellipsoid center.

14. DIFFERENCE POLYGON SURFACE QUADRATIC SURFACE
BASIC REPRESENTATION Objects are represented using a collection of flat
polygons (usually triangles or quadrilaterals).
These equations define smooth surfaces
and can represent a wide range of
shapes, including curves and more
complex surfaces.
SMOOTHNESS Smooth surfaces may require a high polygon
count.
represent curves and surfaces with high
precision without the need for a large
number of data points.
COMPLEXITY Well-suited for representing simple shapes and
objects with flat or planar surfaces
More versatile in representing complex
shapes, including curved surfaces and
objects with intricate geometries
RENDERING PERFORMANCE Generally faster to render, especially in real-time
applications, due to their simple representation
May require more computational
resources for rendering, especially for
complex surfaces.
EDITING AND MODELLING Easy to edit and model, especially for artists and
designers.
Editing and modeling may involve
manipulating mathematical equations,
which can be less intuitive for artists.

15. THANK YOU